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Hindawi Publishing CorporationJournal of Complex SystemsVolume 2013 Article ID 972352 13 pageshttpdxdoiorg1011552013972352

Research ArticleDiffusion Models for Information Dissemination Dynamics inWireless Complex Communication Networks

Shin-Ming Cheng1 Vasileios Karyotis2 Pin-Yu Chen3

Kwang-Cheng Chen4 and Symeon Papavassiliou2

1 National Taiwan University of Science and Technology Taipei 106 Taiwan2 School of Electrical and Computer Engineering National Technical University of Athens (NTUA) 15780 Zografou Athens Greece3 Department of EECS University of Michigan Ann Arbor MI 48109 USA4Graduate Institute of Communication Engineering National Taiwan University Taipei 106 Taiwan

Correspondence should be addressed to Vasileios Karyotis vassilisnetmodentuagr

Received 1 May 2013 Revised 18 July 2013 Accepted 29 July 2013

Academic Editor Jinhui Zhang

Copyright copy 2013 Shin-Ming Cheng et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Information dissemination has become one of the most important services of communication networks Modeling the diffusion ofinformation through suchnetworks is crucial for ourmodern information societies In thiswork novelmodels segregating betweenuseful and malicious types of information are introduced in order to better study Information Dissemination Dynamics (IDD) inwireless complex communication networks and eventually allow taking into account special network features in IDD Accordingto the proposed models and inspired from epidemiology we investigate the IDD in various complex network types through theuse of the Susceptible-Infected (SI) paradigm for useful information dissemination and the Susceptible-Infected-Susceptible (SIS)paradigm for malicious information spreading We provide analysis and simulation results for both types of diffused informationin order to identify performance and robustness potentials for each dissemination process with respect to the characteristics of theunderlying complex networking infrastructures We demonstrate that the proposed approach can generically characterize IDD inwireless complex networks and reveal salient features of dissemination dynamics in each network type which could eventually aidin the design of more advanced robust and efficient networks and services

1 Introduction

Information dissemination is a key social process in moderninformation-centric societies and most of the communica-tion infrastructures have been developed in the last thirtyyears mainly to allow transferring diverse types of infor-mation Different information types range in scope (egacademic educational financial and military) criticality(eg confidential sensitive public information etc) andvalue (eg useful harmful and indifferent)

Recent advances in networking have been stimulatedin order to accommodate emerging trends of increasingvolumes and service demands of disseminated informationIn general information may be distinguished in three typescharacterized by useful malicious or indifferent contentThe first may consist of news multimedia or financial data

People are willing to accept such information and usuallysuch data is stored for further use for example e-booksConsequently it is obtained once and a user experiencesa single transition from the state of not having it to thestate of having received the information On the other handusers may get malicious information such as malware whichhowever are in principle reluctant to accept andor useUnfortunately sources of malicious information manage todevise newways of spreading such data for example throughemails viruses and trapped websites so that maliciouscontent is characterized by recurrent behavior which coststime and money In addition the latter holds for indifferenttypes of information such as spam email The user usuallydiscards such information but relevant messages are ofrecurrent nature for example consisting of repeated or newerphishing messages

2 Journal of Complex Systems

Sensor network

Socialnetwork

Infrastructure network

Ad hoc network

Wireless LAN

Wirednetwork

Satellitenetwork

Vehicular network

Backhaul network

Physical contactSocial contact

Figure 1 Contemporary wireless complex communication network architecture presenting cumulatively all considered types of networksincluding interconnections to wired backhauls

In this work we focus especially on information dis-semination in wireless complex communication networksModern networks consist of various complex subnetworks[1] which merge as a heterogeneous large-scale networkpossibly connecting to a wired infrastructure (Figure 1)Contemporary users equipped with devices having variousradio interfaces can communicate with each other in morecomplicated ways than in the past For example a user maycommunicate with another user via mobile phone over cellu-lar networks while also communicating with a different userin hisher geographic vicinity via WiFi [2] or Bluetooth [3]To evaluate the information dissemination dynamics (IDD)in such heterogeneous complex communication networkswhere communications are affected by both social relationsand physical proximity we establish analytical models withparameters representing IDD for different types of underly-ing communication networks and exploit them to study thedynamics of IDD and obtain means of IDD control

