Research ArticleCooperative and Competitive Dynamics Model forInformation Propagation in Online Social Networks

Yaming Zhang1 Chaosheng Tang1 and Li Weigang2

1 School of Economics and Management Yanshan University Qinhuangdao 066004 China2Department of Computer Science University of Brasilia 70910-900 Brasilia DF Brazil

Correspondence should be addressed to Yaming Zhang yaming99ysueducn

Received 1 May 2014 Revised 23 June 2014 Accepted 23 June 2014 Published 13 July 2014

Academic Editor Francisco J Marcellan

Copyright copy 2014 Yaming Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Traditional empirical models of propagation consider individual contagion as an independent process thus spreading in isolationmanner In this paper we study how different contagions interact with each other as they spread through the network in order topropose an alternative dynamics model for information propagationThe proposedmodel is a novel combination of Lotka-Volterracooperative model and competitive model It is assumed that the interaction of one message on another is flexible instead of alwaysnegative We prove that the impact of competition depends on the critical speed of the messages By analyzing the differentialequations one or two stable equilibrium points can be found under certain conditions Simulation results not only show thecorrectness of our theoretical analyses but also provide a more attractive conclusion Different types of messages could coexistin the condition of high critical speed and intense competitive environment or vice versa The messages will benefit from the highcritical speed when they are both competitive and adopting a Tit-for-Tat strategy is necessary during the process of informationpropagation

1 Introduction

With the rapid development of information technology andthe widespread use of intelligent communication tools weprogress further into the age of information explosion andhave been bombarded with massive amounts of data Forinstance there are more than 368 million users in SinaWeibo and 100 million messages are distributed every day[1] while over 500 million registered users and more than200 million tweets are posted every signal day in Twitter [2]As the Twitter and other online social networking (OSN)applications provide great facility by their convenient andefficient way of information dissemination various studiesfocus on the issue of many complex phenomena duringthe process of the spreading of information over networksBecause of the complexity of the system OSN is generallycharacterized as being complex networks in which individ-uals are represented by nodes and information interactionsbetween individuals are represented by links between nodes[3] Since the propagation of information and spreading ofepidemic infection have a lot in common many epidemic

models have been used to describe and explore the evolutionprocess of information For instance the process of rumorpropagation can be viewed as a process of the individualsbeing infected After being infected the ignorants becomespreader with a probability and then the spreaders canbecome the stiflers which is similar as the infected personscan recover over a period of time [4]

Many researches have been devoted to the propagationdynamics The significant model for information or rumorspreading was introduced by Daley and Kendall many yearsago [5 6] Other variants based on Daley-Kendall (DK)model such as Maki-Thompson (MK) model [7] havebeen widely used in the past for quantitative study of themechanism of information dissemination In DK model aclosed and homogeneously mixed population is classifiedinto three main groups ignorants spreaders and stiflersIgnorants are susceptible to rumor or other informationSpreaders play an important role in accelerating informationdiffusion Stiflers are the individuals who get tired of rumorsand refuse to propagate them [8] In DK model the rumoris propagated through the population by pairwise contacts

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 610382 12 pageshttpdxdoiorg1011552014610382

2 Journal of Applied Mathematics

between spreaders and others while it is spread by directedcontacts of the spreaders with others in MK model

As the continuous development of research on complexnetworks a number of recent studies confirm that thenetwork topology has a marked influence on the wholediffusion process In the implementation of the MK modelZanette [9] showed that the simulation results exhibited atransition between regimes of localization and propagationat a finite value of the network randomness Moreno et al[10] studied the dynamics of epidemic spreading processesby Monte Carlo simulations and gave a numerical solutionof rumor diffusion mean-field equations [11] Zhou et al[12] focused on the influence of network structure on rumorpropagation and concluded that the total final infected nodeswill decrease when the structure changes from random toscale-free network Some studies described a formulation ofthe DK model on complex networks in terms of interactingMarkov chains Nekovee et al [13] derived mean-field equa-tions for the dynamics of rumor spreading on the arbitrarydegree correlated networks and drew a conclusion that scale-free social networks are prone to the spreading of rumorsThese studies revealed a complex interplay between thenetwork topology and found that the final infected density ofpopulation with degree 119896 is related to the network structureWhen the structure changes from random networks to scale-free networks the rumor is more easily spread

Besides the epidemic dynamicsmodelsmentioned aboveother widely used models describing information propaga-tion are diffusionmodels in social networks For instance thethreshold models developed by Granovetter [14] describedcollective social behavior and the simplest version of linearthreshold model presented by Kempe et al [15] has takensocial network structure into account during the process ofinformation propagation Those models characterized theinformation propagation based on a directed graph whichoriginated from social networks and the inactive nodes couldturn into active ones with a certain probability under theinfluence of a small number of initially activated nodes

Although extensive progress has been made in under-standing the dynamics of the information spreading moststudies focused on one type of information or topic througha complex network in the context of dynamics of informationspreading Only a few of unsystematic studies concern thecase that more than one type of information coexists andspreads among the population such as two rumors withdifferent probabilities of acceptance spreading among nodes[16] two types of information spreading among a populationof individuals [17] competing ideas in complex social systems[18] and the propagation of competing memes on compositenetworks [19 20]

Nevertheless all of these studies explored different con-tagions competing with each other as they spread over thenetwork but neglected the phenomena of the cooperationas they mutually help each other in spreading In fact thereare multiple pieces of information about an event spreadingthrough the OSN simultaneously Moreover these piecesof information do not spread in isolation manner from allother information currently diffusing in the network Thebuilt-in mechanism underlying these complex interactions

between different types of contagions makes a noticeableimpact on dynamic spreading For instance competing con-tagions decrease each otherrsquos probability of spreading whilecooperating contagions help each other in being adoptedthroughout the network [21] Thus it is necessary for usto model the interaction between contagions and not justconsider each contagion in isolation manner separated fromothers

As the main contribution of this paper we propose acooperative and competitive model for the different conta-gions of information diffusion in OSN First of all the pro-posed model is based on the population dynamics models todescribe the development tendency and interaction betweentwo different types of information It provides amore realisticdescription of this process comparing with previous modelsby Lotka-Volterra and others Secondly we use differentialequations of stability theory to analyze the proposed modelBy means of the approximate analytical and exact numericalsolutions of these equations we examine the steady-state ofinformation propagation in OSN Finally the analytic resultswith the real data collected from SinaWeibo indicate that theproposedmodel is effective for understanding and explainingsome social phenomena presented in the process of onlineinformation spreading especially the vital role of the cooper-ation and competition in the information propagation

The rest of this paper is organized as follows In Section 2a formulation of the information spreading model withinthe framework of biological models is proposed Section 3 isdevoted to analyzing the cooperative and competitive modeland explaining the results of the steady-state behavior InSection 4 the numerical simulation with the data from SinaWeibo demonstrates the investigations of the steady state anddynamics of the model The last section concludes the paperwith the direction of further research

2 Model

In this section we describe the basic information spreadingsystem and then establish a general co-competition modelbased on Lotka-Volterra model

21 System Description In OSNs websites once a user pub-lishes a message on Twitter or other microblog websitesthe message will be transmitted to hisher followers Ifthe message is so interesting that some followers decideto repost it [3] this message can be read by more usersand the information could thus reach beyond the networkof the original author We define the user behaviors suchas retweet review and mention as a growth factor Thegreater this value is the faster the message spreads and moreusers see the message Meanwhile the individuals may alsocease spreading a message spontaneously at a certain rateinfluenced by time decaying effect [22] This phenomenoncan be considered as decay factor of the message

Considering the special time attributes of the informationspreading in OSNs we choose the user growth rate V(119905) asdynamical variables instead of the users themselves In OSNsthe spread of messages is primarily determined by the repostnumber in time interval For example when we describe

Journal of Applied Mathematics 3

the spread of a tweet in Twitter it is usually defined that thismessage spread more than119873 times per hour or the messagehas been reposted a total number of nearly119873 times by time 119905However the latter is a cumulative value that cannot reflect acertain number of changes at some points Furthermore themessage spread process has strong timeliness and variation inOSNs and each message has a life cycle that spreads slowlyat beginning and then breaks out and vanishes finally Thevariables V(119905) can reflect the life circle curve of the messagewhile the variables119873 cannot

Assume that there is a message published at time 119905 inTwitter then users repost this tweet and the message couldspread at speed V(119905) As mentioned above we set 120583(V) asa function of the speed V(119905) However the speed could notimmeasurably increase and there should be a maximumvalue 119881 When the speed V(119905) achieves 119881 its value willdecrease We suppose that the spread of messages obeyslogistic growth pattern in OSNs and set

120583 (V) = 120575 sdot (1 minusV (119905)119881

) (1)

Here 120575 is a constant denoting the growth factor influence onspeed The speed V(119905) can be denoted as V(119905) = 119889119873(119905)119889119905119873(119905) denotes the total number of people who have heardthe message up to time 119905 and V(119905) denote the variationnumber in time interval [119905 minus 1 119905] Meanwhile the spread ofmessage in OSNs will be influenced by growth factor 120579(119905) anddecay factor 120582 we can establish an equation to describe thisdynamic process

119889V (119905)119889119905

= 120575 sdot (1 minusV (119905)119881

) sdot 120579 (119905) minus 120582V (119905) (2)

We discuss the main features of (2) as follows(i) When 120579(119905) = 0 or the speed V(119905) reaches its maximum

value 119881 (2) becomes

119889V (119905)119889119905

= minus120582V (119905) (3)

In (3) V(119905) is a reduction function that the speed will decreasewith time

(ii) When 120579(119905) is a nonzero constant we introducevariables 120574 and 120581 set

120574 =120575 sdot 120579 (119905)

119881+ 120582

120581 = 120575 sdot 120579 (119905)

(4)

then (2) becomes

119889V (119905)119889119905

= 120581 minus 120574V (119905) (5)

Set V(0) = V0

V (119905) =120581

120574(1 minus 119890

minus120574119905) + V0119890minus120574119905 (6)

Equation (8) denotes the function of V(119905) when 120579(119905) is anonzero constant

(iii) When the speed remains unchanged 119889V(119905)119889119905 = 0(2) becomes

120575 sdot (1 minusV (119905)119881

) sdot 120579 (119905) minus 120582V (119905) = 0 (7)

then we obtain

V (119905) =120575119881120579 (119905)

120575120579 (119905) + 120582119881 (8)

Based on the above analysis we prove that the speed V(119905)is a function of the growth factor 120579(119905) In next section wefurther discuss the situation that different types of messagesare spreading over networks

22 Co-Competition Lotka-Volterra Model The spreading ofabundance of information to which we are exposed throughonline social networks is a complex sociopsychological pro-cess In the real world these contagions not only propagateat the same time but also interplay with each other asthey spread over the networks This phenomenon is similarto the mutualism-competition interaction among multiplespecies During the spreading of pieces of information thecooperation happens when the different types of informationare at low speed and the competition happens when theyare at high speed Hence according to [23] the generalcooperation-competitionmodel of 119899 types of information canbe presented as follows 119889V

119894119889119905 = V

119894119891119894(V1 V2 V

119899) where

V119894denotes the speed of the 119894th messages and 119891

119894(V1 V2 V

119899)

represents its growth rate The partial derivatives of thefunction 119891

119894(V1 V2 V

119899) can be written as 120597119891

119894120597V119895 119894 119895 =

1 2 119899 The cooperation-competition interaction is set by(9) by satisfying the condition that for each 119894 1 le 119894 le 119899 thereis 119887119894gt 0 Consider

120597119891119895

120597V119894

gt 0 0 lt V119894lt 119887119894

120597119891119895

120597V119894

lt 0 V119894gt 119887119894

(9)

where 119894 = 119895 119894 119895 = 1 2 119899 According to the ecologicaltheory we can infer that the 119894th message has a positive effecton other messages as it is at low density (0 lt V

119894lt 119887119894) while it

has negative effect on them at high density (V119894gt 119887119894)

We study the common case that two different typesof information interact with each other and then make anatural extension of the classical two-species Lotka-Volterracompetitive model and Lotka-Volterra cooperative modelWe assume that the spread of each message is affected by theinternal attractiveness itself and the external pressure amongmessages simultaneously Considering two different types ofinformation on the same event spreading around the sametime in Twitter or other OSNs they will compete with eachother for the reason that users only choose to believe one ofthem In the meantime if all the messages are so importantor interesting users would be more likely to adopt and sharethem It is considered that they mutually help each otherin spreading through the network We term the two types

4 Journal of Applied Mathematics

of information as message 1 and message 2 and they willinteract with each other when they spread at the same timeThe propagation process and the interactions between thetwo types of information can be governed by the followingset of rules

(i) Each message has a relatively stable natural growthcoefficient 120583

119894and a decay coefficient 120582

119894 which repre-

sent the growth factor and decay factor respectivelyand 120583

119894gt 120582119894

(ii) Since a signal message could not spread indefinitelythere exists a maximum speed119881

119894during the propaga-

tion process(iii) Two messages will compete or cooperate with one

another and take negative or positive effect repre-sented by 120590

119894119895

Supported by the rules (i) and (ii) we have the followingmodel

119889V1 (119905)

119889119905= 1205831V1(1 minus

V1

1198811

) minus 1205821V1

119889V2 (119905)

119889119905= 1205832V2(1 minus

V2

1198812

) minus 1205822V2

(10)

Here 120583119894denotes the growth coefficient of the 119894th message

and 120583119894isin [0 1] 120582

119894denotes the decay coefficient of the 119894th

message and 120582119894isin [0 1] 119894 = 1 2

According to the rule (iii) when the messages 1 and 2coexist in the network the interactive influence among thetwo messages produces positive effect or negative effect asthey cooperate or compete with each other Based on theanalysis above (10) can be written as

119889V1 (119905)

119889119905= 1205831V1(1 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

) minus 1205821V1

119889V2(119905)

119889119905= 1205832V2(1 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

) minus 1205822V2

(11)

In order to facilitate the analysis set 119886 = 1 minus 12058211205831 119887 =

1 minus 12058221205832 (11) can be written as

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)

119889V2 (119905)

119889119905= 1205832V2(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)

(12)

Equation (12) can be considered as a competition andcooperation multisystem Let V

2= 0 in the first equation

of the system (12) we have 119889V1119889119905 = 120583

1V1(1198861198811minus V1)1198811

then the parameter 1198861198811is the speed of the message 1 when it

disseminates independently form message 2 Let 1198621(V1 V2) =

119886+12059021(1205731198812)minusV11198811minus12059021(|V2minus 120573|lowast1198812) the function |V

2minus 120573|lowast

can be defined as the absolute function |V2minus 120573| while the

function |V2minus 120573| is smoothed in a very small neighborhood

2

2120573+aV212059021

bV2

120573

L12

K2

K3

D2

D1

K1

L22

O 120572 aV1 12120572 + bV112059012

L1

L2

L21

L11

Figure 1 The smoothed parts of the isoclines 1198711and 119871

2

of its vertex (0 120573) Thus the isocline 1198711 1198621(V1 V2) = 0 is

smooth as shown in Figure 1 Then the function 1198911(V1 V2) =

1205831V11198621(V1 V2) and satisfies

1205971198911

120597V2

=12059021

1198812

1205831V1 0 lt V

2lt 120573

1205971198911

120597V2

= minus12059021

1198812

1205831V1 V2gt 120573

(13)

As shown in (13) the positive or negative effect actingon message 1 depends on the speed of message 2 Whenmessage 2 travels slowly (V

2lt 120573) it takes positive effect on

message 1 otherwise it produces negative effect on message1 at high speed (V

2gt 120573) The parameter 120573 signifies the

critical speed of message 2 near which the effect of message2 acting on message 1 would turn positive effect into negativeeffect Furthermore according to the Lotka-Volterra modelsthe parameter 120590

21represents both the cooperation (at low

speed) and competition (at high speed) level of message 2to message 1 Hence the low speed of message 2 0 lt V

2lt

120573 represents the scope of cooperation of message 2 withmessage 1 A similar conclusion can be obtained that thelow speed of message 1 0 lt V

1lt 120572 represents the region

of net cooperation message 1 with message 2 The isocline1198712 1198622(V1 V2) = 119887+120590

12(1205721198811)minusV21198812minus12059012(|V1minus 120572|lowast1198811) = 0

is smooth as shown in Figure 1Since the system (12) can exhibit the features of the Lotka-

Volterra cooperative model and competitive model it canbe named co-competition Lotka-Volterra model Obviouslywhen the messages are at low speed for instance V

1lt 120572

Journal of Applied Mathematics 5

The isocline of x1

+

+

minus

x2

0

0

Direction of speed change Positive partNeutral partNegative partminus

(a)

The isocline of x2

x1

minus+

0

+

0

Direction of speed change Positive partNeutral partNegative partminus

(b)

Figure 2 The parabolic zero growth isoclines of message 1 (a) and message 2 (b)

and V2lt 120573 the system (12) turns into the Lotka-Volterra

cooperative system

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

V2

1198812

minusV1

1198811

)

119889V2(119905)

119889119905= 1205832V2(119887 + 120590

12

V1

1198811

minusV2

1198812

)

(14)

Conversely when the messages are at high speed forinstance V

1gt 120572 and V

2gt 120573 the system (12) turns into the

Lotka-Volterra competitive model

119889V1(119905)

119889119905= 1205831V1(119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

)

119889V2(119905)

119889119905= 1205832V2(119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

)

(15)

By systematic comparison of the system (12) with theLotka-Volterra competitive system (15) an important conclu-sion is reached that the cooperation at low speed promotes thecoexistence level of the two messages in co-opetition Lotka-Volterra model When message 2 is at a low speed (V

2lt 120573)

it follows the first equation of system (15) that the effect ofmessage 2 acting on message 1 is negative (minus120590

21V21198812lt 0)

while it follows the first equation of system (12) that the effectof message 2 acting on message 1 is positive (120590

21V21198812gt 0)

which originates from the cooperation of message 2 at lowspeed As a consequence message 1 has survived for a longtime when message 2 is at low speed When message 2 is athigh speed (V

2gt 120573) the negative effect produced by message

2 acting on message 1 in system (15) is (minus12059021V21198812) while that

in system (12) is minus12059021(V2minus 120573)119881

2 Since minus120590

21(V2minus 120573)119881

2gt

minus12059021V21198812 the negative effect is lessened for the cooperation

among messages in system (12) Similarly the cooperation insystem (12) promotes the existence of message 1 In a wordthe cooperation in system (12) facilitates the coexistence ofboth the two messages

3 Model Analysis

In this section we discuss the intersection of two isoclines1198711and 119871

2by a symmetrical hypothesis and then analyze the

dynamics of the model and the global stability of the system

31 Basic Analysis In order to show the existence of intersec-tion of two isoclines 119871

1and 119871

2 we assume that the positive or

negative effects produced by messages 1 and 2 are almost thesame as each other In the condition we can set hypothesisas follows 119886 = 119887 = 119911 120590

21= 12059012

= 120590 1198811= 1198812= 119881

and 120572 = 120573 = 120577 then the values of 1205831and 120583

2can reflect the

difference of the growth factors among the twomessages For|V1minus120572| and |V

2minus120573| are smoothed in a very small neighborhood

of their vertexes (0 120572) and (0 120573) respectively the system (12)can be written as

119889V1 (119905)

119889119905= 1205831V1(119911 + 120590

120577

119881minusV1

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V2

120577minus 1

10038161003816100381610038161003816100381610038161003816)

119889V2(119905)

119889119905= 1205832V2(119911 + 120590

120577

119881minusV2

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V1

120577minus 1

10038161003816100381610038161003816100381610038161003816)

(16)

set 1199091= V1120577 and 119909

2= V2120577 then

1198891199091

119889119905= 1205831V1(119911 + 120590

120577

119881minus1205771199091

119881minus120590120577

119881

10038161003816100381610038161199092 minus 11003816100381610038161003816)

1198891199092

119889119905= 1205832V2(119911 + 120590

120577

119881minus1205771199092

119881minus120590120577

119881

10038161003816100381610038161199091 minus 11003816100381610038161003816)

(17)

In int1198772+ the zero growth isoclines of 119909

1and 1199092are as follows

1199091= minus120590

10038161003816100381610038161199092 minus 11003816100381610038161003816 + 119862

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(18)

where 119862 = 119881119911120577 + 120590According to [24] the isoclines include three parts as

shown in Figure 2 positive part (+) neutral part (0) and

6 Journal of Applied Mathematics

negative part (minus) The positive part neutral part or negativepart is the part of isoclines when the slope (for message1 the slope is 119889V

1(119905)119889V

2(119905) for message 2 the slope is

119889V2(119905)119889V

1(119905)) is positive part (+) neutral part (0) and

negative part (minus) The two isoclines are symmetry around theline 119909

1= 1199092 Then we only need to discuss

1199091= 1199092

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(19)

Obviously (19) always have solutions we can also obverse thefollowing aspects

(i) There are two different roots (120590 minus 119862)(120590 minus 1) and (120590 +119862)(120590 + 1) when 1205902 = 0 1 and 119862 = 1

(ii) There is double root 119909 = 1 or 119909 = 119862 when 1205902 = 1 and119862 = 1 or 120590 = 0

(iii) There is root (120590 minus 119862)(120590 minus 1) or (120590 + 119862)(120590 + 1) when1205902= 1 and 119862 = 1

Based on the above analysis we can conclude thattrajectories in phase space always intersect isoclines either inthe horizontal or in the vertical direction In the next sectionwe make further analysis about the stability of intersection

32 Equilibrium In order to study the final results of interac-tive effect between two different types of messages we needto conduct a comprehensive analysis for the stability of theequilibrium points of simultaneous differential equations

Theorem 1 System (12) admits no periodic orbit

Proof Let

119861 (V1 V2) =

1

V1V2

(V1 V2) isin int1198772

+

(20)

then we have

120597

120597V1

[1205831(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)]

+120597

120597V2

[1205832(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)]

= minus1205831

V21198811

minus1205832

V11198812

lt 0

(21)

According to the Bendixson-Dulac theorem [25] system(12) admits no periodic orbit

We study the evolution of V1(119905) and V

2(119905) as time 119905

approaches infinity As shown in Figure 1 there are threeequilibrium points in system (12) 119874

1(0 0) 119874

2(1198861198811 0) and

1198743(0 119887119881

2) To determine the stability of the equilibrium

points let

119872 = [119891V1

119891V2119892V1

119892V2

] (22)

we have

119901 = minus(119891V1

+ 119892V2)10038161003816100381610038161003816119901119894

119902 = det119872|119901119894

(23)

Then the stability conditions of equilibrium point 119874119894are

119901 gt 0 and 119902 gt 0 The Jacobin matrix of the system (12) atequilibrium state is as follows

(i) For the equilibrium point 1198741(0 0) we get

119872(1198741) = [

1198861205831

0

0 1198871205832

]

1199011= minus (119886120583

1+ 1198871205832) 119902

1= 119886119887120583

11205832

(24)

Based on the assumption that 119886 = 1 minus 12058211205831gt 0 and 119887 =

1 minus 12058221205832gt 0 we have 119901

1lt 0 and 119902

1gt 0 the point 119874

1(0 0)

is an unstable nodeSince our aim is to show the coexistence of competition

and cooperation by the introduction of 120572 and 120573 we supposethat the condition (25) is always satisfied in the rest of thepaper Consider

119887 + 12059012(2120572

1198811

minus 119886) gt 0

119886 + 12059021(2120573

1198812

minus 119887) gt 0

(25)

(ii) For the equilibrium point 1198742(1198861198811 0) we get

119872(1198742) =

[[[

[

minus1198861205831

1198861198811120583112059021

1198812

0 1205832[119887 + (

2120572

1198811

minus 119886)12059012]

]]]

]

1199012= 1198861205831minus 1205832[12059012(2120572

1198811

minus 119886) + 119887]

1199022=11988612058311205832(minus2120572120590

12+ 119886120590121198811minus 1198871198811)

1198811

(26)

The stability condition of equilibrium point 1198742(1198861198811 0) is

12059012gt 1198871198811(1198861198811minus 2120572)

Condition 12059012

gt 1198871198811(1198861198811minus 2120572) means that message

1 is more competitive than message 2 It implies that thepersonswho should have adoptedmessage 2 choose to believemessage 1 Consequently message 2 will eventually becomeextinct in the process of competition while message 1 cancontinue to diffuse and reach the maximum speed 119881

1

(iii) For the equilibrium point 1198743(0 119887119881

2) we get

119872(1198743) =

[[[

[

1205831[119886 + (

2120573

1198812

minus 119887)12059021] 0

1198871198812120583212059012

1198811

minus1198871205832

]]]

]

1199013= 1198871205832minus 1205831[119886 + 120590

21(2120573

1198812

minus 119887)]

1199023=11988712058311205832(119887120590211198812minus 1198861198812minus 2120573120590

21)

1198812

(27)

Journal of Applied Mathematics 7

The stability condition of equilibrium point 1198743(0 119887119881

2) is

12059021gt 1198861198812(1198871198812minus 2120573)

Condition 12059021gt 1198861198812(1198871198812minus 2120573) shows that message 2 is

more competitive thanmessage 1This just means that peopleaccepted message 2 even though they should have believedmessage 1 Therefore during the process of competitionmessage 2 can survive and reach a maximum speed119881

2 while

message 1 will eventually become extinct finally

33 Dynamics Assume that the conditions in (25) are satis-fied the twomessages in system (12) can coexistThe isocline1198711ofmessage 1 can be subdivided into 119871

11and 119871

12as follows

11987111 119886 + 120590

21

V2

1198812

minusV1

1198811

0 lt V2lt 120573

11987112 119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

V2gt 120573

(28)

where 1198711is smoothed in a small neighborhood of the vertex

1198631(1198861198811+ 1205731205902111988111198812 120573) as shown in Figure 1 Similarly we

divide the isocline1198712ofmessage 2 into119871

21and119871

22as follows

11987121 119887 + 120590

12

V1

1198811

minusV2

1198812

0 lt V1lt 120572

11987122 119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

V1gt 120572

(29)

where 1198712is smoothed in a small neighborhood of the vertex

1198632(120572 119887119881

2+ 1205721205901211988121198811) as shown in Figure 1

While the line segments 11987111and 119871

21are restricted to the

intervals (1198861198811 1198861198811+ 1205731205902111988111198812) and (0 120572) respectively they

are impossible to intersect in the interval 120572 le 1198861198811Then there

is a maximum of three isolated positive equilibrium points insystem (12) As shown in Figure 1 let 119870

1(11989611 11989612) represent

the intersection of 11987111and 119871

22 let 119870

2(11989621 11989622) represent the

intersection of 11987121and 119871

12 and let119870

3(11989631 11989632) represent the

intersection of 11987112and 119871

22 Then we have

11989611=1198811(119886 + 119887120590

21) + 2120572120590

1212059021

1205901212059021+ 1

11989612=1198812(1198811(119887 minus 119886120590

12) + 2120572120590

12)

(1205901212059021+ 1)119881

1

11989621=1198811(1198812(119886 minus 119887120590

21) + 2120573120590

21)

(1205901212059021+ 1)119881

2

11989622=1198812(11988612059012+ 119887) + 2120573120590

1212059021

1205901212059021+ 1

11989631=1198811(1198812(11988712059021minus 119886) minus 2120573120590

21) + 2120572120590

12120590211198812

(1205901212059021minus 1)119881

2

11989632=1198811(1198812(11988612059012minus 119887) + 2120573120590

1212059021) minus 2120572120590

121198812

(1205901212059021minus 1)119881

1

(30)

2

2120573+aV212059021

bV2

K2

K3

K1

O aV112120572 + bV112059012

L1

L2

Figure 3 Three positive equilibria 1198701 1198702 and 119870

3

Suppose that 1198661and 119866

2represent the slope of the lines

11987112and 119871

22 respectively Suppose that119866 denotes the slope of

the line11986311198632 Then we have

1198661= minus

1198812

119881112059021

lt 0

1198662= minus

119881212059012

1198811

lt 0

119866 =1198812(11988711988121198811+ 120572120590121198812minus 1205731198811)

1198811(11988611988111198812minus 1205721198812+ 120573120590211198811)lt 0

(31)

Dynamics of system (12) can be displayed in five caseswhile we discuss the situation of |119866

1| lt |119866

2| in cases 1ndash4 and

that of |1198661| gt |119866

2| in case 5

(i) Case 1 (|1198661| lt |119866| lt |119866

2|) In this case there are three

positive equilibria 1198701 1198702 and 119870

3as shown in Figure 3 The

Jacobian matrix of system (12) at 1198701is

119872(1198701) =

[[[

[

minus119896111205831

1198811

11989611120583112059021

1198812

minus11989612120583212059012

1198811

minus119896121205832

1198812

]]]

]

(32)

then 119901(119872(1198701)) = 119896

1112058311198811+ 1205832119896121198812gt 0 and 119902(119872(119870

1)) =

119896111198961212058311205832(1205901212059021+1)119881

11198812gt 0That is119870

1is asymptotically

stable Then we obtain that the stability condition of equilib-rium point119870

1is

0 lt 12059012lt

12058311198811(1205822minus 1205832)

1205832(21205721205831+ 12058211198811minus 12058311198811)

12059021gt 0 119881

1gt

21205721205831

1205831minus 1205821

(33)

8 Journal of Applied Mathematics

2

2120573+aV212059021

bV2

K2

K1

O aV112120572 + bV112059012

L1

L2

Figure 4 Two positive equilibria 1198701and 119870

2

Condition 0 lt 12059012

lt 12058311198811(1205822minus 1205832)1205832(21205721205831+ 12058211198811minus

12058311198811)means thatmessage 1 is less competitive thanmessage 2

When (33) are satisfied bothmessage 1 andmessage 2 coexistand the speed of message 1 and message 2 will tend to reach anonzero finite value

Similarly The Jacobian matrix of system (12) at 1198702is

119872(1198702) =

[[[

[

minus119896211205831

1198811

minus11989621120583112059021

1198812

11989622120583212059012

1198811

minus119896221205832

1198812

]]]

]

(34)

then the 119901(119872(1198702)) = 119896

2112058311198811+ 1198962212058321198812

gt 0 and119902(119872(119870

2)) = 119896

211198962212058311205832(1205901212059021+ 1)119881

11198812gt 0 That is

1198702is asymptotically stable Then we obtain that the stability

condition of equilibrium point1198702is

0 lt 12059021lt

12058321198812(1205821minus 1205831)

1205831(21205731205832+ 12058221198812minus 12058321198812)

12059012gt 0 119881

2gt

21205731205832

1205832minus 1205822

(35)

The condition 0 lt 12059021lt 12058321198812(1205821minus 1205831)1205831(21205731205832+ 12058221198812minus

12058321198812)means thatmessage 2 is less competitive thanmessage 1

When (35) are satisfied bothmessage 1 andmessage 2 coexistand their speed will change over time to reach a stable state

The Jacobian matrix of system (12) at 1198703is

119872(1198703) =

[[[

[

minus119896311205831

1198811

minus11989631120583112059021

1198812

minus11989632120583212059012

1198811

minus119896321205832

1198812

]]]

]

(36)

then the 119901(119872(1198703)) = 119896

3112058311198811+ 1198963212058321198812

gt 0 and119902(119872(119870

3)) = 119896

311198963212058311205832(1 minus 120590

1212059021)11988111198812 By |119866

1| lt |119866

2|

2

2120573+aV212059021

bV2

K2

O aV112120572 + bV112059012

L1

L2

Figure 5 The unique positive equilibrium 1198702

we have 1205901212059021gt 1 and 119902(119872(119870

3)) lt 0 It means that 119870

3is a

saddle point In int 1198772+ all orbits but 119870

3and the separatrixes

of1198703converge to equilibria119870

1and119870

2as shown in Figure 3

(ii) Case 2 (|1198661| lt |119866| = |119866

2|) As can be seen in Figure 4

there are two positive equilibrium points 1198701and 119870

2 It is

important to note that however 1198701and 119870

3in Figure 3

coincide Similar to the analysis in case 11198702is asymptotically

stable and the stability condition of equilibrium point 1198702

satisfies (35)Since 119901(119872(119870

1)) = 119896

1112058311198811+ 1205832119896121198812

gt 0 and119902(119872(119870

1)) = 0 it can be determined that 119870

1is a saddle node

by the saddle node rule [26] According toTheorem 1 we candraw the dynamics of system (12) as shown in Figure 4 Thesame approach also applies to the case |119866

1| = |119866| lt |119866

2| the

difference is that1198701is asymptotically stable and119870

2is a saddle

node equilibrium and all orbits converge to equilibria 1198701in

int 1198772+except 119870

2and the separatrixes of 119870

2

(iii) Case 3 (|1198661| le |119866

2| lt |119866|) In case 3 there is

only one positive equilibrium point 1198702 By Theorem 1 119870

2is

asymptotically stable as shown in Figure 5 and the stabilitycondition of equilibrium point 119870

2satisfies (33) Similar

discussions can be given for the case |119866| lt |1198661| le |119866

2| in

which 1198701is asymptotically stable

(iv) Case 4 (|1198661| = |119866

2| = |119866|) In this case the isoclines 119871

12

and 11987122coincide and all the positive equilibrium points dis-

tribute on the line segment11986311198632 According toTheorem 1 in

int 1198772+ all orbits of system (12) converge to the line segment

as shown in Figure 6

(v) Case 5 (|1198662| lt |119866

1|) In this situation we have similar

discussions with the case 3 and globally stable is1198702as shown

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

between spreaders and others while it is spread by directedcontacts of the spreaders with others in MK model

As the continuous development of research on complexnetworks a number of recent studies confirm that thenetwork topology has a marked influence on the wholediffusion process In the implementation of the MK modelZanette [9] showed that the simulation results exhibited atransition between regimes of localization and propagationat a finite value of the network randomness Moreno et al[10] studied the dynamics of epidemic spreading processesby Monte Carlo simulations and gave a numerical solutionof rumor diffusion mean-field equations [11] Zhou et al[12] focused on the influence of network structure on rumorpropagation and concluded that the total final infected nodeswill decrease when the structure changes from random toscale-free network Some studies described a formulation ofthe DK model on complex networks in terms of interactingMarkov chains Nekovee et al [13] derived mean-field equa-tions for the dynamics of rumor spreading on the arbitrarydegree correlated networks and drew a conclusion that scale-free social networks are prone to the spreading of rumorsThese studies revealed a complex interplay between thenetwork topology and found that the final infected density ofpopulation with degree 119896 is related to the network structureWhen the structure changes from random networks to scale-free networks the rumor is more easily spread

Besides the epidemic dynamicsmodelsmentioned aboveother widely used models describing information propaga-tion are diffusionmodels in social networks For instance thethreshold models developed by Granovetter [14] describedcollective social behavior and the simplest version of linearthreshold model presented by Kempe et al [15] has takensocial network structure into account during the process ofinformation propagation Those models characterized theinformation propagation based on a directed graph whichoriginated from social networks and the inactive nodes couldturn into active ones with a certain probability under theinfluence of a small number of initially activated nodes

Although extensive progress has been made in under-standing the dynamics of the information spreading moststudies focused on one type of information or topic througha complex network in the context of dynamics of informationspreading Only a few of unsystematic studies concern thecase that more than one type of information coexists andspreads among the population such as two rumors withdifferent probabilities of acceptance spreading among nodes[16] two types of information spreading among a populationof individuals [17] competing ideas in complex social systems[18] and the propagation of competing memes on compositenetworks [19 20]

Nevertheless all of these studies explored different con-tagions competing with each other as they spread over thenetwork but neglected the phenomena of the cooperationas they mutually help each other in spreading In fact thereare multiple pieces of information about an event spreadingthrough the OSN simultaneously Moreover these piecesof information do not spread in isolation manner from allother information currently diffusing in the network Thebuilt-in mechanism underlying these complex interactions

between different types of contagions makes a noticeableimpact on dynamic spreading For instance competing con-tagions decrease each otherrsquos probability of spreading whilecooperating contagions help each other in being adoptedthroughout the network [21] Thus it is necessary for usto model the interaction between contagions and not justconsider each contagion in isolation manner separated fromothers

As the main contribution of this paper we propose acooperative and competitive model for the different conta-gions of information diffusion in OSN First of all the pro-posed model is based on the population dynamics models todescribe the development tendency and interaction betweentwo different types of information It provides amore realisticdescription of this process comparing with previous modelsby Lotka-Volterra and others Secondly we use differentialequations of stability theory to analyze the proposed modelBy means of the approximate analytical and exact numericalsolutions of these equations we examine the steady-state ofinformation propagation in OSN Finally the analytic resultswith the real data collected from SinaWeibo indicate that theproposedmodel is effective for understanding and explainingsome social phenomena presented in the process of onlineinformation spreading especially the vital role of the cooper-ation and competition in the information propagation

The rest of this paper is organized as follows In Section 2a formulation of the information spreading model withinthe framework of biological models is proposed Section 3 isdevoted to analyzing the cooperative and competitive modeland explaining the results of the steady-state behavior InSection 4 the numerical simulation with the data from SinaWeibo demonstrates the investigations of the steady state anddynamics of the model The last section concludes the paperwith the direction of further research

2 Model

In this section we describe the basic information spreadingsystem and then establish a general co-competition modelbased on Lotka-Volterra model

21 System Description In OSNs websites once a user pub-lishes a message on Twitter or other microblog websitesthe message will be transmitted to hisher followers Ifthe message is so interesting that some followers decideto repost it [3] this message can be read by more usersand the information could thus reach beyond the networkof the original author We define the user behaviors suchas retweet review and mention as a growth factor Thegreater this value is the faster the message spreads and moreusers see the message Meanwhile the individuals may alsocease spreading a message spontaneously at a certain rateinfluenced by time decaying effect [22] This phenomenoncan be considered as decay factor of the message

Considering the special time attributes of the informationspreading in OSNs we choose the user growth rate V(119905) asdynamical variables instead of the users themselves In OSNsthe spread of messages is primarily determined by the repostnumber in time interval For example when we describe

Journal of Applied Mathematics 3

the spread of a tweet in Twitter it is usually defined that thismessage spread more than119873 times per hour or the messagehas been reposted a total number of nearly119873 times by time 119905However the latter is a cumulative value that cannot reflect acertain number of changes at some points Furthermore themessage spread process has strong timeliness and variation inOSNs and each message has a life cycle that spreads slowlyat beginning and then breaks out and vanishes finally Thevariables V(119905) can reflect the life circle curve of the messagewhile the variables119873 cannot

Assume that there is a message published at time 119905 inTwitter then users repost this tweet and the message couldspread at speed V(119905) As mentioned above we set 120583(V) asa function of the speed V(119905) However the speed could notimmeasurably increase and there should be a maximumvalue 119881 When the speed V(119905) achieves 119881 its value willdecrease We suppose that the spread of messages obeyslogistic growth pattern in OSNs and set

120583 (V) = 120575 sdot (1 minusV (119905)119881

) (1)

Here 120575 is a constant denoting the growth factor influence onspeed The speed V(119905) can be denoted as V(119905) = 119889119873(119905)119889119905119873(119905) denotes the total number of people who have heardthe message up to time 119905 and V(119905) denote the variationnumber in time interval [119905 minus 1 119905] Meanwhile the spread ofmessage in OSNs will be influenced by growth factor 120579(119905) anddecay factor 120582 we can establish an equation to describe thisdynamic process

119889V (119905)119889119905

= 120575 sdot (1 minusV (119905)119881

) sdot 120579 (119905) minus 120582V (119905) (2)

We discuss the main features of (2) as follows(i) When 120579(119905) = 0 or the speed V(119905) reaches its maximum

value 119881 (2) becomes

119889V (119905)119889119905

= minus120582V (119905) (3)

In (3) V(119905) is a reduction function that the speed will decreasewith time

(ii) When 120579(119905) is a nonzero constant we introducevariables 120574 and 120581 set

120574 =120575 sdot 120579 (119905)

119881+ 120582

120581 = 120575 sdot 120579 (119905)

(4)

then (2) becomes

119889V (119905)119889119905

= 120581 minus 120574V (119905) (5)

Set V(0) = V0

V (119905) =120581

120574(1 minus 119890

minus120574119905) + V0119890minus120574119905 (6)

Equation (8) denotes the function of V(119905) when 120579(119905) is anonzero constant

(iii) When the speed remains unchanged 119889V(119905)119889119905 = 0(2) becomes

120575 sdot (1 minusV (119905)119881

) sdot 120579 (119905) minus 120582V (119905) = 0 (7)

then we obtain

V (119905) =120575119881120579 (119905)

120575120579 (119905) + 120582119881 (8)

Based on the above analysis we prove that the speed V(119905)is a function of the growth factor 120579(119905) In next section wefurther discuss the situation that different types of messagesare spreading over networks

22 Co-Competition Lotka-Volterra Model The spreading ofabundance of information to which we are exposed throughonline social networks is a complex sociopsychological pro-cess In the real world these contagions not only propagateat the same time but also interplay with each other asthey spread over the networks This phenomenon is similarto the mutualism-competition interaction among multiplespecies During the spreading of pieces of information thecooperation happens when the different types of informationare at low speed and the competition happens when theyare at high speed Hence according to [23] the generalcooperation-competitionmodel of 119899 types of information canbe presented as follows 119889V

119894119889119905 = V

119894119891119894(V1 V2 V

119899) where

V119894denotes the speed of the 119894th messages and 119891

119894(V1 V2 V

119899)

represents its growth rate The partial derivatives of thefunction 119891

119894(V1 V2 V

119899) can be written as 120597119891

119894120597V119895 119894 119895 =

1 2 119899 The cooperation-competition interaction is set by(9) by satisfying the condition that for each 119894 1 le 119894 le 119899 thereis 119887119894gt 0 Consider

120597119891119895

120597V119894

gt 0 0 lt V119894lt 119887119894

120597119891119895

120597V119894

lt 0 V119894gt 119887119894

(9)

where 119894 = 119895 119894 119895 = 1 2 119899 According to the ecologicaltheory we can infer that the 119894th message has a positive effecton other messages as it is at low density (0 lt V

119894lt 119887119894) while it

has negative effect on them at high density (V119894gt 119887119894)

We study the common case that two different typesof information interact with each other and then make anatural extension of the classical two-species Lotka-Volterracompetitive model and Lotka-Volterra cooperative modelWe assume that the spread of each message is affected by theinternal attractiveness itself and the external pressure amongmessages simultaneously Considering two different types ofinformation on the same event spreading around the sametime in Twitter or other OSNs they will compete with eachother for the reason that users only choose to believe one ofthem In the meantime if all the messages are so importantor interesting users would be more likely to adopt and sharethem It is considered that they mutually help each otherin spreading through the network We term the two types

4 Journal of Applied Mathematics

of information as message 1 and message 2 and they willinteract with each other when they spread at the same timeThe propagation process and the interactions between thetwo types of information can be governed by the followingset of rules

(i) Each message has a relatively stable natural growthcoefficient 120583

119894and a decay coefficient 120582

119894 which repre-

sent the growth factor and decay factor respectivelyand 120583

119894gt 120582119894

(ii) Since a signal message could not spread indefinitelythere exists a maximum speed119881

119894during the propaga-

tion process(iii) Two messages will compete or cooperate with one

another and take negative or positive effect repre-sented by 120590

119894119895

Supported by the rules (i) and (ii) we have the followingmodel

119889V1 (119905)

119889119905= 1205831V1(1 minus

V1

1198811

) minus 1205821V1

119889V2 (119905)

119889119905= 1205832V2(1 minus

V2

1198812

) minus 1205822V2

(10)

Here 120583119894denotes the growth coefficient of the 119894th message

and 120583119894isin [0 1] 120582

119894denotes the decay coefficient of the 119894th

message and 120582119894isin [0 1] 119894 = 1 2

According to the rule (iii) when the messages 1 and 2coexist in the network the interactive influence among thetwo messages produces positive effect or negative effect asthey cooperate or compete with each other Based on theanalysis above (10) can be written as

119889V1 (119905)

119889119905= 1205831V1(1 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

) minus 1205821V1

119889V2(119905)

119889119905= 1205832V2(1 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

) minus 1205822V2

(11)

In order to facilitate the analysis set 119886 = 1 minus 12058211205831 119887 =

1 minus 12058221205832 (11) can be written as

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)

119889V2 (119905)

119889119905= 1205832V2(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)

(12)

Equation (12) can be considered as a competition andcooperation multisystem Let V

2= 0 in the first equation

of the system (12) we have 119889V1119889119905 = 120583

1V1(1198861198811minus V1)1198811

then the parameter 1198861198811is the speed of the message 1 when it

disseminates independently form message 2 Let 1198621(V1 V2) =

119886+12059021(1205731198812)minusV11198811minus12059021(|V2minus 120573|lowast1198812) the function |V

2minus 120573|lowast

can be defined as the absolute function |V2minus 120573| while the

function |V2minus 120573| is smoothed in a very small neighborhood

2

2120573+aV212059021

bV2

120573

L12

K2

K3

D2

D1

K1

L22

O 120572 aV1 12120572 + bV112059012

L1

L2

L21

L11

Figure 1 The smoothed parts of the isoclines 1198711and 119871

2

of its vertex (0 120573) Thus the isocline 1198711 1198621(V1 V2) = 0 is

smooth as shown in Figure 1 Then the function 1198911(V1 V2) =

1205831V11198621(V1 V2) and satisfies

1205971198911

120597V2

=12059021

1198812

1205831V1 0 lt V

2lt 120573

1205971198911

120597V2

= minus12059021

1198812

1205831V1 V2gt 120573

(13)

As shown in (13) the positive or negative effect actingon message 1 depends on the speed of message 2 Whenmessage 2 travels slowly (V

2lt 120573) it takes positive effect on

message 1 otherwise it produces negative effect on message1 at high speed (V

2gt 120573) The parameter 120573 signifies the

critical speed of message 2 near which the effect of message2 acting on message 1 would turn positive effect into negativeeffect Furthermore according to the Lotka-Volterra modelsthe parameter 120590

21represents both the cooperation (at low

speed) and competition (at high speed) level of message 2to message 1 Hence the low speed of message 2 0 lt V

2lt

120573 represents the scope of cooperation of message 2 withmessage 1 A similar conclusion can be obtained that thelow speed of message 1 0 lt V

1lt 120572 represents the region

of net cooperation message 1 with message 2 The isocline1198712 1198622(V1 V2) = 119887+120590

12(1205721198811)minusV21198812minus12059012(|V1minus 120572|lowast1198811) = 0

is smooth as shown in Figure 1Since the system (12) can exhibit the features of the Lotka-

Volterra cooperative model and competitive model it canbe named co-competition Lotka-Volterra model Obviouslywhen the messages are at low speed for instance V

1lt 120572

Journal of Applied Mathematics 5

The isocline of x1

+

+

minus

x2

0

0

Direction of speed change Positive partNeutral partNegative partminus

(a)

The isocline of x2

x1

minus+

0

+

0

Direction of speed change Positive partNeutral partNegative partminus

(b)

Figure 2 The parabolic zero growth isoclines of message 1 (a) and message 2 (b)

and V2lt 120573 the system (12) turns into the Lotka-Volterra

cooperative system

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

V2

1198812

minusV1

1198811

)

119889V2(119905)

119889119905= 1205832V2(119887 + 120590

12

V1

1198811

minusV2

1198812

)

(14)

Conversely when the messages are at high speed forinstance V

1gt 120572 and V

2gt 120573 the system (12) turns into the

Lotka-Volterra competitive model

119889V1(119905)

119889119905= 1205831V1(119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

)

119889V2(119905)

119889119905= 1205832V2(119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

)

(15)

By systematic comparison of the system (12) with theLotka-Volterra competitive system (15) an important conclu-sion is reached that the cooperation at low speed promotes thecoexistence level of the two messages in co-opetition Lotka-Volterra model When message 2 is at a low speed (V

2lt 120573)

it follows the first equation of system (15) that the effect ofmessage 2 acting on message 1 is negative (minus120590

21V21198812lt 0)

while it follows the first equation of system (12) that the effectof message 2 acting on message 1 is positive (120590

21V21198812gt 0)

which originates from the cooperation of message 2 at lowspeed As a consequence message 1 has survived for a longtime when message 2 is at low speed When message 2 is athigh speed (V

2gt 120573) the negative effect produced by message

2 acting on message 1 in system (15) is (minus12059021V21198812) while that

in system (12) is minus12059021(V2minus 120573)119881

2 Since minus120590

21(V2minus 120573)119881

2gt

minus12059021V21198812 the negative effect is lessened for the cooperation

among messages in system (12) Similarly the cooperation insystem (12) promotes the existence of message 1 In a wordthe cooperation in system (12) facilitates the coexistence ofboth the two messages

3 Model Analysis

In this section we discuss the intersection of two isoclines1198711and 119871

2by a symmetrical hypothesis and then analyze the

dynamics of the model and the global stability of the system

31 Basic Analysis In order to show the existence of intersec-tion of two isoclines 119871

1and 119871

2 we assume that the positive or

negative effects produced by messages 1 and 2 are almost thesame as each other In the condition we can set hypothesisas follows 119886 = 119887 = 119911 120590

21= 12059012

= 120590 1198811= 1198812= 119881

and 120572 = 120573 = 120577 then the values of 1205831and 120583

2can reflect the

difference of the growth factors among the twomessages For|V1minus120572| and |V

2minus120573| are smoothed in a very small neighborhood

of their vertexes (0 120572) and (0 120573) respectively the system (12)can be written as

119889V1 (119905)

119889119905= 1205831V1(119911 + 120590

120577

119881minusV1

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V2

120577minus 1

10038161003816100381610038161003816100381610038161003816)

119889V2(119905)

119889119905= 1205832V2(119911 + 120590

120577

119881minusV2

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V1

120577minus 1

10038161003816100381610038161003816100381610038161003816)

(16)

set 1199091= V1120577 and 119909

2= V2120577 then

1198891199091

119889119905= 1205831V1(119911 + 120590

120577

119881minus1205771199091

119881minus120590120577

119881

10038161003816100381610038161199092 minus 11003816100381610038161003816)

1198891199092

119889119905= 1205832V2(119911 + 120590

120577

119881minus1205771199092

119881minus120590120577

119881

10038161003816100381610038161199091 minus 11003816100381610038161003816)

(17)

In int1198772+ the zero growth isoclines of 119909

1and 1199092are as follows

1199091= minus120590

10038161003816100381610038161199092 minus 11003816100381610038161003816 + 119862

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(18)

where 119862 = 119881119911120577 + 120590According to [24] the isoclines include three parts as

shown in Figure 2 positive part (+) neutral part (0) and

6 Journal of Applied Mathematics

negative part (minus) The positive part neutral part or negativepart is the part of isoclines when the slope (for message1 the slope is 119889V

1(119905)119889V

2(119905) for message 2 the slope is

119889V2(119905)119889V

1(119905)) is positive part (+) neutral part (0) and

negative part (minus) The two isoclines are symmetry around theline 119909

1= 1199092 Then we only need to discuss

1199091= 1199092

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(19)

Obviously (19) always have solutions we can also obverse thefollowing aspects

(i) There are two different roots (120590 minus 119862)(120590 minus 1) and (120590 +119862)(120590 + 1) when 1205902 = 0 1 and 119862 = 1

(ii) There is double root 119909 = 1 or 119909 = 119862 when 1205902 = 1 and119862 = 1 or 120590 = 0

(iii) There is root (120590 minus 119862)(120590 minus 1) or (120590 + 119862)(120590 + 1) when1205902= 1 and 119862 = 1

Based on the above analysis we can conclude thattrajectories in phase space always intersect isoclines either inthe horizontal or in the vertical direction In the next sectionwe make further analysis about the stability of intersection

32 Equilibrium In order to study the final results of interac-tive effect between two different types of messages we needto conduct a comprehensive analysis for the stability of theequilibrium points of simultaneous differential equations

Theorem 1 System (12) admits no periodic orbit

Proof Let

119861 (V1 V2) =

1

V1V2

(V1 V2) isin int1198772

+

(20)

then we have

120597

120597V1

[1205831(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)]

+120597

120597V2

[1205832(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)]

= minus1205831

V21198811

minus1205832

V11198812

lt 0

(21)

According to the Bendixson-Dulac theorem [25] system(12) admits no periodic orbit

We study the evolution of V1(119905) and V

2(119905) as time 119905

approaches infinity As shown in Figure 1 there are threeequilibrium points in system (12) 119874

1(0 0) 119874

2(1198861198811 0) and

1198743(0 119887119881

2) To determine the stability of the equilibrium

points let

119872 = [119891V1

119891V2119892V1

119892V2

] (22)

we have

119901 = minus(119891V1

+ 119892V2)10038161003816100381610038161003816119901119894

119902 = det119872|119901119894

(23)

Then the stability conditions of equilibrium point 119874119894are

119901 gt 0 and 119902 gt 0 The Jacobin matrix of the system (12) atequilibrium state is as follows

(i) For the equilibrium point 1198741(0 0) we get

119872(1198741) = [

1198861205831

0

0 1198871205832

]

1199011= minus (119886120583

1+ 1198871205832) 119902

1= 119886119887120583

11205832

(24)

Based on the assumption that 119886 = 1 minus 12058211205831gt 0 and 119887 =

1 minus 12058221205832gt 0 we have 119901

1lt 0 and 119902

1gt 0 the point 119874

1(0 0)

is an unstable nodeSince our aim is to show the coexistence of competition

and cooperation by the introduction of 120572 and 120573 we supposethat the condition (25) is always satisfied in the rest of thepaper Consider

119887 + 12059012(2120572

1198811

minus 119886) gt 0

119886 + 12059021(2120573

1198812

minus 119887) gt 0

(25)

(ii) For the equilibrium point 1198742(1198861198811 0) we get

119872(1198742) =

[[[

[

minus1198861205831

1198861198811120583112059021

1198812

0 1205832[119887 + (

2120572

1198811

minus 119886)12059012]

]]]

]

1199012= 1198861205831minus 1205832[12059012(2120572

1198811

minus 119886) + 119887]

1199022=11988612058311205832(minus2120572120590

12+ 119886120590121198811minus 1198871198811)

1198811

(26)

The stability condition of equilibrium point 1198742(1198861198811 0) is

12059012gt 1198871198811(1198861198811minus 2120572)

Condition 12059012

gt 1198871198811(1198861198811minus 2120572) means that message

1 is more competitive than message 2 It implies that thepersonswho should have adoptedmessage 2 choose to believemessage 1 Consequently message 2 will eventually becomeextinct in the process of competition while message 1 cancontinue to diffuse and reach the maximum speed 119881

1

(iii) For the equilibrium point 1198743(0 119887119881

2) we get

119872(1198743) =

[[[

[

1205831[119886 + (

2120573

1198812

minus 119887)12059021] 0

1198871198812120583212059012

1198811

minus1198871205832

]]]

]

1199013= 1198871205832minus 1205831[119886 + 120590

21(2120573

1198812

minus 119887)]

1199023=11988712058311205832(119887120590211198812minus 1198861198812minus 2120573120590

21)

1198812

(27)

Journal of Applied Mathematics 7

The stability condition of equilibrium point 1198743(0 119887119881

2) is

12059021gt 1198861198812(1198871198812minus 2120573)

Condition 12059021gt 1198861198812(1198871198812minus 2120573) shows that message 2 is

more competitive thanmessage 1This just means that peopleaccepted message 2 even though they should have believedmessage 1 Therefore during the process of competitionmessage 2 can survive and reach a maximum speed119881

2 while

message 1 will eventually become extinct finally

33 Dynamics Assume that the conditions in (25) are satis-fied the twomessages in system (12) can coexistThe isocline1198711ofmessage 1 can be subdivided into 119871

11and 119871

12as follows

11987111 119886 + 120590

21

V2

1198812

minusV1

1198811

0 lt V2lt 120573

11987112 119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

V2gt 120573

(28)

where 1198711is smoothed in a small neighborhood of the vertex

1198631(1198861198811+ 1205731205902111988111198812 120573) as shown in Figure 1 Similarly we

divide the isocline1198712ofmessage 2 into119871

21and119871

22as follows

11987121 119887 + 120590

12

V1

1198811

minusV2

1198812

0 lt V1lt 120572

11987122 119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

V1gt 120572

(29)

where 1198712is smoothed in a small neighborhood of the vertex

1198632(120572 119887119881

2+ 1205721205901211988121198811) as shown in Figure 1

While the line segments 11987111and 119871

21are restricted to the

intervals (1198861198811 1198861198811+ 1205731205902111988111198812) and (0 120572) respectively they

are impossible to intersect in the interval 120572 le 1198861198811Then there

is a maximum of three isolated positive equilibrium points insystem (12) As shown in Figure 1 let 119870

1(11989611 11989612) represent

the intersection of 11987111and 119871

22 let 119870

2(11989621 11989622) represent the

intersection of 11987121and 119871

12 and let119870

3(11989631 11989632) represent the

intersection of 11987112and 119871

22 Then we have

11989611=1198811(119886 + 119887120590

21) + 2120572120590

1212059021

1205901212059021+ 1

11989612=1198812(1198811(119887 minus 119886120590

12) + 2120572120590

12)

(1205901212059021+ 1)119881

1

11989621=1198811(1198812(119886 minus 119887120590

21) + 2120573120590

21)

(1205901212059021+ 1)119881

2

11989622=1198812(11988612059012+ 119887) + 2120573120590

1212059021

1205901212059021+ 1

11989631=1198811(1198812(11988712059021minus 119886) minus 2120573120590

21) + 2120572120590

12120590211198812

(1205901212059021minus 1)119881

2

11989632=1198811(1198812(11988612059012minus 119887) + 2120573120590

1212059021) minus 2120572120590

121198812

(1205901212059021minus 1)119881

1

(30)

2

2120573+aV212059021

bV2

K2

K3

K1

O aV112120572 + bV112059012

L1

L2

Figure 3 Three positive equilibria 1198701 1198702 and 119870

3

Suppose that 1198661and 119866

2represent the slope of the lines

11987112and 119871

22 respectively Suppose that119866 denotes the slope of

the line11986311198632 Then we have

1198661= minus

1198812

119881112059021

lt 0

1198662= minus

119881212059012

1198811

lt 0

119866 =1198812(11988711988121198811+ 120572120590121198812minus 1205731198811)

1198811(11988611988111198812minus 1205721198812+ 120573120590211198811)lt 0

(31)

Dynamics of system (12) can be displayed in five caseswhile we discuss the situation of |119866

1| lt |119866

2| in cases 1ndash4 and

that of |1198661| gt |119866

2| in case 5

(i) Case 1 (|1198661| lt |119866| lt |119866

2|) In this case there are three

positive equilibria 1198701 1198702 and 119870

3as shown in Figure 3 The

Jacobian matrix of system (12) at 1198701is

119872(1198701) =

[[[

[

minus119896111205831

1198811

11989611120583112059021

1198812

minus11989612120583212059012

1198811

minus119896121205832

1198812

]]]

]

(32)

then 119901(119872(1198701)) = 119896

1112058311198811+ 1205832119896121198812gt 0 and 119902(119872(119870

1)) =

119896111198961212058311205832(1205901212059021+1)119881

11198812gt 0That is119870

1is asymptotically

stable Then we obtain that the stability condition of equilib-rium point119870

1is

0 lt 12059012lt

12058311198811(1205822minus 1205832)

1205832(21205721205831+ 12058211198811minus 12058311198811)

12059021gt 0 119881

1gt

21205721205831

1205831minus 1205821

(33)

8 Journal of Applied Mathematics

2

2120573+aV212059021

bV2

K2

K1

O aV112120572 + bV112059012

L1

L2

Figure 4 Two positive equilibria 1198701and 119870

2

Condition 0 lt 12059012

lt 12058311198811(1205822minus 1205832)1205832(21205721205831+ 12058211198811minus

12058311198811)means thatmessage 1 is less competitive thanmessage 2

When (33) are satisfied bothmessage 1 andmessage 2 coexistand the speed of message 1 and message 2 will tend to reach anonzero finite value

Similarly The Jacobian matrix of system (12) at 1198702is

119872(1198702) =

[[[

[

minus119896211205831

1198811

minus11989621120583112059021

1198812

11989622120583212059012

1198811

minus119896221205832

1198812

]]]

]

(34)

then the 119901(119872(1198702)) = 119896

2112058311198811+ 1198962212058321198812

gt 0 and119902(119872(119870

2)) = 119896

211198962212058311205832(1205901212059021+ 1)119881

11198812gt 0 That is

1198702is asymptotically stable Then we obtain that the stability

condition of equilibrium point1198702is

0 lt 12059021lt

12058321198812(1205821minus 1205831)

1205831(21205731205832+ 12058221198812minus 12058321198812)

12059012gt 0 119881

2gt

21205731205832

1205832minus 1205822

(35)

The condition 0 lt 12059021lt 12058321198812(1205821minus 1205831)1205831(21205731205832+ 12058221198812minus

12058321198812)means thatmessage 2 is less competitive thanmessage 1

When (35) are satisfied bothmessage 1 andmessage 2 coexistand their speed will change over time to reach a stable state

The Jacobian matrix of system (12) at 1198703is

119872(1198703) =

[[[

[

minus119896311205831

1198811

minus11989631120583112059021

1198812

minus11989632120583212059012

1198811

minus119896321205832

1198812

]]]

]

(36)

then the 119901(119872(1198703)) = 119896

3112058311198811+ 1198963212058321198812

gt 0 and119902(119872(119870

3)) = 119896

311198963212058311205832(1 minus 120590

1212059021)11988111198812 By |119866

1| lt |119866

2|

2

2120573+aV212059021

bV2

K2

O aV112120572 + bV112059012

L1

L2

Figure 5 The unique positive equilibrium 1198702

we have 1205901212059021gt 1 and 119902(119872(119870

3)) lt 0 It means that 119870

3is a

saddle point In int 1198772+ all orbits but 119870

3and the separatrixes

of1198703converge to equilibria119870

1and119870

2as shown in Figure 3

(ii) Case 2 (|1198661| lt |119866| = |119866

2|) As can be seen in Figure 4

there are two positive equilibrium points 1198701and 119870

2 It is

important to note that however 1198701and 119870

3in Figure 3

coincide Similar to the analysis in case 11198702is asymptotically

stable and the stability condition of equilibrium point 1198702

satisfies (35)Since 119901(119872(119870

1)) = 119896

1112058311198811+ 1205832119896121198812

gt 0 and119902(119872(119870

1)) = 0 it can be determined that 119870

1is a saddle node

by the saddle node rule [26] According toTheorem 1 we candraw the dynamics of system (12) as shown in Figure 4 Thesame approach also applies to the case |119866

1| = |119866| lt |119866

2| the

difference is that1198701is asymptotically stable and119870

2is a saddle

node equilibrium and all orbits converge to equilibria 1198701in

int 1198772+except 119870

2and the separatrixes of 119870

2

(iii) Case 3 (|1198661| le |119866

2| lt |119866|) In case 3 there is

only one positive equilibrium point 1198702 By Theorem 1 119870

2is

asymptotically stable as shown in Figure 5 and the stabilitycondition of equilibrium point 119870

2satisfies (33) Similar

discussions can be given for the case |119866| lt |1198661| le |119866

2| in

which 1198701is asymptotically stable

(iv) Case 4 (|1198661| = |119866

2| = |119866|) In this case the isoclines 119871

12

and 11987122coincide and all the positive equilibrium points dis-

tribute on the line segment11986311198632 According toTheorem 1 in

int 1198772+ all orbits of system (12) converge to the line segment

as shown in Figure 6

(v) Case 5 (|1198662| lt |119866

1|) In this situation we have similar

discussions with the case 3 and globally stable is1198702as shown

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

the spread of a tweet in Twitter it is usually defined that thismessage spread more than119873 times per hour or the messagehas been reposted a total number of nearly119873 times by time 119905However the latter is a cumulative value that cannot reflect acertain number of changes at some points Furthermore themessage spread process has strong timeliness and variation inOSNs and each message has a life cycle that spreads slowlyat beginning and then breaks out and vanishes finally Thevariables V(119905) can reflect the life circle curve of the messagewhile the variables119873 cannot

Assume that there is a message published at time 119905 inTwitter then users repost this tweet and the message couldspread at speed V(119905) As mentioned above we set 120583(V) asa function of the speed V(119905) However the speed could notimmeasurably increase and there should be a maximumvalue 119881 When the speed V(119905) achieves 119881 its value willdecrease We suppose that the spread of messages obeyslogistic growth pattern in OSNs and set

120583 (V) = 120575 sdot (1 minusV (119905)119881

) (1)

Here 120575 is a constant denoting the growth factor influence onspeed The speed V(119905) can be denoted as V(119905) = 119889119873(119905)119889119905119873(119905) denotes the total number of people who have heardthe message up to time 119905 and V(119905) denote the variationnumber in time interval [119905 minus 1 119905] Meanwhile the spread ofmessage in OSNs will be influenced by growth factor 120579(119905) anddecay factor 120582 we can establish an equation to describe thisdynamic process

119889V (119905)119889119905

= 120575 sdot (1 minusV (119905)119881

) sdot 120579 (119905) minus 120582V (119905) (2)

We discuss the main features of (2) as follows(i) When 120579(119905) = 0 or the speed V(119905) reaches its maximum

value 119881 (2) becomes

119889V (119905)119889119905

= minus120582V (119905) (3)

In (3) V(119905) is a reduction function that the speed will decreasewith time

(ii) When 120579(119905) is a nonzero constant we introducevariables 120574 and 120581 set

120574 =120575 sdot 120579 (119905)

119881+ 120582

120581 = 120575 sdot 120579 (119905)

(4)

then (2) becomes

119889V (119905)119889119905

= 120581 minus 120574V (119905) (5)

Set V(0) = V0

V (119905) =120581

120574(1 minus 119890

minus120574119905) + V0119890minus120574119905 (6)

Equation (8) denotes the function of V(119905) when 120579(119905) is anonzero constant

(iii) When the speed remains unchanged 119889V(119905)119889119905 = 0(2) becomes

120575 sdot (1 minusV (119905)119881

) sdot 120579 (119905) minus 120582V (119905) = 0 (7)

then we obtain

V (119905) =120575119881120579 (119905)

120575120579 (119905) + 120582119881 (8)

Based on the above analysis we prove that the speed V(119905)is a function of the growth factor 120579(119905) In next section wefurther discuss the situation that different types of messagesare spreading over networks

22 Co-Competition Lotka-Volterra Model The spreading ofabundance of information to which we are exposed throughonline social networks is a complex sociopsychological pro-cess In the real world these contagions not only propagateat the same time but also interplay with each other asthey spread over the networks This phenomenon is similarto the mutualism-competition interaction among multiplespecies During the spreading of pieces of information thecooperation happens when the different types of informationare at low speed and the competition happens when theyare at high speed Hence according to [23] the generalcooperation-competitionmodel of 119899 types of information canbe presented as follows 119889V

119894119889119905 = V

119894119891119894(V1 V2 V

119899) where

V119894denotes the speed of the 119894th messages and 119891

119894(V1 V2 V

119899)

represents its growth rate The partial derivatives of thefunction 119891

119894(V1 V2 V

119899) can be written as 120597119891

119894120597V119895 119894 119895 =

1 2 119899 The cooperation-competition interaction is set by(9) by satisfying the condition that for each 119894 1 le 119894 le 119899 thereis 119887119894gt 0 Consider

120597119891119895

120597V119894

gt 0 0 lt V119894lt 119887119894

120597119891119895

120597V119894

lt 0 V119894gt 119887119894

(9)

where 119894 = 119895 119894 119895 = 1 2 119899 According to the ecologicaltheory we can infer that the 119894th message has a positive effecton other messages as it is at low density (0 lt V

119894lt 119887119894) while it

has negative effect on them at high density (V119894gt 119887119894)

We study the common case that two different typesof information interact with each other and then make anatural extension of the classical two-species Lotka-Volterracompetitive model and Lotka-Volterra cooperative modelWe assume that the spread of each message is affected by theinternal attractiveness itself and the external pressure amongmessages simultaneously Considering two different types ofinformation on the same event spreading around the sametime in Twitter or other OSNs they will compete with eachother for the reason that users only choose to believe one ofthem In the meantime if all the messages are so importantor interesting users would be more likely to adopt and sharethem It is considered that they mutually help each otherin spreading through the network We term the two types

4 Journal of Applied Mathematics

of information as message 1 and message 2 and they willinteract with each other when they spread at the same timeThe propagation process and the interactions between thetwo types of information can be governed by the followingset of rules

(i) Each message has a relatively stable natural growthcoefficient 120583

119894and a decay coefficient 120582

119894 which repre-

sent the growth factor and decay factor respectivelyand 120583

119894gt 120582119894

(ii) Since a signal message could not spread indefinitelythere exists a maximum speed119881

119894during the propaga-

tion process(iii) Two messages will compete or cooperate with one

another and take negative or positive effect repre-sented by 120590

119894119895

Supported by the rules (i) and (ii) we have the followingmodel

119889V1 (119905)

119889119905= 1205831V1(1 minus

V1

1198811

) minus 1205821V1

119889V2 (119905)

119889119905= 1205832V2(1 minus

V2

1198812

) minus 1205822V2

(10)

Here 120583119894denotes the growth coefficient of the 119894th message

and 120583119894isin [0 1] 120582

119894denotes the decay coefficient of the 119894th

message and 120582119894isin [0 1] 119894 = 1 2

According to the rule (iii) when the messages 1 and 2coexist in the network the interactive influence among thetwo messages produces positive effect or negative effect asthey cooperate or compete with each other Based on theanalysis above (10) can be written as

119889V1 (119905)

119889119905= 1205831V1(1 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

) minus 1205821V1

119889V2(119905)

119889119905= 1205832V2(1 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

) minus 1205822V2

(11)

In order to facilitate the analysis set 119886 = 1 minus 12058211205831 119887 =

1 minus 12058221205832 (11) can be written as

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)

119889V2 (119905)

119889119905= 1205832V2(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)

(12)

Equation (12) can be considered as a competition andcooperation multisystem Let V

2= 0 in the first equation

of the system (12) we have 119889V1119889119905 = 120583

1V1(1198861198811minus V1)1198811

then the parameter 1198861198811is the speed of the message 1 when it

disseminates independently form message 2 Let 1198621(V1 V2) =

119886+12059021(1205731198812)minusV11198811minus12059021(|V2minus 120573|lowast1198812) the function |V

2minus 120573|lowast

can be defined as the absolute function |V2minus 120573| while the

function |V2minus 120573| is smoothed in a very small neighborhood

2

2120573+aV212059021

bV2

120573

L12

K2

K3

D2

D1

K1

L22

O 120572 aV1 12120572 + bV112059012

L1

L2

L21

L11

Figure 1 The smoothed parts of the isoclines 1198711and 119871

2

of its vertex (0 120573) Thus the isocline 1198711 1198621(V1 V2) = 0 is

smooth as shown in Figure 1 Then the function 1198911(V1 V2) =

1205831V11198621(V1 V2) and satisfies

1205971198911

120597V2

=12059021

1198812

1205831V1 0 lt V

2lt 120573

1205971198911

120597V2

= minus12059021

1198812

1205831V1 V2gt 120573

(13)

As shown in (13) the positive or negative effect actingon message 1 depends on the speed of message 2 Whenmessage 2 travels slowly (V

2lt 120573) it takes positive effect on

message 1 otherwise it produces negative effect on message1 at high speed (V

2gt 120573) The parameter 120573 signifies the

critical speed of message 2 near which the effect of message2 acting on message 1 would turn positive effect into negativeeffect Furthermore according to the Lotka-Volterra modelsthe parameter 120590

21represents both the cooperation (at low

speed) and competition (at high speed) level of message 2to message 1 Hence the low speed of message 2 0 lt V

2lt

120573 represents the scope of cooperation of message 2 withmessage 1 A similar conclusion can be obtained that thelow speed of message 1 0 lt V

1lt 120572 represents the region

of net cooperation message 1 with message 2 The isocline1198712 1198622(V1 V2) = 119887+120590

12(1205721198811)minusV21198812minus12059012(|V1minus 120572|lowast1198811) = 0

is smooth as shown in Figure 1Since the system (12) can exhibit the features of the Lotka-

Volterra cooperative model and competitive model it canbe named co-competition Lotka-Volterra model Obviouslywhen the messages are at low speed for instance V

1lt 120572

Journal of Applied Mathematics 5

The isocline of x1

+

+

minus

x2

0

0

Direction of speed change Positive partNeutral partNegative partminus

(a)

The isocline of x2

x1

minus+

0

+

0

Direction of speed change Positive partNeutral partNegative partminus

(b)

Figure 2 The parabolic zero growth isoclines of message 1 (a) and message 2 (b)

and V2lt 120573 the system (12) turns into the Lotka-Volterra

cooperative system

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

V2

1198812

minusV1

1198811

)

119889V2(119905)

119889119905= 1205832V2(119887 + 120590

12

V1

1198811

minusV2

1198812

)

(14)

Conversely when the messages are at high speed forinstance V

1gt 120572 and V

2gt 120573 the system (12) turns into the

Lotka-Volterra competitive model

119889V1(119905)

119889119905= 1205831V1(119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

)

119889V2(119905)

119889119905= 1205832V2(119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

)

(15)

By systematic comparison of the system (12) with theLotka-Volterra competitive system (15) an important conclu-sion is reached that the cooperation at low speed promotes thecoexistence level of the two messages in co-opetition Lotka-Volterra model When message 2 is at a low speed (V

2lt 120573)

it follows the first equation of system (15) that the effect ofmessage 2 acting on message 1 is negative (minus120590

21V21198812lt 0)

while it follows the first equation of system (12) that the effectof message 2 acting on message 1 is positive (120590

21V21198812gt 0)

which originates from the cooperation of message 2 at lowspeed As a consequence message 1 has survived for a longtime when message 2 is at low speed When message 2 is athigh speed (V

2gt 120573) the negative effect produced by message

2 acting on message 1 in system (15) is (minus12059021V21198812) while that

in system (12) is minus12059021(V2minus 120573)119881

2 Since minus120590

21(V2minus 120573)119881

2gt

minus12059021V21198812 the negative effect is lessened for the cooperation

among messages in system (12) Similarly the cooperation insystem (12) promotes the existence of message 1 In a wordthe cooperation in system (12) facilitates the coexistence ofboth the two messages

3 Model Analysis

In this section we discuss the intersection of two isoclines1198711and 119871

2by a symmetrical hypothesis and then analyze the

dynamics of the model and the global stability of the system

31 Basic Analysis In order to show the existence of intersec-tion of two isoclines 119871

1and 119871

2 we assume that the positive or

negative effects produced by messages 1 and 2 are almost thesame as each other In the condition we can set hypothesisas follows 119886 = 119887 = 119911 120590

21= 12059012

= 120590 1198811= 1198812= 119881

and 120572 = 120573 = 120577 then the values of 1205831and 120583

2can reflect the

difference of the growth factors among the twomessages For|V1minus120572| and |V

2minus120573| are smoothed in a very small neighborhood

of their vertexes (0 120572) and (0 120573) respectively the system (12)can be written as

119889V1 (119905)

119889119905= 1205831V1(119911 + 120590

120577

119881minusV1

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V2

120577minus 1

10038161003816100381610038161003816100381610038161003816)

119889V2(119905)

119889119905= 1205832V2(119911 + 120590

120577

119881minusV2

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V1

120577minus 1

10038161003816100381610038161003816100381610038161003816)

(16)

set 1199091= V1120577 and 119909

2= V2120577 then

1198891199091

119889119905= 1205831V1(119911 + 120590

120577

119881minus1205771199091

119881minus120590120577

119881

10038161003816100381610038161199092 minus 11003816100381610038161003816)

1198891199092

119889119905= 1205832V2(119911 + 120590

120577

119881minus1205771199092

119881minus120590120577

119881

10038161003816100381610038161199091 minus 11003816100381610038161003816)

(17)

In int1198772+ the zero growth isoclines of 119909

1and 1199092are as follows

1199091= minus120590

10038161003816100381610038161199092 minus 11003816100381610038161003816 + 119862

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(18)

where 119862 = 119881119911120577 + 120590According to [24] the isoclines include three parts as

shown in Figure 2 positive part (+) neutral part (0) and

6 Journal of Applied Mathematics

negative part (minus) The positive part neutral part or negativepart is the part of isoclines when the slope (for message1 the slope is 119889V

1(119905)119889V

2(119905) for message 2 the slope is

119889V2(119905)119889V

1(119905)) is positive part (+) neutral part (0) and

negative part (minus) The two isoclines are symmetry around theline 119909

1= 1199092 Then we only need to discuss

1199091= 1199092

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(19)

Obviously (19) always have solutions we can also obverse thefollowing aspects

(i) There are two different roots (120590 minus 119862)(120590 minus 1) and (120590 +119862)(120590 + 1) when 1205902 = 0 1 and 119862 = 1

(ii) There is double root 119909 = 1 or 119909 = 119862 when 1205902 = 1 and119862 = 1 or 120590 = 0

(iii) There is root (120590 minus 119862)(120590 minus 1) or (120590 + 119862)(120590 + 1) when1205902= 1 and 119862 = 1

Based on the above analysis we can conclude thattrajectories in phase space always intersect isoclines either inthe horizontal or in the vertical direction In the next sectionwe make further analysis about the stability of intersection

32 Equilibrium In order to study the final results of interac-tive effect between two different types of messages we needto conduct a comprehensive analysis for the stability of theequilibrium points of simultaneous differential equations

Theorem 1 System (12) admits no periodic orbit

Proof Let

119861 (V1 V2) =

1

V1V2

(V1 V2) isin int1198772

+

(20)

then we have

120597

120597V1

[1205831(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)]

+120597

120597V2

[1205832(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)]

= minus1205831

V21198811

minus1205832

V11198812

lt 0

(21)

According to the Bendixson-Dulac theorem [25] system(12) admits no periodic orbit

We study the evolution of V1(119905) and V

2(119905) as time 119905

approaches infinity As shown in Figure 1 there are threeequilibrium points in system (12) 119874

1(0 0) 119874

2(1198861198811 0) and

1198743(0 119887119881

2) To determine the stability of the equilibrium

points let

119872 = [119891V1

119891V2119892V1

119892V2

] (22)

we have

119901 = minus(119891V1

+ 119892V2)10038161003816100381610038161003816119901119894

119902 = det119872|119901119894

(23)

Then the stability conditions of equilibrium point 119874119894are

119901 gt 0 and 119902 gt 0 The Jacobin matrix of the system (12) atequilibrium state is as follows

(i) For the equilibrium point 1198741(0 0) we get

119872(1198741) = [

1198861205831

0

0 1198871205832

]

1199011= minus (119886120583

1+ 1198871205832) 119902

1= 119886119887120583

11205832

(24)

Based on the assumption that 119886 = 1 minus 12058211205831gt 0 and 119887 =

1 minus 12058221205832gt 0 we have 119901

1lt 0 and 119902

1gt 0 the point 119874

1(0 0)

is an unstable nodeSince our aim is to show the coexistence of competition

and cooperation by the introduction of 120572 and 120573 we supposethat the condition (25) is always satisfied in the rest of thepaper Consider

119887 + 12059012(2120572

1198811

minus 119886) gt 0

119886 + 12059021(2120573

1198812

minus 119887) gt 0

(25)

(ii) For the equilibrium point 1198742(1198861198811 0) we get

119872(1198742) =

[[[

[

minus1198861205831

1198861198811120583112059021

1198812

0 1205832[119887 + (

2120572

1198811

minus 119886)12059012]

]]]

]

1199012= 1198861205831minus 1205832[12059012(2120572

1198811

minus 119886) + 119887]

1199022=11988612058311205832(minus2120572120590

12+ 119886120590121198811minus 1198871198811)

1198811

(26)

The stability condition of equilibrium point 1198742(1198861198811 0) is

12059012gt 1198871198811(1198861198811minus 2120572)

Condition 12059012

gt 1198871198811(1198861198811minus 2120572) means that message

1 is more competitive than message 2 It implies that thepersonswho should have adoptedmessage 2 choose to believemessage 1 Consequently message 2 will eventually becomeextinct in the process of competition while message 1 cancontinue to diffuse and reach the maximum speed 119881

1

(iii) For the equilibrium point 1198743(0 119887119881

2) we get

119872(1198743) =

[[[

[

1205831[119886 + (

2120573

1198812

minus 119887)12059021] 0

1198871198812120583212059012

1198811

minus1198871205832

]]]

]

1199013= 1198871205832minus 1205831[119886 + 120590

21(2120573

1198812

minus 119887)]

1199023=11988712058311205832(119887120590211198812minus 1198861198812minus 2120573120590

21)

1198812

(27)

Journal of Applied Mathematics 7

The stability condition of equilibrium point 1198743(0 119887119881

2) is

12059021gt 1198861198812(1198871198812minus 2120573)

Condition 12059021gt 1198861198812(1198871198812minus 2120573) shows that message 2 is

more competitive thanmessage 1This just means that peopleaccepted message 2 even though they should have believedmessage 1 Therefore during the process of competitionmessage 2 can survive and reach a maximum speed119881

2 while

message 1 will eventually become extinct finally

33 Dynamics Assume that the conditions in (25) are satis-fied the twomessages in system (12) can coexistThe isocline1198711ofmessage 1 can be subdivided into 119871

11and 119871

12as follows

11987111 119886 + 120590

21

V2

1198812

minusV1

1198811

0 lt V2lt 120573

11987112 119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

V2gt 120573

(28)

where 1198711is smoothed in a small neighborhood of the vertex

1198631(1198861198811+ 1205731205902111988111198812 120573) as shown in Figure 1 Similarly we

divide the isocline1198712ofmessage 2 into119871

21and119871

22as follows

11987121 119887 + 120590

12

V1

1198811

minusV2

1198812

0 lt V1lt 120572

11987122 119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

V1gt 120572

(29)

where 1198712is smoothed in a small neighborhood of the vertex

1198632(120572 119887119881

2+ 1205721205901211988121198811) as shown in Figure 1

While the line segments 11987111and 119871

21are restricted to the

intervals (1198861198811 1198861198811+ 1205731205902111988111198812) and (0 120572) respectively they

are impossible to intersect in the interval 120572 le 1198861198811Then there

is a maximum of three isolated positive equilibrium points insystem (12) As shown in Figure 1 let 119870

1(11989611 11989612) represent

the intersection of 11987111and 119871

22 let 119870

2(11989621 11989622) represent the

intersection of 11987121and 119871

12 and let119870

3(11989631 11989632) represent the

intersection of 11987112and 119871

22 Then we have

11989611=1198811(119886 + 119887120590

21) + 2120572120590

1212059021

1205901212059021+ 1

11989612=1198812(1198811(119887 minus 119886120590

12) + 2120572120590

12)

(1205901212059021+ 1)119881

1

11989621=1198811(1198812(119886 minus 119887120590

21) + 2120573120590

21)

(1205901212059021+ 1)119881

2

11989622=1198812(11988612059012+ 119887) + 2120573120590

1212059021

1205901212059021+ 1

11989631=1198811(1198812(11988712059021minus 119886) minus 2120573120590

21) + 2120572120590

12120590211198812

(1205901212059021minus 1)119881

2

11989632=1198811(1198812(11988612059012minus 119887) + 2120573120590

1212059021) minus 2120572120590

121198812

(1205901212059021minus 1)119881

1

(30)

2

2120573+aV212059021

bV2

K2

K3

K1

O aV112120572 + bV112059012

L1

L2

Figure 3 Three positive equilibria 1198701 1198702 and 119870

3

Suppose that 1198661and 119866

2represent the slope of the lines

11987112and 119871

22 respectively Suppose that119866 denotes the slope of

the line11986311198632 Then we have

1198661= minus

1198812

119881112059021

lt 0

1198662= minus

119881212059012

1198811

lt 0

119866 =1198812(11988711988121198811+ 120572120590121198812minus 1205731198811)

1198811(11988611988111198812minus 1205721198812+ 120573120590211198811)lt 0

(31)

Dynamics of system (12) can be displayed in five caseswhile we discuss the situation of |119866

1| lt |119866

2| in cases 1ndash4 and

that of |1198661| gt |119866

2| in case 5

(i) Case 1 (|1198661| lt |119866| lt |119866

2|) In this case there are three

positive equilibria 1198701 1198702 and 119870

3as shown in Figure 3 The

Jacobian matrix of system (12) at 1198701is

119872(1198701) =

[[[

[

minus119896111205831

1198811

11989611120583112059021

1198812

minus11989612120583212059012

1198811

minus119896121205832

1198812

]]]

]

(32)

then 119901(119872(1198701)) = 119896

1112058311198811+ 1205832119896121198812gt 0 and 119902(119872(119870

1)) =

119896111198961212058311205832(1205901212059021+1)119881

11198812gt 0That is119870

1is asymptotically

stable Then we obtain that the stability condition of equilib-rium point119870

1is

0 lt 12059012lt

12058311198811(1205822minus 1205832)

1205832(21205721205831+ 12058211198811minus 12058311198811)

12059021gt 0 119881

1gt

21205721205831

1205831minus 1205821

(33)

8 Journal of Applied Mathematics

2

2120573+aV212059021

bV2

K2

K1

O aV112120572 + bV112059012

L1

L2

Figure 4 Two positive equilibria 1198701and 119870

2

Condition 0 lt 12059012

lt 12058311198811(1205822minus 1205832)1205832(21205721205831+ 12058211198811minus

12058311198811)means thatmessage 1 is less competitive thanmessage 2

When (33) are satisfied bothmessage 1 andmessage 2 coexistand the speed of message 1 and message 2 will tend to reach anonzero finite value

Similarly The Jacobian matrix of system (12) at 1198702is

119872(1198702) =

[[[

[

minus119896211205831

1198811

minus11989621120583112059021

1198812

11989622120583212059012

1198811

minus119896221205832

1198812

]]]

]

(34)

then the 119901(119872(1198702)) = 119896

2112058311198811+ 1198962212058321198812

gt 0 and119902(119872(119870

2)) = 119896

211198962212058311205832(1205901212059021+ 1)119881

11198812gt 0 That is

1198702is asymptotically stable Then we obtain that the stability

condition of equilibrium point1198702is

0 lt 12059021lt

12058321198812(1205821minus 1205831)

1205831(21205731205832+ 12058221198812minus 12058321198812)

12059012gt 0 119881

2gt

21205731205832

1205832minus 1205822

(35)

The condition 0 lt 12059021lt 12058321198812(1205821minus 1205831)1205831(21205731205832+ 12058221198812minus

12058321198812)means thatmessage 2 is less competitive thanmessage 1

When (35) are satisfied bothmessage 1 andmessage 2 coexistand their speed will change over time to reach a stable state

The Jacobian matrix of system (12) at 1198703is

119872(1198703) =

[[[

[

minus119896311205831

1198811

minus11989631120583112059021

1198812

minus11989632120583212059012

1198811

minus119896321205832

1198812

]]]

]

(36)

then the 119901(119872(1198703)) = 119896

3112058311198811+ 1198963212058321198812

gt 0 and119902(119872(119870

3)) = 119896

311198963212058311205832(1 minus 120590

1212059021)11988111198812 By |119866

1| lt |119866

2|

2

2120573+aV212059021

bV2

K2

O aV112120572 + bV112059012

L1

L2

Figure 5 The unique positive equilibrium 1198702

we have 1205901212059021gt 1 and 119902(119872(119870

3)) lt 0 It means that 119870

3is a

saddle point In int 1198772+ all orbits but 119870

3and the separatrixes

of1198703converge to equilibria119870

1and119870

2as shown in Figure 3

(ii) Case 2 (|1198661| lt |119866| = |119866

2|) As can be seen in Figure 4

there are two positive equilibrium points 1198701and 119870

2 It is

important to note that however 1198701and 119870

3in Figure 3

coincide Similar to the analysis in case 11198702is asymptotically

stable and the stability condition of equilibrium point 1198702

satisfies (35)Since 119901(119872(119870

1)) = 119896

1112058311198811+ 1205832119896121198812

gt 0 and119902(119872(119870

1)) = 0 it can be determined that 119870

1is a saddle node

by the saddle node rule [26] According toTheorem 1 we candraw the dynamics of system (12) as shown in Figure 4 Thesame approach also applies to the case |119866

1| = |119866| lt |119866

2| the

difference is that1198701is asymptotically stable and119870

2is a saddle

node equilibrium and all orbits converge to equilibria 1198701in

int 1198772+except 119870

2and the separatrixes of 119870

2

(iii) Case 3 (|1198661| le |119866

2| lt |119866|) In case 3 there is

only one positive equilibrium point 1198702 By Theorem 1 119870

2is

asymptotically stable as shown in Figure 5 and the stabilitycondition of equilibrium point 119870

2satisfies (33) Similar

discussions can be given for the case |119866| lt |1198661| le |119866

2| in

which 1198701is asymptotically stable

(iv) Case 4 (|1198661| = |119866

2| = |119866|) In this case the isoclines 119871

12

and 11987122coincide and all the positive equilibrium points dis-

tribute on the line segment11986311198632 According toTheorem 1 in

int 1198772+ all orbits of system (12) converge to the line segment

as shown in Figure 6

(v) Case 5 (|1198662| lt |119866

1|) In this situation we have similar

discussions with the case 3 and globally stable is1198702as shown

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

of information as message 1 and message 2 and they willinteract with each other when they spread at the same timeThe propagation process and the interactions between thetwo types of information can be governed by the followingset of rules

(i) Each message has a relatively stable natural growthcoefficient 120583

119894and a decay coefficient 120582

119894 which repre-

sent the growth factor and decay factor respectivelyand 120583

119894gt 120582119894

(ii) Since a signal message could not spread indefinitelythere exists a maximum speed119881

119894during the propaga-

tion process(iii) Two messages will compete or cooperate with one

another and take negative or positive effect repre-sented by 120590

119894119895

Supported by the rules (i) and (ii) we have the followingmodel

119889V1 (119905)

119889119905= 1205831V1(1 minus

V1

1198811

) minus 1205821V1

119889V2 (119905)

119889119905= 1205832V2(1 minus

V2

1198812

) minus 1205822V2

(10)

Here 120583119894denotes the growth coefficient of the 119894th message

and 120583119894isin [0 1] 120582

119894denotes the decay coefficient of the 119894th

message and 120582119894isin [0 1] 119894 = 1 2

According to the rule (iii) when the messages 1 and 2coexist in the network the interactive influence among thetwo messages produces positive effect or negative effect asthey cooperate or compete with each other Based on theanalysis above (10) can be written as

119889V1 (119905)

119889119905= 1205831V1(1 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

) minus 1205821V1

119889V2(119905)

119889119905= 1205832V2(1 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

) minus 1205822V2

(11)

In order to facilitate the analysis set 119886 = 1 minus 12058211205831 119887 =

1 minus 12058221205832 (11) can be written as

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)

119889V2 (119905)

119889119905= 1205832V2(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)

(12)

Equation (12) can be considered as a competition andcooperation multisystem Let V

2= 0 in the first equation

of the system (12) we have 119889V1119889119905 = 120583

1V1(1198861198811minus V1)1198811

then the parameter 1198861198811is the speed of the message 1 when it

disseminates independently form message 2 Let 1198621(V1 V2) =

119886+12059021(1205731198812)minusV11198811minus12059021(|V2minus 120573|lowast1198812) the function |V

2minus 120573|lowast

can be defined as the absolute function |V2minus 120573| while the

function |V2minus 120573| is smoothed in a very small neighborhood

2

2120573+aV212059021

bV2

120573

L12

K2

K3

D2

D1

K1

L22

O 120572 aV1 12120572 + bV112059012

L1

L2

L21

L11

Figure 1 The smoothed parts of the isoclines 1198711and 119871

2

of its vertex (0 120573) Thus the isocline 1198711 1198621(V1 V2) = 0 is

smooth as shown in Figure 1 Then the function 1198911(V1 V2) =

1205831V11198621(V1 V2) and satisfies

1205971198911

120597V2

=12059021

1198812

1205831V1 0 lt V

2lt 120573

1205971198911

120597V2

= minus12059021

1198812

1205831V1 V2gt 120573

(13)

As shown in (13) the positive or negative effect actingon message 1 depends on the speed of message 2 Whenmessage 2 travels slowly (V

2lt 120573) it takes positive effect on

message 1 otherwise it produces negative effect on message1 at high speed (V

2gt 120573) The parameter 120573 signifies the

critical speed of message 2 near which the effect of message2 acting on message 1 would turn positive effect into negativeeffect Furthermore according to the Lotka-Volterra modelsthe parameter 120590

21represents both the cooperation (at low

speed) and competition (at high speed) level of message 2to message 1 Hence the low speed of message 2 0 lt V

2lt

120573 represents the scope of cooperation of message 2 withmessage 1 A similar conclusion can be obtained that thelow speed of message 1 0 lt V

1lt 120572 represents the region

of net cooperation message 1 with message 2 The isocline1198712 1198622(V1 V2) = 119887+120590

12(1205721198811)minusV21198812minus12059012(|V1minus 120572|lowast1198811) = 0

is smooth as shown in Figure 1Since the system (12) can exhibit the features of the Lotka-

Volterra cooperative model and competitive model it canbe named co-competition Lotka-Volterra model Obviouslywhen the messages are at low speed for instance V

1lt 120572

Journal of Applied Mathematics 5

The isocline of x1

+

+

minus

x2

0

0

Direction of speed change Positive partNeutral partNegative partminus

(a)

The isocline of x2

x1

minus+

0

+

0

Direction of speed change Positive partNeutral partNegative partminus

(b)

Figure 2 The parabolic zero growth isoclines of message 1 (a) and message 2 (b)

and V2lt 120573 the system (12) turns into the Lotka-Volterra

cooperative system

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

V2

1198812

minusV1

1198811

)

119889V2(119905)

119889119905= 1205832V2(119887 + 120590

12

V1

1198811

minusV2

1198812

)

(14)

Conversely when the messages are at high speed forinstance V

1gt 120572 and V

2gt 120573 the system (12) turns into the

Lotka-Volterra competitive model

119889V1(119905)

119889119905= 1205831V1(119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

)

119889V2(119905)

119889119905= 1205832V2(119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

)

(15)

By systematic comparison of the system (12) with theLotka-Volterra competitive system (15) an important conclu-sion is reached that the cooperation at low speed promotes thecoexistence level of the two messages in co-opetition Lotka-Volterra model When message 2 is at a low speed (V

2lt 120573)

it follows the first equation of system (15) that the effect ofmessage 2 acting on message 1 is negative (minus120590

21V21198812lt 0)

while it follows the first equation of system (12) that the effectof message 2 acting on message 1 is positive (120590

21V21198812gt 0)

which originates from the cooperation of message 2 at lowspeed As a consequence message 1 has survived for a longtime when message 2 is at low speed When message 2 is athigh speed (V

2gt 120573) the negative effect produced by message

2 acting on message 1 in system (15) is (minus12059021V21198812) while that

in system (12) is minus12059021(V2minus 120573)119881

2 Since minus120590

21(V2minus 120573)119881

2gt

minus12059021V21198812 the negative effect is lessened for the cooperation

among messages in system (12) Similarly the cooperation insystem (12) promotes the existence of message 1 In a wordthe cooperation in system (12) facilitates the coexistence ofboth the two messages

3 Model Analysis

In this section we discuss the intersection of two isoclines1198711and 119871

2by a symmetrical hypothesis and then analyze the

dynamics of the model and the global stability of the system

31 Basic Analysis In order to show the existence of intersec-tion of two isoclines 119871

1and 119871

2 we assume that the positive or

negative effects produced by messages 1 and 2 are almost thesame as each other In the condition we can set hypothesisas follows 119886 = 119887 = 119911 120590

21= 12059012

= 120590 1198811= 1198812= 119881

and 120572 = 120573 = 120577 then the values of 1205831and 120583

2can reflect the

difference of the growth factors among the twomessages For|V1minus120572| and |V

2minus120573| are smoothed in a very small neighborhood

of their vertexes (0 120572) and (0 120573) respectively the system (12)can be written as

119889V1 (119905)

119889119905= 1205831V1(119911 + 120590

120577

119881minusV1

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V2

120577minus 1

10038161003816100381610038161003816100381610038161003816)

119889V2(119905)

119889119905= 1205832V2(119911 + 120590

120577

119881minusV2

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V1

120577minus 1

10038161003816100381610038161003816100381610038161003816)

(16)

set 1199091= V1120577 and 119909

2= V2120577 then

1198891199091

119889119905= 1205831V1(119911 + 120590

120577

119881minus1205771199091

119881minus120590120577

119881

10038161003816100381610038161199092 minus 11003816100381610038161003816)

1198891199092

119889119905= 1205832V2(119911 + 120590

120577

119881minus1205771199092

119881minus120590120577

119881

10038161003816100381610038161199091 minus 11003816100381610038161003816)

(17)

In int1198772+ the zero growth isoclines of 119909

1and 1199092are as follows

1199091= minus120590

10038161003816100381610038161199092 minus 11003816100381610038161003816 + 119862

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(18)

where 119862 = 119881119911120577 + 120590According to [24] the isoclines include three parts as

shown in Figure 2 positive part (+) neutral part (0) and

6 Journal of Applied Mathematics

negative part (minus) The positive part neutral part or negativepart is the part of isoclines when the slope (for message1 the slope is 119889V

1(119905)119889V

2(119905) for message 2 the slope is

119889V2(119905)119889V

1(119905)) is positive part (+) neutral part (0) and

negative part (minus) The two isoclines are symmetry around theline 119909

1= 1199092 Then we only need to discuss

1199091= 1199092

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(19)

Obviously (19) always have solutions we can also obverse thefollowing aspects

(i) There are two different roots (120590 minus 119862)(120590 minus 1) and (120590 +119862)(120590 + 1) when 1205902 = 0 1 and 119862 = 1

(ii) There is double root 119909 = 1 or 119909 = 119862 when 1205902 = 1 and119862 = 1 or 120590 = 0

(iii) There is root (120590 minus 119862)(120590 minus 1) or (120590 + 119862)(120590 + 1) when1205902= 1 and 119862 = 1

Based on the above analysis we can conclude thattrajectories in phase space always intersect isoclines either inthe horizontal or in the vertical direction In the next sectionwe make further analysis about the stability of intersection

32 Equilibrium In order to study the final results of interac-tive effect between two different types of messages we needto conduct a comprehensive analysis for the stability of theequilibrium points of simultaneous differential equations

Theorem 1 System (12) admits no periodic orbit

Proof Let

119861 (V1 V2) =

1

V1V2

(V1 V2) isin int1198772

+

(20)

then we have

120597

120597V1

[1205831(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)]

+120597

120597V2

[1205832(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)]

= minus1205831

V21198811

minus1205832

V11198812

lt 0

(21)

According to the Bendixson-Dulac theorem [25] system(12) admits no periodic orbit

We study the evolution of V1(119905) and V

2(119905) as time 119905

approaches infinity As shown in Figure 1 there are threeequilibrium points in system (12) 119874

1(0 0) 119874

2(1198861198811 0) and

1198743(0 119887119881

2) To determine the stability of the equilibrium

points let

119872 = [119891V1

119891V2119892V1

119892V2

] (22)

we have

119901 = minus(119891V1

+ 119892V2)10038161003816100381610038161003816119901119894

119902 = det119872|119901119894

(23)

Then the stability conditions of equilibrium point 119874119894are

119901 gt 0 and 119902 gt 0 The Jacobin matrix of the system (12) atequilibrium state is as follows

(i) For the equilibrium point 1198741(0 0) we get

119872(1198741) = [

1198861205831

0

0 1198871205832

]

1199011= minus (119886120583

1+ 1198871205832) 119902

1= 119886119887120583

11205832

(24)

Based on the assumption that 119886 = 1 minus 12058211205831gt 0 and 119887 =

1 minus 12058221205832gt 0 we have 119901

1lt 0 and 119902

1gt 0 the point 119874

1(0 0)

is an unstable nodeSince our aim is to show the coexistence of competition

and cooperation by the introduction of 120572 and 120573 we supposethat the condition (25) is always satisfied in the rest of thepaper Consider

119887 + 12059012(2120572

1198811

minus 119886) gt 0

119886 + 12059021(2120573

1198812

minus 119887) gt 0

(25)

(ii) For the equilibrium point 1198742(1198861198811 0) we get

119872(1198742) =

[[[

[

minus1198861205831

1198861198811120583112059021

1198812

0 1205832[119887 + (

2120572

1198811

minus 119886)12059012]

]]]

]

1199012= 1198861205831minus 1205832[12059012(2120572

1198811

minus 119886) + 119887]

1199022=11988612058311205832(minus2120572120590

12+ 119886120590121198811minus 1198871198811)

1198811

(26)

The stability condition of equilibrium point 1198742(1198861198811 0) is

12059012gt 1198871198811(1198861198811minus 2120572)

Condition 12059012

gt 1198871198811(1198861198811minus 2120572) means that message

1 is more competitive than message 2 It implies that thepersonswho should have adoptedmessage 2 choose to believemessage 1 Consequently message 2 will eventually becomeextinct in the process of competition while message 1 cancontinue to diffuse and reach the maximum speed 119881

1

(iii) For the equilibrium point 1198743(0 119887119881

2) we get

119872(1198743) =

[[[

[

1205831[119886 + (

2120573

1198812

minus 119887)12059021] 0

1198871198812120583212059012

1198811

minus1198871205832

]]]

]

1199013= 1198871205832minus 1205831[119886 + 120590

21(2120573

1198812

minus 119887)]

1199023=11988712058311205832(119887120590211198812minus 1198861198812minus 2120573120590

21)

1198812

(27)

Journal of Applied Mathematics 7

The stability condition of equilibrium point 1198743(0 119887119881

2) is

12059021gt 1198861198812(1198871198812minus 2120573)

Condition 12059021gt 1198861198812(1198871198812minus 2120573) shows that message 2 is

more competitive thanmessage 1This just means that peopleaccepted message 2 even though they should have believedmessage 1 Therefore during the process of competitionmessage 2 can survive and reach a maximum speed119881

2 while

message 1 will eventually become extinct finally

33 Dynamics Assume that the conditions in (25) are satis-fied the twomessages in system (12) can coexistThe isocline1198711ofmessage 1 can be subdivided into 119871

11and 119871

12as follows

11987111 119886 + 120590

21

V2

1198812

minusV1

1198811

0 lt V2lt 120573

11987112 119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

V2gt 120573

(28)

where 1198711is smoothed in a small neighborhood of the vertex

1198631(1198861198811+ 1205731205902111988111198812 120573) as shown in Figure 1 Similarly we

divide the isocline1198712ofmessage 2 into119871

21and119871

22as follows

11987121 119887 + 120590

12

V1

1198811

minusV2

1198812

0 lt V1lt 120572

11987122 119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

V1gt 120572

(29)

where 1198712is smoothed in a small neighborhood of the vertex

1198632(120572 119887119881

2+ 1205721205901211988121198811) as shown in Figure 1

While the line segments 11987111and 119871

21are restricted to the

intervals (1198861198811 1198861198811+ 1205731205902111988111198812) and (0 120572) respectively they

are impossible to intersect in the interval 120572 le 1198861198811Then there

is a maximum of three isolated positive equilibrium points insystem (12) As shown in Figure 1 let 119870

1(11989611 11989612) represent

the intersection of 11987111and 119871

22 let 119870

2(11989621 11989622) represent the

intersection of 11987121and 119871

12 and let119870

3(11989631 11989632) represent the

intersection of 11987112and 119871

22 Then we have

11989611=1198811(119886 + 119887120590

21) + 2120572120590

1212059021

1205901212059021+ 1

11989612=1198812(1198811(119887 minus 119886120590

12) + 2120572120590

12)

(1205901212059021+ 1)119881

1

11989621=1198811(1198812(119886 minus 119887120590

21) + 2120573120590

21)

(1205901212059021+ 1)119881

2

11989622=1198812(11988612059012+ 119887) + 2120573120590

1212059021

1205901212059021+ 1

11989631=1198811(1198812(11988712059021minus 119886) minus 2120573120590

21) + 2120572120590

12120590211198812

(1205901212059021minus 1)119881

2

11989632=1198811(1198812(11988612059012minus 119887) + 2120573120590

1212059021) minus 2120572120590

121198812

(1205901212059021minus 1)119881

1

(30)

2

2120573+aV212059021

bV2

K2

K3

K1

O aV112120572 + bV112059012

L1

L2

Figure 3 Three positive equilibria 1198701 1198702 and 119870

3

Suppose that 1198661and 119866

2represent the slope of the lines

11987112and 119871

22 respectively Suppose that119866 denotes the slope of

the line11986311198632 Then we have

1198661= minus

1198812

119881112059021

lt 0

1198662= minus

119881212059012

1198811

lt 0

119866 =1198812(11988711988121198811+ 120572120590121198812minus 1205731198811)

1198811(11988611988111198812minus 1205721198812+ 120573120590211198811)lt 0

(31)

Dynamics of system (12) can be displayed in five caseswhile we discuss the situation of |119866

1| lt |119866

2| in cases 1ndash4 and

that of |1198661| gt |119866

2| in case 5

(i) Case 1 (|1198661| lt |119866| lt |119866

2|) In this case there are three

positive equilibria 1198701 1198702 and 119870

3as shown in Figure 3 The

Jacobian matrix of system (12) at 1198701is

119872(1198701) =

[[[

[

minus119896111205831

1198811

11989611120583112059021

1198812

minus11989612120583212059012

1198811

minus119896121205832

1198812

]]]

]

(32)

then 119901(119872(1198701)) = 119896

1112058311198811+ 1205832119896121198812gt 0 and 119902(119872(119870

1)) =

119896111198961212058311205832(1205901212059021+1)119881

11198812gt 0That is119870

1is asymptotically

stable Then we obtain that the stability condition of equilib-rium point119870

1is

0 lt 12059012lt

12058311198811(1205822minus 1205832)

1205832(21205721205831+ 12058211198811minus 12058311198811)

12059021gt 0 119881

1gt

21205721205831

1205831minus 1205821

(33)

8 Journal of Applied Mathematics

2

2120573+aV212059021

bV2

K2

K1

O aV112120572 + bV112059012

L1

L2

Figure 4 Two positive equilibria 1198701and 119870

2

Condition 0 lt 12059012

lt 12058311198811(1205822minus 1205832)1205832(21205721205831+ 12058211198811minus

12058311198811)means thatmessage 1 is less competitive thanmessage 2

When (33) are satisfied bothmessage 1 andmessage 2 coexistand the speed of message 1 and message 2 will tend to reach anonzero finite value

Similarly The Jacobian matrix of system (12) at 1198702is

119872(1198702) =

[[[

[

minus119896211205831

1198811

minus11989621120583112059021

1198812

11989622120583212059012

1198811

minus119896221205832

1198812

]]]

]

(34)

then the 119901(119872(1198702)) = 119896

2112058311198811+ 1198962212058321198812

gt 0 and119902(119872(119870

2)) = 119896

211198962212058311205832(1205901212059021+ 1)119881

11198812gt 0 That is

1198702is asymptotically stable Then we obtain that the stability

condition of equilibrium point1198702is

0 lt 12059021lt

12058321198812(1205821minus 1205831)

1205831(21205731205832+ 12058221198812minus 12058321198812)

12059012gt 0 119881

2gt

21205731205832

1205832minus 1205822

(35)

The condition 0 lt 12059021lt 12058321198812(1205821minus 1205831)1205831(21205731205832+ 12058221198812minus

12058321198812)means thatmessage 2 is less competitive thanmessage 1

When (35) are satisfied bothmessage 1 andmessage 2 coexistand their speed will change over time to reach a stable state

The Jacobian matrix of system (12) at 1198703is

119872(1198703) =

[[[

[

minus119896311205831

1198811

minus11989631120583112059021

1198812

minus11989632120583212059012

1198811

minus119896321205832

1198812

]]]

]

(36)

then the 119901(119872(1198703)) = 119896

3112058311198811+ 1198963212058321198812

gt 0 and119902(119872(119870

3)) = 119896

311198963212058311205832(1 minus 120590

1212059021)11988111198812 By |119866

1| lt |119866

2|

2

2120573+aV212059021

bV2

K2

O aV112120572 + bV112059012

L1

L2

Figure 5 The unique positive equilibrium 1198702

we have 1205901212059021gt 1 and 119902(119872(119870

3)) lt 0 It means that 119870

3is a

saddle point In int 1198772+ all orbits but 119870

3and the separatrixes

of1198703converge to equilibria119870

1and119870

2as shown in Figure 3

(ii) Case 2 (|1198661| lt |119866| = |119866

2|) As can be seen in Figure 4

there are two positive equilibrium points 1198701and 119870

2 It is

important to note that however 1198701and 119870

3in Figure 3

coincide Similar to the analysis in case 11198702is asymptotically

stable and the stability condition of equilibrium point 1198702

satisfies (35)Since 119901(119872(119870

1)) = 119896

1112058311198811+ 1205832119896121198812

gt 0 and119902(119872(119870

1)) = 0 it can be determined that 119870

1is a saddle node

by the saddle node rule [26] According toTheorem 1 we candraw the dynamics of system (12) as shown in Figure 4 Thesame approach also applies to the case |119866

1| = |119866| lt |119866

2| the

difference is that1198701is asymptotically stable and119870

2is a saddle

node equilibrium and all orbits converge to equilibria 1198701in

int 1198772+except 119870

2and the separatrixes of 119870

2

(iii) Case 3 (|1198661| le |119866

2| lt |119866|) In case 3 there is

only one positive equilibrium point 1198702 By Theorem 1 119870

2is

asymptotically stable as shown in Figure 5 and the stabilitycondition of equilibrium point 119870

2satisfies (33) Similar

discussions can be given for the case |119866| lt |1198661| le |119866

2| in

which 1198701is asymptotically stable

(iv) Case 4 (|1198661| = |119866

2| = |119866|) In this case the isoclines 119871

12

and 11987122coincide and all the positive equilibrium points dis-

tribute on the line segment11986311198632 According toTheorem 1 in

int 1198772+ all orbits of system (12) converge to the line segment

as shown in Figure 6

(v) Case 5 (|1198662| lt |119866

1|) In this situation we have similar

discussions with the case 3 and globally stable is1198702as shown

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

The isocline of x1

+

+

minus

x2

0

0

Direction of speed change Positive partNeutral partNegative partminus

(a)

The isocline of x2

x1

minus+

0

+

0

Direction of speed change Positive partNeutral partNegative partminus

(b)

Figure 2 The parabolic zero growth isoclines of message 1 (a) and message 2 (b)

and V2lt 120573 the system (12) turns into the Lotka-Volterra

cooperative system

119889V1(119905)

119889119905= 1205831V1(119886 + 120590

21

V2

1198812

minusV1

1198811

)

119889V2(119905)

119889119905= 1205832V2(119887 + 120590

12

V1

1198811

minusV2

1198812

)

(14)

Conversely when the messages are at high speed forinstance V

1gt 120572 and V

2gt 120573 the system (12) turns into the

Lotka-Volterra competitive model

119889V1(119905)

119889119905= 1205831V1(119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

)

119889V2(119905)

119889119905= 1205832V2(119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

)

(15)

By systematic comparison of the system (12) with theLotka-Volterra competitive system (15) an important conclu-sion is reached that the cooperation at low speed promotes thecoexistence level of the two messages in co-opetition Lotka-Volterra model When message 2 is at a low speed (V

2lt 120573)

it follows the first equation of system (15) that the effect ofmessage 2 acting on message 1 is negative (minus120590

21V21198812lt 0)

while it follows the first equation of system (12) that the effectof message 2 acting on message 1 is positive (120590

21V21198812gt 0)

which originates from the cooperation of message 2 at lowspeed As a consequence message 1 has survived for a longtime when message 2 is at low speed When message 2 is athigh speed (V

2gt 120573) the negative effect produced by message

2 acting on message 1 in system (15) is (minus12059021V21198812) while that

in system (12) is minus12059021(V2minus 120573)119881

2 Since minus120590

21(V2minus 120573)119881

2gt

minus12059021V21198812 the negative effect is lessened for the cooperation

among messages in system (12) Similarly the cooperation insystem (12) promotes the existence of message 1 In a wordthe cooperation in system (12) facilitates the coexistence ofboth the two messages

3 Model Analysis

In this section we discuss the intersection of two isoclines1198711and 119871

2by a symmetrical hypothesis and then analyze the

dynamics of the model and the global stability of the system

31 Basic Analysis In order to show the existence of intersec-tion of two isoclines 119871

1and 119871

2 we assume that the positive or

negative effects produced by messages 1 and 2 are almost thesame as each other In the condition we can set hypothesisas follows 119886 = 119887 = 119911 120590

21= 12059012

= 120590 1198811= 1198812= 119881

and 120572 = 120573 = 120577 then the values of 1205831and 120583

2can reflect the

difference of the growth factors among the twomessages For|V1minus120572| and |V

2minus120573| are smoothed in a very small neighborhood

of their vertexes (0 120572) and (0 120573) respectively the system (12)can be written as

119889V1 (119905)

119889119905= 1205831V1(119911 + 120590

120577

119881minusV1

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V2

120577minus 1

10038161003816100381610038161003816100381610038161003816)

119889V2(119905)

119889119905= 1205832V2(119911 + 120590

120577

119881minusV2

119881minus120590120577

119881

10038161003816100381610038161003816100381610038161003816

V1

120577minus 1

10038161003816100381610038161003816100381610038161003816)

(16)

set 1199091= V1120577 and 119909

2= V2120577 then

1198891199091

119889119905= 1205831V1(119911 + 120590

120577

119881minus1205771199091

119881minus120590120577

119881

10038161003816100381610038161199092 minus 11003816100381610038161003816)

1198891199092

119889119905= 1205832V2(119911 + 120590

120577

119881minus1205771199092

119881minus120590120577

119881

10038161003816100381610038161199091 minus 11003816100381610038161003816)

(17)

In int1198772+ the zero growth isoclines of 119909

1and 1199092are as follows

1199091= minus120590

10038161003816100381610038161199092 minus 11003816100381610038161003816 + 119862

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(18)

where 119862 = 119881119911120577 + 120590According to [24] the isoclines include three parts as

shown in Figure 2 positive part (+) neutral part (0) and

6 Journal of Applied Mathematics

negative part (minus) The positive part neutral part or negativepart is the part of isoclines when the slope (for message1 the slope is 119889V

1(119905)119889V

2(119905) for message 2 the slope is

119889V2(119905)119889V

1(119905)) is positive part (+) neutral part (0) and

negative part (minus) The two isoclines are symmetry around theline 119909

1= 1199092 Then we only need to discuss

1199091= 1199092

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(19)

Obviously (19) always have solutions we can also obverse thefollowing aspects

(i) There are two different roots (120590 minus 119862)(120590 minus 1) and (120590 +119862)(120590 + 1) when 1205902 = 0 1 and 119862 = 1

(ii) There is double root 119909 = 1 or 119909 = 119862 when 1205902 = 1 and119862 = 1 or 120590 = 0

(iii) There is root (120590 minus 119862)(120590 minus 1) or (120590 + 119862)(120590 + 1) when1205902= 1 and 119862 = 1

Based on the above analysis we can conclude thattrajectories in phase space always intersect isoclines either inthe horizontal or in the vertical direction In the next sectionwe make further analysis about the stability of intersection

32 Equilibrium In order to study the final results of interac-tive effect between two different types of messages we needto conduct a comprehensive analysis for the stability of theequilibrium points of simultaneous differential equations

Theorem 1 System (12) admits no periodic orbit

Proof Let

119861 (V1 V2) =

1

V1V2

(V1 V2) isin int1198772

+

(20)

then we have

120597

120597V1

[1205831(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)]

+120597

120597V2

[1205832(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)]

= minus1205831

V21198811

minus1205832

V11198812

lt 0

(21)

According to the Bendixson-Dulac theorem [25] system(12) admits no periodic orbit

We study the evolution of V1(119905) and V

2(119905) as time 119905

approaches infinity As shown in Figure 1 there are threeequilibrium points in system (12) 119874

1(0 0) 119874

2(1198861198811 0) and

1198743(0 119887119881

2) To determine the stability of the equilibrium

points let

119872 = [119891V1

119891V2119892V1

119892V2

] (22)

we have

119901 = minus(119891V1

+ 119892V2)10038161003816100381610038161003816119901119894

119902 = det119872|119901119894

(23)

Then the stability conditions of equilibrium point 119874119894are

119901 gt 0 and 119902 gt 0 The Jacobin matrix of the system (12) atequilibrium state is as follows

(i) For the equilibrium point 1198741(0 0) we get

119872(1198741) = [

1198861205831

0

0 1198871205832

]

1199011= minus (119886120583

1+ 1198871205832) 119902

1= 119886119887120583

11205832

(24)

Based on the assumption that 119886 = 1 minus 12058211205831gt 0 and 119887 =

1 minus 12058221205832gt 0 we have 119901

1lt 0 and 119902

1gt 0 the point 119874

1(0 0)

is an unstable nodeSince our aim is to show the coexistence of competition

and cooperation by the introduction of 120572 and 120573 we supposethat the condition (25) is always satisfied in the rest of thepaper Consider

119887 + 12059012(2120572

1198811

minus 119886) gt 0

119886 + 12059021(2120573

1198812

minus 119887) gt 0

(25)

(ii) For the equilibrium point 1198742(1198861198811 0) we get

119872(1198742) =

[[[

[

minus1198861205831

1198861198811120583112059021

1198812

0 1205832[119887 + (

2120572

1198811

minus 119886)12059012]

]]]

]

1199012= 1198861205831minus 1205832[12059012(2120572

1198811

minus 119886) + 119887]

1199022=11988612058311205832(minus2120572120590

12+ 119886120590121198811minus 1198871198811)

1198811

(26)

The stability condition of equilibrium point 1198742(1198861198811 0) is

12059012gt 1198871198811(1198861198811minus 2120572)

Condition 12059012

gt 1198871198811(1198861198811minus 2120572) means that message

1 is more competitive than message 2 It implies that thepersonswho should have adoptedmessage 2 choose to believemessage 1 Consequently message 2 will eventually becomeextinct in the process of competition while message 1 cancontinue to diffuse and reach the maximum speed 119881

1

(iii) For the equilibrium point 1198743(0 119887119881

2) we get

119872(1198743) =

[[[

[

1205831[119886 + (

2120573

1198812

minus 119887)12059021] 0

1198871198812120583212059012

1198811

minus1198871205832

]]]

]

1199013= 1198871205832minus 1205831[119886 + 120590

21(2120573

1198812

minus 119887)]

1199023=11988712058311205832(119887120590211198812minus 1198861198812minus 2120573120590

21)

1198812

(27)

Journal of Applied Mathematics 7

The stability condition of equilibrium point 1198743(0 119887119881

2) is

12059021gt 1198861198812(1198871198812minus 2120573)

Condition 12059021gt 1198861198812(1198871198812minus 2120573) shows that message 2 is

more competitive thanmessage 1This just means that peopleaccepted message 2 even though they should have believedmessage 1 Therefore during the process of competitionmessage 2 can survive and reach a maximum speed119881

2 while

message 1 will eventually become extinct finally

33 Dynamics Assume that the conditions in (25) are satis-fied the twomessages in system (12) can coexistThe isocline1198711ofmessage 1 can be subdivided into 119871

11and 119871

12as follows

11987111 119886 + 120590

21

V2

1198812

minusV1

1198811

0 lt V2lt 120573

11987112 119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

V2gt 120573

(28)

where 1198711is smoothed in a small neighborhood of the vertex

1198631(1198861198811+ 1205731205902111988111198812 120573) as shown in Figure 1 Similarly we

divide the isocline1198712ofmessage 2 into119871

21and119871

22as follows

11987121 119887 + 120590

12

V1

1198811

minusV2

1198812

0 lt V1lt 120572

11987122 119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

V1gt 120572

(29)

where 1198712is smoothed in a small neighborhood of the vertex

1198632(120572 119887119881

2+ 1205721205901211988121198811) as shown in Figure 1

While the line segments 11987111and 119871

21are restricted to the

intervals (1198861198811 1198861198811+ 1205731205902111988111198812) and (0 120572) respectively they

are impossible to intersect in the interval 120572 le 1198861198811Then there

is a maximum of three isolated positive equilibrium points insystem (12) As shown in Figure 1 let 119870

1(11989611 11989612) represent

the intersection of 11987111and 119871

22 let 119870

2(11989621 11989622) represent the

intersection of 11987121and 119871

12 and let119870

3(11989631 11989632) represent the

intersection of 11987112and 119871

22 Then we have

11989611=1198811(119886 + 119887120590

21) + 2120572120590

1212059021

1205901212059021+ 1

11989612=1198812(1198811(119887 minus 119886120590

12) + 2120572120590

12)

(1205901212059021+ 1)119881

1

11989621=1198811(1198812(119886 minus 119887120590

21) + 2120573120590

21)

(1205901212059021+ 1)119881

2

11989622=1198812(11988612059012+ 119887) + 2120573120590

1212059021

1205901212059021+ 1

11989631=1198811(1198812(11988712059021minus 119886) minus 2120573120590

21) + 2120572120590

12120590211198812

(1205901212059021minus 1)119881

2

11989632=1198811(1198812(11988612059012minus 119887) + 2120573120590

1212059021) minus 2120572120590

121198812

(1205901212059021minus 1)119881

1

(30)

2

2120573+aV212059021

bV2

K2

K3

K1

O aV112120572 + bV112059012

L1

L2

Figure 3 Three positive equilibria 1198701 1198702 and 119870

3

Suppose that 1198661and 119866

2represent the slope of the lines

11987112and 119871

22 respectively Suppose that119866 denotes the slope of

the line11986311198632 Then we have

1198661= minus

1198812

119881112059021

lt 0

1198662= minus

119881212059012

1198811

lt 0

119866 =1198812(11988711988121198811+ 120572120590121198812minus 1205731198811)

1198811(11988611988111198812minus 1205721198812+ 120573120590211198811)lt 0

(31)

Dynamics of system (12) can be displayed in five caseswhile we discuss the situation of |119866

1| lt |119866

2| in cases 1ndash4 and

that of |1198661| gt |119866

2| in case 5

(i) Case 1 (|1198661| lt |119866| lt |119866

2|) In this case there are three

positive equilibria 1198701 1198702 and 119870

3as shown in Figure 3 The

Jacobian matrix of system (12) at 1198701is

119872(1198701) =

[[[

[

minus119896111205831

1198811

11989611120583112059021

1198812

minus11989612120583212059012

1198811

minus119896121205832

1198812

]]]

]

(32)

then 119901(119872(1198701)) = 119896

1112058311198811+ 1205832119896121198812gt 0 and 119902(119872(119870

1)) =

119896111198961212058311205832(1205901212059021+1)119881

11198812gt 0That is119870

1is asymptotically

stable Then we obtain that the stability condition of equilib-rium point119870

1is

0 lt 12059012lt

12058311198811(1205822minus 1205832)

1205832(21205721205831+ 12058211198811minus 12058311198811)

12059021gt 0 119881

1gt

21205721205831

1205831minus 1205821

(33)

8 Journal of Applied Mathematics

2

2120573+aV212059021

bV2

K2

K1

O aV112120572 + bV112059012

L1

L2

Figure 4 Two positive equilibria 1198701and 119870

2

Condition 0 lt 12059012

lt 12058311198811(1205822minus 1205832)1205832(21205721205831+ 12058211198811minus

12058311198811)means thatmessage 1 is less competitive thanmessage 2

When (33) are satisfied bothmessage 1 andmessage 2 coexistand the speed of message 1 and message 2 will tend to reach anonzero finite value

Similarly The Jacobian matrix of system (12) at 1198702is

119872(1198702) =

[[[

[

minus119896211205831

1198811

minus11989621120583112059021

1198812

11989622120583212059012

1198811

minus119896221205832

1198812

]]]

]

(34)

then the 119901(119872(1198702)) = 119896

2112058311198811+ 1198962212058321198812

gt 0 and119902(119872(119870

2)) = 119896

211198962212058311205832(1205901212059021+ 1)119881

11198812gt 0 That is

1198702is asymptotically stable Then we obtain that the stability

condition of equilibrium point1198702is

0 lt 12059021lt

12058321198812(1205821minus 1205831)

1205831(21205731205832+ 12058221198812minus 12058321198812)

12059012gt 0 119881

2gt

21205731205832

1205832minus 1205822

(35)

The condition 0 lt 12059021lt 12058321198812(1205821minus 1205831)1205831(21205731205832+ 12058221198812minus

12058321198812)means thatmessage 2 is less competitive thanmessage 1

When (35) are satisfied bothmessage 1 andmessage 2 coexistand their speed will change over time to reach a stable state

The Jacobian matrix of system (12) at 1198703is

119872(1198703) =

[[[

[

minus119896311205831

1198811

minus11989631120583112059021

1198812

minus11989632120583212059012

1198811

minus119896321205832

1198812

]]]

]

(36)

then the 119901(119872(1198703)) = 119896

3112058311198811+ 1198963212058321198812

gt 0 and119902(119872(119870

3)) = 119896

311198963212058311205832(1 minus 120590

1212059021)11988111198812 By |119866

1| lt |119866

2|

2

2120573+aV212059021

bV2

K2

O aV112120572 + bV112059012

L1

L2

Figure 5 The unique positive equilibrium 1198702

we have 1205901212059021gt 1 and 119902(119872(119870

3)) lt 0 It means that 119870

3is a

saddle point In int 1198772+ all orbits but 119870

3and the separatrixes

of1198703converge to equilibria119870

1and119870

2as shown in Figure 3

(ii) Case 2 (|1198661| lt |119866| = |119866

2|) As can be seen in Figure 4

there are two positive equilibrium points 1198701and 119870

2 It is

important to note that however 1198701and 119870

3in Figure 3

coincide Similar to the analysis in case 11198702is asymptotically

stable and the stability condition of equilibrium point 1198702

satisfies (35)Since 119901(119872(119870

1)) = 119896

1112058311198811+ 1205832119896121198812

gt 0 and119902(119872(119870

1)) = 0 it can be determined that 119870

1is a saddle node

by the saddle node rule [26] According toTheorem 1 we candraw the dynamics of system (12) as shown in Figure 4 Thesame approach also applies to the case |119866

1| = |119866| lt |119866

2| the

difference is that1198701is asymptotically stable and119870

2is a saddle

node equilibrium and all orbits converge to equilibria 1198701in

int 1198772+except 119870

2and the separatrixes of 119870

2

(iii) Case 3 (|1198661| le |119866

2| lt |119866|) In case 3 there is

only one positive equilibrium point 1198702 By Theorem 1 119870

2is

asymptotically stable as shown in Figure 5 and the stabilitycondition of equilibrium point 119870

2satisfies (33) Similar

discussions can be given for the case |119866| lt |1198661| le |119866

2| in

which 1198701is asymptotically stable

(iv) Case 4 (|1198661| = |119866

2| = |119866|) In this case the isoclines 119871

12

and 11987122coincide and all the positive equilibrium points dis-

tribute on the line segment11986311198632 According toTheorem 1 in

int 1198772+ all orbits of system (12) converge to the line segment

as shown in Figure 6

(v) Case 5 (|1198662| lt |119866

1|) In this situation we have similar

discussions with the case 3 and globally stable is1198702as shown

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

negative part (minus) The positive part neutral part or negativepart is the part of isoclines when the slope (for message1 the slope is 119889V

1(119905)119889V

2(119905) for message 2 the slope is

119889V2(119905)119889V

1(119905)) is positive part (+) neutral part (0) and

negative part (minus) The two isoclines are symmetry around theline 119909

1= 1199092 Then we only need to discuss

1199091= 1199092

1199092= minus120590

10038161003816100381610038161199091 minus 11003816100381610038161003816 + 119862

(19)

Obviously (19) always have solutions we can also obverse thefollowing aspects

(i) There are two different roots (120590 minus 119862)(120590 minus 1) and (120590 +119862)(120590 + 1) when 1205902 = 0 1 and 119862 = 1

(ii) There is double root 119909 = 1 or 119909 = 119862 when 1205902 = 1 and119862 = 1 or 120590 = 0

(iii) There is root (120590 minus 119862)(120590 minus 1) or (120590 + 119862)(120590 + 1) when1205902= 1 and 119862 = 1

Based on the above analysis we can conclude thattrajectories in phase space always intersect isoclines either inthe horizontal or in the vertical direction In the next sectionwe make further analysis about the stability of intersection

32 Equilibrium In order to study the final results of interac-tive effect between two different types of messages we needto conduct a comprehensive analysis for the stability of theequilibrium points of simultaneous differential equations

Theorem 1 System (12) admits no periodic orbit

Proof Let

119861 (V1 V2) =

1

V1V2

(V1 V2) isin int1198772

+

(20)

then we have

120597

120597V1

[1205831(119886 + 120590

21

120573

1198812

minusV1

1198811

minus 12059021

1003816100381610038161003816V2 minus 1205731003816100381610038161003816lowast

1198812

)]

+120597

120597V2

[1205832(119887 + 120590

12

120572

1198811

minusV2

1198812

minus 12059012

1003816100381610038161003816V1 minus 1205721003816100381610038161003816lowast

1198811

)]

= minus1205831

V21198811

minus1205832

V11198812

lt 0

(21)

According to the Bendixson-Dulac theorem [25] system(12) admits no periodic orbit

We study the evolution of V1(119905) and V

2(119905) as time 119905

approaches infinity As shown in Figure 1 there are threeequilibrium points in system (12) 119874

1(0 0) 119874

2(1198861198811 0) and

1198743(0 119887119881

2) To determine the stability of the equilibrium

points let

119872 = [119891V1

119891V2119892V1

119892V2

] (22)

we have

119901 = minus(119891V1

+ 119892V2)10038161003816100381610038161003816119901119894

119902 = det119872|119901119894

(23)

Then the stability conditions of equilibrium point 119874119894are

119901 gt 0 and 119902 gt 0 The Jacobin matrix of the system (12) atequilibrium state is as follows

(i) For the equilibrium point 1198741(0 0) we get

119872(1198741) = [

1198861205831

0

0 1198871205832

]

1199011= minus (119886120583

1+ 1198871205832) 119902

1= 119886119887120583

11205832

(24)

Based on the assumption that 119886 = 1 minus 12058211205831gt 0 and 119887 =

1 minus 12058221205832gt 0 we have 119901

1lt 0 and 119902

1gt 0 the point 119874

1(0 0)

is an unstable nodeSince our aim is to show the coexistence of competition

and cooperation by the introduction of 120572 and 120573 we supposethat the condition (25) is always satisfied in the rest of thepaper Consider

119887 + 12059012(2120572

1198811

minus 119886) gt 0

119886 + 12059021(2120573

1198812

minus 119887) gt 0

(25)

(ii) For the equilibrium point 1198742(1198861198811 0) we get

119872(1198742) =

[[[

[

minus1198861205831

1198861198811120583112059021

1198812

0 1205832[119887 + (

2120572

1198811

minus 119886)12059012]

]]]

]

1199012= 1198861205831minus 1205832[12059012(2120572

1198811

minus 119886) + 119887]

1199022=11988612058311205832(minus2120572120590

12+ 119886120590121198811minus 1198871198811)

1198811

(26)

The stability condition of equilibrium point 1198742(1198861198811 0) is

12059012gt 1198871198811(1198861198811minus 2120572)

Condition 12059012

gt 1198871198811(1198861198811minus 2120572) means that message

1 is more competitive than message 2 It implies that thepersonswho should have adoptedmessage 2 choose to believemessage 1 Consequently message 2 will eventually becomeextinct in the process of competition while message 1 cancontinue to diffuse and reach the maximum speed 119881

1

(iii) For the equilibrium point 1198743(0 119887119881

2) we get

119872(1198743) =

[[[

[

1205831[119886 + (

2120573

1198812

minus 119887)12059021] 0

1198871198812120583212059012

1198811

minus1198871205832

]]]

]

1199013= 1198871205832minus 1205831[119886 + 120590

21(2120573

1198812

minus 119887)]

1199023=11988712058311205832(119887120590211198812minus 1198861198812minus 2120573120590

21)

1198812

(27)

Journal of Applied Mathematics 7

The stability condition of equilibrium point 1198743(0 119887119881

2) is

12059021gt 1198861198812(1198871198812minus 2120573)

Condition 12059021gt 1198861198812(1198871198812minus 2120573) shows that message 2 is

more competitive thanmessage 1This just means that peopleaccepted message 2 even though they should have believedmessage 1 Therefore during the process of competitionmessage 2 can survive and reach a maximum speed119881

2 while

message 1 will eventually become extinct finally

33 Dynamics Assume that the conditions in (25) are satis-fied the twomessages in system (12) can coexistThe isocline1198711ofmessage 1 can be subdivided into 119871

11and 119871

12as follows

11987111 119886 + 120590

21

V2

1198812

minusV1

1198811

0 lt V2lt 120573

11987112 119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

V2gt 120573

(28)

where 1198711is smoothed in a small neighborhood of the vertex

1198631(1198861198811+ 1205731205902111988111198812 120573) as shown in Figure 1 Similarly we

divide the isocline1198712ofmessage 2 into119871

21and119871

22as follows

11987121 119887 + 120590

12

V1

1198811

minusV2

1198812

0 lt V1lt 120572

11987122 119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

V1gt 120572

(29)

where 1198712is smoothed in a small neighborhood of the vertex

1198632(120572 119887119881

2+ 1205721205901211988121198811) as shown in Figure 1

While the line segments 11987111and 119871

21are restricted to the

intervals (1198861198811 1198861198811+ 1205731205902111988111198812) and (0 120572) respectively they

are impossible to intersect in the interval 120572 le 1198861198811Then there

is a maximum of three isolated positive equilibrium points insystem (12) As shown in Figure 1 let 119870

1(11989611 11989612) represent

the intersection of 11987111and 119871

22 let 119870

2(11989621 11989622) represent the

intersection of 11987121and 119871

12 and let119870

3(11989631 11989632) represent the

intersection of 11987112and 119871

22 Then we have

11989611=1198811(119886 + 119887120590

21) + 2120572120590

1212059021

1205901212059021+ 1

11989612=1198812(1198811(119887 minus 119886120590

12) + 2120572120590

12)

(1205901212059021+ 1)119881

1

11989621=1198811(1198812(119886 minus 119887120590

21) + 2120573120590

21)

(1205901212059021+ 1)119881

2

11989622=1198812(11988612059012+ 119887) + 2120573120590

1212059021

1205901212059021+ 1

11989631=1198811(1198812(11988712059021minus 119886) minus 2120573120590

21) + 2120572120590

12120590211198812

(1205901212059021minus 1)119881

2

11989632=1198811(1198812(11988612059012minus 119887) + 2120573120590

1212059021) minus 2120572120590

121198812

(1205901212059021minus 1)119881

1

(30)

2

2120573+aV212059021

bV2

K2

K3

K1

O aV112120572 + bV112059012

L1

L2

Figure 3 Three positive equilibria 1198701 1198702 and 119870

3

Suppose that 1198661and 119866

2represent the slope of the lines

11987112and 119871

22 respectively Suppose that119866 denotes the slope of

the line11986311198632 Then we have

1198661= minus

1198812

119881112059021

lt 0

1198662= minus

119881212059012

1198811

lt 0

119866 =1198812(11988711988121198811+ 120572120590121198812minus 1205731198811)

1198811(11988611988111198812minus 1205721198812+ 120573120590211198811)lt 0

(31)

Dynamics of system (12) can be displayed in five caseswhile we discuss the situation of |119866

1| lt |119866

2| in cases 1ndash4 and

that of |1198661| gt |119866

2| in case 5

(i) Case 1 (|1198661| lt |119866| lt |119866

2|) In this case there are three

positive equilibria 1198701 1198702 and 119870

3as shown in Figure 3 The

Jacobian matrix of system (12) at 1198701is

119872(1198701) =

[[[

[

minus119896111205831

1198811

11989611120583112059021

1198812

minus11989612120583212059012

1198811

minus119896121205832

1198812

]]]

]

(32)

then 119901(119872(1198701)) = 119896

1112058311198811+ 1205832119896121198812gt 0 and 119902(119872(119870

1)) =

119896111198961212058311205832(1205901212059021+1)119881

11198812gt 0That is119870

1is asymptotically

stable Then we obtain that the stability condition of equilib-rium point119870

1is

0 lt 12059012lt

12058311198811(1205822minus 1205832)

1205832(21205721205831+ 12058211198811minus 12058311198811)

12059021gt 0 119881

1gt

21205721205831

1205831minus 1205821

(33)

8 Journal of Applied Mathematics

2

2120573+aV212059021

bV2

K2

K1

O aV112120572 + bV112059012

L1

L2

Figure 4 Two positive equilibria 1198701and 119870

2

Condition 0 lt 12059012

lt 12058311198811(1205822minus 1205832)1205832(21205721205831+ 12058211198811minus

12058311198811)means thatmessage 1 is less competitive thanmessage 2

When (33) are satisfied bothmessage 1 andmessage 2 coexistand the speed of message 1 and message 2 will tend to reach anonzero finite value

Similarly The Jacobian matrix of system (12) at 1198702is

119872(1198702) =

[[[

[

minus119896211205831

1198811

minus11989621120583112059021

1198812

11989622120583212059012

1198811

minus119896221205832

1198812

]]]

]

(34)

then the 119901(119872(1198702)) = 119896

2112058311198811+ 1198962212058321198812

gt 0 and119902(119872(119870

2)) = 119896

211198962212058311205832(1205901212059021+ 1)119881

11198812gt 0 That is

1198702is asymptotically stable Then we obtain that the stability

condition of equilibrium point1198702is

0 lt 12059021lt

12058321198812(1205821minus 1205831)

1205831(21205731205832+ 12058221198812minus 12058321198812)

12059012gt 0 119881

2gt

21205731205832

1205832minus 1205822

(35)

The condition 0 lt 12059021lt 12058321198812(1205821minus 1205831)1205831(21205731205832+ 12058221198812minus

12058321198812)means thatmessage 2 is less competitive thanmessage 1

When (35) are satisfied bothmessage 1 andmessage 2 coexistand their speed will change over time to reach a stable state

The Jacobian matrix of system (12) at 1198703is

119872(1198703) =

[[[

[

minus119896311205831

1198811

minus11989631120583112059021

1198812

minus11989632120583212059012

1198811

minus119896321205832

1198812

]]]

]

(36)

then the 119901(119872(1198703)) = 119896

3112058311198811+ 1198963212058321198812

gt 0 and119902(119872(119870

3)) = 119896

311198963212058311205832(1 minus 120590

1212059021)11988111198812 By |119866

1| lt |119866

2|

2

2120573+aV212059021

bV2

K2

O aV112120572 + bV112059012

L1

L2

Figure 5 The unique positive equilibrium 1198702

we have 1205901212059021gt 1 and 119902(119872(119870

3)) lt 0 It means that 119870

3is a

saddle point In int 1198772+ all orbits but 119870

3and the separatrixes

of1198703converge to equilibria119870

1and119870

2as shown in Figure 3

(ii) Case 2 (|1198661| lt |119866| = |119866

2|) As can be seen in Figure 4

there are two positive equilibrium points 1198701and 119870

2 It is

important to note that however 1198701and 119870

3in Figure 3

coincide Similar to the analysis in case 11198702is asymptotically

stable and the stability condition of equilibrium point 1198702

satisfies (35)Since 119901(119872(119870

1)) = 119896

1112058311198811+ 1205832119896121198812

gt 0 and119902(119872(119870

1)) = 0 it can be determined that 119870

1is a saddle node

by the saddle node rule [26] According toTheorem 1 we candraw the dynamics of system (12) as shown in Figure 4 Thesame approach also applies to the case |119866

1| = |119866| lt |119866

2| the

difference is that1198701is asymptotically stable and119870

2is a saddle

node equilibrium and all orbits converge to equilibria 1198701in

int 1198772+except 119870

2and the separatrixes of 119870

2

(iii) Case 3 (|1198661| le |119866

2| lt |119866|) In case 3 there is

only one positive equilibrium point 1198702 By Theorem 1 119870

2is

asymptotically stable as shown in Figure 5 and the stabilitycondition of equilibrium point 119870

2satisfies (33) Similar

discussions can be given for the case |119866| lt |1198661| le |119866

2| in

which 1198701is asymptotically stable

(iv) Case 4 (|1198661| = |119866

2| = |119866|) In this case the isoclines 119871

12

and 11987122coincide and all the positive equilibrium points dis-

tribute on the line segment11986311198632 According toTheorem 1 in

int 1198772+ all orbits of system (12) converge to the line segment

as shown in Figure 6

(v) Case 5 (|1198662| lt |119866

1|) In this situation we have similar

discussions with the case 3 and globally stable is1198702as shown

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

The stability condition of equilibrium point 1198743(0 119887119881

2) is

12059021gt 1198861198812(1198871198812minus 2120573)

Condition 12059021gt 1198861198812(1198871198812minus 2120573) shows that message 2 is

more competitive thanmessage 1This just means that peopleaccepted message 2 even though they should have believedmessage 1 Therefore during the process of competitionmessage 2 can survive and reach a maximum speed119881

2 while

message 1 will eventually become extinct finally

33 Dynamics Assume that the conditions in (25) are satis-fied the twomessages in system (12) can coexistThe isocline1198711ofmessage 1 can be subdivided into 119871

11and 119871

12as follows

11987111 119886 + 120590

21

V2

1198812

minusV1

1198811

0 lt V2lt 120573

11987112 119886 + 2120590

21

120573

1198812

minusV1

1198811

minus 12059021

V2

1198812

V2gt 120573

(28)

where 1198711is smoothed in a small neighborhood of the vertex

1198631(1198861198811+ 1205731205902111988111198812 120573) as shown in Figure 1 Similarly we

divide the isocline1198712ofmessage 2 into119871

21and119871

22as follows

11987121 119887 + 120590

12

V1

1198811

minusV2

1198812

0 lt V1lt 120572

11987122 119887 + 2120590

12

120572

1198811

minusV2

1198812

minus 12059012

V1

1198811

V1gt 120572

(29)

where 1198712is smoothed in a small neighborhood of the vertex

1198632(120572 119887119881

2+ 1205721205901211988121198811) as shown in Figure 1

While the line segments 11987111and 119871

21are restricted to the

intervals (1198861198811 1198861198811+ 1205731205902111988111198812) and (0 120572) respectively they

are impossible to intersect in the interval 120572 le 1198861198811Then there

is a maximum of three isolated positive equilibrium points insystem (12) As shown in Figure 1 let 119870

1(11989611 11989612) represent

the intersection of 11987111and 119871

22 let 119870

2(11989621 11989622) represent the

intersection of 11987121and 119871

12 and let119870

3(11989631 11989632) represent the

intersection of 11987112and 119871

22 Then we have

11989611=1198811(119886 + 119887120590

21) + 2120572120590

1212059021

1205901212059021+ 1

11989612=1198812(1198811(119887 minus 119886120590

12) + 2120572120590

12)

(1205901212059021+ 1)119881

1

11989621=1198811(1198812(119886 minus 119887120590

21) + 2120573120590

21)

(1205901212059021+ 1)119881

2

11989622=1198812(11988612059012+ 119887) + 2120573120590

1212059021

1205901212059021+ 1

11989631=1198811(1198812(11988712059021minus 119886) minus 2120573120590

21) + 2120572120590

12120590211198812

(1205901212059021minus 1)119881

2

11989632=1198811(1198812(11988612059012minus 119887) + 2120573120590

1212059021) minus 2120572120590

121198812

(1205901212059021minus 1)119881

1

(30)

2

2120573+aV212059021

bV2

K2

K3

K1

O aV112120572 + bV112059012

L1

L2

Figure 3 Three positive equilibria 1198701 1198702 and 119870

3

Suppose that 1198661and 119866

2represent the slope of the lines

11987112and 119871

22 respectively Suppose that119866 denotes the slope of

the line11986311198632 Then we have

1198661= minus

1198812

119881112059021

lt 0

1198662= minus

119881212059012

1198811

lt 0

119866 =1198812(11988711988121198811+ 120572120590121198812minus 1205731198811)

1198811(11988611988111198812minus 1205721198812+ 120573120590211198811)lt 0

(31)

Dynamics of system (12) can be displayed in five caseswhile we discuss the situation of |119866

1| lt |119866

2| in cases 1ndash4 and

that of |1198661| gt |119866

2| in case 5

(i) Case 1 (|1198661| lt |119866| lt |119866

2|) In this case there are three

positive equilibria 1198701 1198702 and 119870

3as shown in Figure 3 The

Jacobian matrix of system (12) at 1198701is

119872(1198701) =

[[[

[

minus119896111205831

1198811

11989611120583112059021

1198812

minus11989612120583212059012

1198811

minus119896121205832

1198812

]]]

]

(32)

then 119901(119872(1198701)) = 119896

1112058311198811+ 1205832119896121198812gt 0 and 119902(119872(119870

1)) =

119896111198961212058311205832(1205901212059021+1)119881

11198812gt 0That is119870

1is asymptotically

stable Then we obtain that the stability condition of equilib-rium point119870

1is

0 lt 12059012lt

12058311198811(1205822minus 1205832)

1205832(21205721205831+ 12058211198811minus 12058311198811)

12059021gt 0 119881

1gt

21205721205831

1205831minus 1205821

(33)

8 Journal of Applied Mathematics

2

2120573+aV212059021

bV2

K2

K1

O aV112120572 + bV112059012

L1

L2

Figure 4 Two positive equilibria 1198701and 119870

2

Condition 0 lt 12059012

lt 12058311198811(1205822minus 1205832)1205832(21205721205831+ 12058211198811minus

12058311198811)means thatmessage 1 is less competitive thanmessage 2

When (33) are satisfied bothmessage 1 andmessage 2 coexistand the speed of message 1 and message 2 will tend to reach anonzero finite value

Similarly The Jacobian matrix of system (12) at 1198702is

119872(1198702) =

[[[

[

minus119896211205831

1198811

minus11989621120583112059021

1198812

11989622120583212059012

1198811

minus119896221205832

1198812

]]]

]

(34)

then the 119901(119872(1198702)) = 119896

2112058311198811+ 1198962212058321198812

gt 0 and119902(119872(119870

2)) = 119896

211198962212058311205832(1205901212059021+ 1)119881

11198812gt 0 That is

1198702is asymptotically stable Then we obtain that the stability

condition of equilibrium point1198702is

0 lt 12059021lt

12058321198812(1205821minus 1205831)

1205831(21205731205832+ 12058221198812minus 12058321198812)

12059012gt 0 119881

2gt

21205731205832

1205832minus 1205822

(35)

The condition 0 lt 12059021lt 12058321198812(1205821minus 1205831)1205831(21205731205832+ 12058221198812minus

12058321198812)means thatmessage 2 is less competitive thanmessage 1

When (35) are satisfied bothmessage 1 andmessage 2 coexistand their speed will change over time to reach a stable state

The Jacobian matrix of system (12) at 1198703is

119872(1198703) =

[[[

[

minus119896311205831

1198811

minus11989631120583112059021

1198812

minus11989632120583212059012

1198811

minus119896321205832

1198812

]]]

]

(36)

then the 119901(119872(1198703)) = 119896

3112058311198811+ 1198963212058321198812

gt 0 and119902(119872(119870

3)) = 119896

311198963212058311205832(1 minus 120590

1212059021)11988111198812 By |119866

1| lt |119866

2|

2

2120573+aV212059021

bV2

K2

O aV112120572 + bV112059012

L1

L2

Figure 5 The unique positive equilibrium 1198702

we have 1205901212059021gt 1 and 119902(119872(119870

3)) lt 0 It means that 119870

3is a

saddle point In int 1198772+ all orbits but 119870

3and the separatrixes

of1198703converge to equilibria119870

1and119870

2as shown in Figure 3

(ii) Case 2 (|1198661| lt |119866| = |119866

2|) As can be seen in Figure 4

there are two positive equilibrium points 1198701and 119870

2 It is

important to note that however 1198701and 119870

3in Figure 3

coincide Similar to the analysis in case 11198702is asymptotically

stable and the stability condition of equilibrium point 1198702

satisfies (35)Since 119901(119872(119870

1)) = 119896

1112058311198811+ 1205832119896121198812

gt 0 and119902(119872(119870

1)) = 0 it can be determined that 119870

1is a saddle node

by the saddle node rule [26] According toTheorem 1 we candraw the dynamics of system (12) as shown in Figure 4 Thesame approach also applies to the case |119866

1| = |119866| lt |119866

2| the

difference is that1198701is asymptotically stable and119870

2is a saddle

node equilibrium and all orbits converge to equilibria 1198701in

int 1198772+except 119870

2and the separatrixes of 119870

2

(iii) Case 3 (|1198661| le |119866

2| lt |119866|) In case 3 there is

only one positive equilibrium point 1198702 By Theorem 1 119870

2is

asymptotically stable as shown in Figure 5 and the stabilitycondition of equilibrium point 119870

2satisfies (33) Similar

discussions can be given for the case |119866| lt |1198661| le |119866

2| in

which 1198701is asymptotically stable

(iv) Case 4 (|1198661| = |119866

2| = |119866|) In this case the isoclines 119871

12

and 11987122coincide and all the positive equilibrium points dis-

tribute on the line segment11986311198632 According toTheorem 1 in

int 1198772+ all orbits of system (12) converge to the line segment

as shown in Figure 6

(v) Case 5 (|1198662| lt |119866

1|) In this situation we have similar

discussions with the case 3 and globally stable is1198702as shown

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

2

2120573+aV212059021

bV2

K2

K1

O aV112120572 + bV112059012

L1

L2

Figure 4 Two positive equilibria 1198701and 119870

2

Condition 0 lt 12059012

lt 12058311198811(1205822minus 1205832)1205832(21205721205831+ 12058211198811minus

12058311198811)means thatmessage 1 is less competitive thanmessage 2

When (33) are satisfied bothmessage 1 andmessage 2 coexistand the speed of message 1 and message 2 will tend to reach anonzero finite value

Similarly The Jacobian matrix of system (12) at 1198702is

119872(1198702) =

[[[

[

minus119896211205831

1198811

minus11989621120583112059021

1198812

11989622120583212059012

1198811

minus119896221205832

1198812

]]]

]

(34)

then the 119901(119872(1198702)) = 119896

2112058311198811+ 1198962212058321198812

gt 0 and119902(119872(119870

2)) = 119896

211198962212058311205832(1205901212059021+ 1)119881

11198812gt 0 That is

1198702is asymptotically stable Then we obtain that the stability

condition of equilibrium point1198702is

0 lt 12059021lt

12058321198812(1205821minus 1205831)

1205831(21205731205832+ 12058221198812minus 12058321198812)

12059012gt 0 119881

2gt

21205731205832

1205832minus 1205822

(35)

The condition 0 lt 12059021lt 12058321198812(1205821minus 1205831)1205831(21205731205832+ 12058221198812minus

12058321198812)means thatmessage 2 is less competitive thanmessage 1

When (35) are satisfied bothmessage 1 andmessage 2 coexistand their speed will change over time to reach a stable state

The Jacobian matrix of system (12) at 1198703is

119872(1198703) =

[[[

[

minus119896311205831

1198811

minus11989631120583112059021

1198812

minus11989632120583212059012

1198811

minus119896321205832

1198812

]]]

]

(36)

then the 119901(119872(1198703)) = 119896

3112058311198811+ 1198963212058321198812

gt 0 and119902(119872(119870

3)) = 119896

311198963212058311205832(1 minus 120590

1212059021)11988111198812 By |119866

1| lt |119866

2|

2

2120573+aV212059021

bV2

K2

O aV112120572 + bV112059012

L1

L2

Figure 5 The unique positive equilibrium 1198702

we have 1205901212059021gt 1 and 119902(119872(119870

3)) lt 0 It means that 119870

3is a

saddle point In int 1198772+ all orbits but 119870

3and the separatrixes

of1198703converge to equilibria119870

1and119870

2as shown in Figure 3

(ii) Case 2 (|1198661| lt |119866| = |119866

2|) As can be seen in Figure 4

there are two positive equilibrium points 1198701and 119870

2 It is

important to note that however 1198701and 119870

3in Figure 3

coincide Similar to the analysis in case 11198702is asymptotically

stable and the stability condition of equilibrium point 1198702

satisfies (35)Since 119901(119872(119870

1)) = 119896

1112058311198811+ 1205832119896121198812

gt 0 and119902(119872(119870

1)) = 0 it can be determined that 119870

1is a saddle node

by the saddle node rule [26] According toTheorem 1 we candraw the dynamics of system (12) as shown in Figure 4 Thesame approach also applies to the case |119866

1| = |119866| lt |119866

2| the

difference is that1198701is asymptotically stable and119870

2is a saddle

node equilibrium and all orbits converge to equilibria 1198701in

int 1198772+except 119870

2and the separatrixes of 119870

2

(iii) Case 3 (|1198661| le |119866

2| lt |119866|) In case 3 there is

only one positive equilibrium point 1198702 By Theorem 1 119870

2is

asymptotically stable as shown in Figure 5 and the stabilitycondition of equilibrium point 119870

2satisfies (33) Similar

discussions can be given for the case |119866| lt |1198661| le |119866

2| in

which 1198701is asymptotically stable

(iv) Case 4 (|1198661| = |119866

2| = |119866|) In this case the isoclines 119871

12

and 11987122coincide and all the positive equilibrium points dis-

tribute on the line segment11986311198632 According toTheorem 1 in

int 1198772+ all orbits of system (12) converge to the line segment

as shown in Figure 6

(v) Case 5 (|1198662| lt |119866

1|) In this situation we have similar

discussions with the case 3 and globally stable is1198702as shown

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

2

2120573+aV212059021

bV2

O aV112120572 + bV112059012

L1

L2

D1

D2

Figure 6 Equilibrium points on the line segment11986311198632

in Figure 5 The same principle applies to the condition of|119866| lt |119866

2| lt |119866

1| that 119870

1is globally stable In the case of

|1198662| le |119866| lt |119866

1| or |119866

2| lt |119866| le |119866

1| there is a unique

positive globally stable point1198703of system (12) in Figure 7

4 Numerical Simulation

To better understand the proposed model we carry outcomputer simulations with real data from Sina Weibo aChinese microblog Our data collected by a crowd sourcingWeibo visual analytic system [27] and the interval time isfrom March 31th 2014 to May 30th 2014 We extract twomessages published by famous Chinese actors Wen ZhangandMa Yili about their marriage In SinaWeibo Wen Zhanghas 53 706 926 fans ranked in 7th place and Ma Yili has 34759 267 fans ranked in 35th place Atmidnight ofMarch 31thWen Zhang apologised to his families and devoted fans andthen his wife Ma Yili forgives him For their action in savingmarriage amount of their fans reposted the two messagesAs the repost actions have occurred mainly before 1400 wespecify a unit time in 15 minutes to statistical analysis among1100ndash1400 in March 31th In order to explore the dynamicsprocess with varied parameters we put the observationsin our model and then further study the influence of thecompetition coefficient (120590

12and 120590

21) in case of messages

propagation with low (or high) critical speed (120572 and 120573)We first consider the case that both messages have less

competitive 12059012lt 1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) at

low critical speed Figure 8(a) shows the evolutionary trend ofmessage 1 and message 2 over time We find that the speed ofthe twomessages can be viewed as a nonmonotonic functionand the velocity curve of message 1 can be divided intothree distinctly different stages In the initial stage message1 appears in OSNs and then attracts more and more users

2

2120573+aV212059021

bV2

K3

O aV112120572 + bV112059012

L1

L2

Figure 7 The unique positive equilibrium 1198703

leading to rapid spread of message 1 However message 1descends slowly followed by the end of initial stage becauseof the natural dying out of its own as well as the competitioncoming from message 2 Specifically since message 1 is lesscompetitive than message 2 the growth of message 2 hindersthe spread of message 1 and message 1 will continuouslydecay due to the competition Finally message 1 arrives atits equilibrium and then converges to a constant value Theevolution process of message 2 is similar to message 1 Aftera slow growth in the initial stage message 2 spreads rapidlyand then slows down till a steady state is reached

The change trends of message 1 and message 2 canbe observed from Figure 8(b) At the beginning the twomessages grow quickly and then slow down and convergeto a nonzero constant This process reflects that with lesscompetitive and low critical speed message 1 and message2 will cooperate with each other in the propagation Bothof them can attract a certain amount of users and they cancoexist in a period of time

In the case of 12059012

lt 1198871198811(1198861198811minus 2120572) and 120590

21gt

1198861198812(1198871198812minus 2120573) the message which is more competitive

than the other will survive at the final state As shown inFigure 9(a) along with the increasing number of people whohave heard message 2 the speed of message 1 increases at thebeginning and then decreases rapidly The main reason forthis phenomenon is that the two messages have cooperatedwith each other when their speed is lower than the criticalspeed at the beginning and then competed against eachother when their speed is higher than the critical speedafter the initial phase Finally only the more competitivemessage (1205909(119886)

21≫ 1205909(119886)

12) survived in the system When 120590

12gt

1198871198811(1198861198811minus2120572) and 120590

21lt 1198861198812(1198871198812minus2120573) we can get a similar

conclusion that message 2 will vanish at stable state That isthe more competitive message can draw the users attention

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

0 20 40 60 80 1000

500

1000

1500

2000

Message 1Message 2

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

(a)

0 5 10 15 200

200

400

600

800

1000

1

2

times102

(b)

Figure 8 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205908(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205908(119886)

21= 08 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

0 10 20 30 40 500

500

1000

1500

Time

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0

200

400

600

800

1000

1200

1

2

0 5 10 15times10

2

(b)

Figure 9 (a) V1and V

2as a function of 119905 with V

1(0) = 2 V

2(0) = 1 119881

1= 3000 120572 = 80 1205909(119886)

12= 06 120583

1= 07 120582

1= 02 119881

2= 1800 120573 = 20

1205909(119886)

21= 2 120583

2= 05 and 120582

2= 015 (b) The development trend of relationship between message 1 and message 2

and spread widely while the message with lesser competitiveis difficult to attract users and vanished after a while

The change trends of message 1 and message 2 inFigure 9(b) is similar to that in Figure 8(b) The difference isthat only message 2 can survive when a final state is reachedUnder the constraint condition of parameters 1205909(119886)

21≫ 1205908(119886)

21

we can make a conclusion that message 2 attracts amount ofusers frommessage 1 at low critical speed and spreads widelyin online social networks

As shown in Figure 10(a) under the condition of 12059012

gt

1198871198811(1198861198811minus 2120572) and 120590

21gt 1198861198812(1198871198812minus 2120573) only message 1

survives and message 2 vanishes in steady state while theycoexist with each other at the final stage in Figure 10(b)It can be inferred that in strong competitive environmentthe high critical speed promotes the coexistence of the twomessages At the same time although restricted by 120590

21lt 12059012

the velocity values of both message 1 and message 2 inFigure 10(b) are greater than that in Figure 10(a) it can beconcluded that when the two messages are both competitivethey will benefit from the high critical speed

Generally more competitive messages disseminate morerapidly than less competitive messages over network Unfor-tunately the situation is often not the case It can be observedthat under the conditions with the same 120590

21 message 2 can

survive and reach its maximum speed in Figure 9(a) but itvanished finally in Figure 10(a) The difference exists becausethe competition coefficient of message 1 (120590

12) is considerably

larger in Figure 10(a) than that in Figure 9(a) (12059010(119886)12

≫

1205909(119886)

12) It means that the advantage of the prior competitive

message will quickly disappear as the competitiveness gapnarrowed Therefore under the condition of low criticalspeed it is necessary to adopt Tit-for-Tat-like strategy to

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

0 5 10 15 20 25 30Time

0

500

1000

1500

2000

Spee

d of

mes

sage

1an

d m

essa

ge2

Message 1Message 2

(a)

0 10 20 30 400

1000

2000

3000

4000

5000

Spee

d of

mes

sage

1an

d m

essa

ge2

Time

Message 1Message 2

(b)

Figure 10 Positive or negative effects for different critical speed with V1(0) = 2 V

2(0) = 1119881

1= 3000 12059010(119886)

12= 3 120583

1= 07 120582

1= 02119881

2= 1800

12059010(119886)

21= 2 120583

2= 05 and 120582

2= 015 (a) V

1and V

2as a function of 119905 with 120572 = 80 120573 = 20 (b) V

1and V

2as a function of 119905 with 120572 = 2000

120573 = 1200

improve their competitiveness when there already exists acompetitive message in online social network

5 Conclusions

The majority of studies on information propagation havefocused on a single contagion spreading through the socialnetwork without considering the influence of other con-tagions In this paper we have explored the informationspreading mechanism impacted by cooperative and com-petitive effectiveness during the process of the propagationof two types of messages over networks We developed anew OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specificcooperation-competition system of these two messages Thestability of the systemwas proved by the differential equationsand the phase trajectory of the stability theory The resultsof stability analysis further indicated that the differentialequations admit no periodic orbit and the system can reachsteady state in certain condition

We proved that two types of messages usually cannotspread together peacefully and only one type message sur-vives till final state Nevertheless it produced stable pointsin the condition of high (or low) critical speed which showsthat it is possible for the coexistence of both messages incompetitive propagation of different contagions Using thereal data collected from Sina Weibo the results validatedthe theoretical analysis and presented another interestingconclusion that the messages will benefit from the highcritical speed when they are both competitive If there alreadyis a competitive message in OSN it is wise to adopt a Tit-for-Tat strategy

As the great massive data are generated every momentin OSN the rapid progress of information technology pro-vides us with an easy way to collect data and observethe phenomenon that various types of information flow inthe form of cascades on Twitter or other OSNs Naturally

the proposed dynamic information propagation modelshould be tested and applied in real OSN with large topologynetwork and vast numbers of the nodes and connectionsThese issues will be addressed in future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas partially supported by theNational Nature Sci-ence Foundation of China (no 71271186) EducationMinistryFund of Humanities and Social Science (no 12YJA630191)Natural Science Foundation of Hebei (no G2013203237) andBrazilian National Council for Scientific and TechnologicalDevelopment-CNPq (no 3049032013-2)

References

[1] W W Wu ldquoThe cooperation-competition model for the hottopics of ChinesemicroblogsrdquoAppliedMechanics andMaterialsvol 380ndash384 pp 2724ndash2727 2013

[2] J C Bosley N W Zhao S Hill et al ldquoDecoding twitterSurveillance and trends for cardiac arrest and resuscitationcommunicationrdquo Resuscitation vol 84 no 2 pp 206ndash212 2013

[3] L Weigang E F O Sandes J Zheng A C M A Melo andL Uden ldquoQuerying dynamic communities in online socialnetworksrdquo Journal of Zhejiang University Science C vol 15 no2 pp 81ndash90 2014

[4] X Zhao and J Wang ldquoDynamical model about rumor spread-ing withmediumrdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 586867 9 pages 2013

[5] D J Daley and D G Kendall ldquoEpidemics and Rumours [49]rdquoNature vol 204 no 4963 p 1118 1964

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

[6] D J Daley and D G Kendall ldquoStochastic rumoursrdquo Journal ofthe Institute of Mathematics and Its Applications vol 1 pp 42ndash55 1965

[7] D P Maki and MThompsonMathematical Models and Appli-cations With Emphasis on the Social Life and ManagementSciences Prentice Hall Englewood Cliffs NJ USA 1973

[8] J R C Piqueira ldquoRumor propagation model an equilibriumstudyrdquoMathematical Problems in Engineering vol 2010 ArticleID 631357 7 pages 2010

[9] DH Zanette ldquoDynamics of rumor propagation on small-worldnetworksrdquo Physical Review E vol 65 no 4 Article ID 0419089 pages 2002

[10] YMorenoMNekovee andA Vespignani ldquoEfficiency and reli-ability of epidemic data dissemination in complex networksrdquoPhysical Review E vol 69 Article ID 055101(R) 2004

[11] YMorenoMNekovee andA F Pacheco ldquoDynamics of rumorspreading in complex networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 69 no 6 Article ID066130 2004

[12] J Zhou Z Liu and B Li ldquoInfluence of network structure onrumor propagationrdquo Physics Letters A vol 368 no 6 pp 458ndash463 2007

[13] MNekovee YMorenoG Bianconi andMMarsili ldquoTheory ofrumour spreading in complex social networksrdquo Physica A vol374 no 1 pp 457ndash470 2007

[14] M Granovetter ldquoThreshold models of collective behaviorrdquoTheAmerican Journal of Sociology pp 1420ndash1443 1978

[15] D Kempe J Kleinberg and E Tardos ldquoMaximizing the spreadof influence through a social networkrdquo in Proceedings of the9th ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining (KDD rsquo03) pp 137ndash146 New YorkNY USA August 2003

[16] D Trpevski W K S Tang and L Kocarev ldquoModel for rumorspreading over networksrdquo Physical Review E vol 81 no 5Article ID 056102 2010

[17] Z Z Liu W Xing-Yuan andWMao-Ji ldquoCompetition betweentwo kinds of information among random-walking individualsrdquoChinese Physics B vol 21 no 4 Article ID 048902 6 pages 2012

[18] X TWei N Valler B A Prakash I NeamtiuM Faloutsos andC Faloutsos ldquoCompeting memes propagation on networks acase study of composite networksrdquo ACM SIGCOMMComputerCommunication Review vol 42 no 5 pp 5ndash12 2012

[19] Y BWangGX Xiao and J Liu ldquoDynamics of competing ideasin complex social systemsrdquoNew Journal of Physics vol 14 no 1Article ID 013015 22 pages 2012

[20] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[21] S A Myers and J Leskovec ldquoClash of the contagions cooper-ation and competition in information diffusionrdquo in Proceedingsof the 12th IEEE International Conference on Data Mining(ICDM rsquo12) pp 539ndash548 Los Alamitos CA USA December2012

[22] L Lu D Chen and T Zhou ldquoThe small world yields the mosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 2011

[23] M J Hernandez and I Barradas ldquoVariation in the outcome ofpopulation interactions bifurcations and catastrophesrdquo Journalof Mathematical Biology vol 46 no 6 pp 571ndash594 2003

[24] Z Zhang ldquoMutualism or cooperation among competitorspromotes coexistence and competitive abilityrdquo Ecological Mod-elling vol 164 no 2-3 pp 271ndash282 2003

[25] J Hofbauer and K Sigmund Evolutionary Games and Popula-tion Dynamics Cambridge University Press Cambridge UK1998

[26] Z Zhang T Ding W Huang and Z Dong Qualitative Theoryof Differential Equations vol 101 ofTranslations ofMathematicalMonographs AMS Providence RI USA 1992

[27] D Ren X Zhang Z Wang J Li and X Yuan ldquoWeiboEventsa crowd sourcing weibo visual analytic systemrdquo in Proceedingsof the IEEE Pacific Visualization Symposium (PacificVis 14) pp330ndash334 Yokohama Japan March 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of