Substantial works [4ndash9] have employed an analogy ofIDD in a single network as infectious spreading diseasesby using ordinary differential equation (ODE) in the fieldof epidemics [10 11] which could act as a quick referenceto efficiently gather approximate knowledge of informationdissemination speed and status with various settings ofaverage node degrees and attain further control [12] in thosenetworks A more thorough summary of such protocolscan be found in [13] However IDD are further complicated

in wireless complex communication networks due to thepresence of dynamic behaviors regarding the variability oftopology and user behavior Randomized approaches basedon both stochastic processes and epidemic models are thosecharacterized as epidemic routing [14ndash16] However all suchworks mainly focus on specific types of information and net-work topology Our work serves as one of the early attemptsto systematically analyze IDD in a generic manner and allowassessing the dissemination and robustness performance inboth current and future wireless complex networks anddifferent types of information

The main contribution of our work is in realizing thatdifferent dynamics are governing the propagation of differenttypes of information and provide appropriatemodels for eachcase Specifically with regard to useful data the objective is tospread such information to as many users as possible so thatuseful information reaches potentially all nodes of a networkOn the contrary in the event of maliciousindifferent infor-mation spreading the objective is to study the robustnessof various network types against harmful or indifferentbut nevertheless network-stressing content To address theabove in this paper we propose two different paradigms thatdescribe the behavior of systems a Susceptible Infected (SI)for the first and Susceptible-Infected-Susceptible (SIS) forthe second and third types of information Both paradigmswere inspired by drawing analogies to the field of epidemics[10] We then propose and analyze two specific approaches

Journal of Complex Systems 3

for obtaining analytically the behavior of each case anddemonstrate how they can be used in order to identify theparameters and properties of the underlyingwireless complexcommunication networks that govern IDD

The rest of this paper is organized as follows In Section 2we present the proposed framework for IDD and discussthe analogy to epidemic models for describing such dynamicoperation while in Section 3 we describe the complex net-works that we will consider and the employed assessmentmetrics In Section 4 an analytical approach for describinguseful information dissemination is presented and evaluatedwhile in Section 5 the case of maliciousindifferent informa-tion spreading is analyzed and evaluated Finally Section 6summarizes the contribution of the paper and discussesemerging trends

2 Epidemic-Based IDD for Wireless ComplexCommunication Networks

Information disseminationmodeling has attracted significantattention the past years with numerous of works attemptingto provide accurate and effective models for modeling thespreading of information many of which have focused onwireless networks from as early as 1999 [17] An indicativecomparison of such protocols may be cumulatively found in[13] However protocols in [17] are not based on epidemicmodels such as those in this work and others in the literaturefor example [18 19] Both [18 19] are involved in the devel-opment of adaptive bioinspired information disseminationmodels for wireless sensor networks However such modelsconsider only this specific type of networks and in additionthey consider one type of information propagating in the net-work Another type of approaches are probabilistic ones suchas those described in [20] and the family of gossip methodssuch as those in [14ndash16] which are based on a combinationof random walk protocol variations and epidemic routingtechniques These randomized techniques have also stirrednovel works in information dissemination in delay-tolerant(intermittently connected) networks such as those presentedin [21 22] where combinations of probabilistic methodsgossip algorithms or other epidemic-based approaches canbe used The main problem with all the above families ofapproaches and individual techniques is that typically onetype of information is considered over a single networktopology and the objective is to spread the information toas many users as possible This is not always the case inarbitrary networks where different types of information maypropagate

More specifically considerably different dynamics governthe propagation of useful and maliciousindifferent types ofinformation Regarding the spreading process the dissemi-nation of useful information resembles that of an epidemicdisease in which population members that get infected by avirus permanently transit to immunization (after recovery)or termination (if the infection is lethal) In epidemics suchmodels are readily referred to as Susceptible Infected (SI)[10] or Susceptible Removed (SR) [23] Such behavior iseffectively described by a two-state model where nodes are

initially prone to receive epidemics (Susceptible) and thenpermanently transit to the Infected state once they receivethe epidemic (SI model) The SR model essentially expressesthe same behaviorwhen nodes are removed from the networkin cases of lethal epidemics The SI (SR) model captures thecharacteristic behavior where for each node only a single andpermanent transition takes place from the susceptible to theinfected (removed) state

Recognizing the similarities between IDD and the spreadof infectious diseases ODE models in the field of epidemic[10 11] are widely adopted as tools to analyze IDD incommunication networks [4ndash9] Since ODE models provideclosed-form formulas for the performance metrics of IDDa wide range of effects can be encompassed by aggregatingindividuals in the network into two states (ie infectedand susceptible) and thus reducing computation time andrequired resources In contrast both agent-based emulationmodel [3] and simulation experiments [24] try to preciselycapture attributes of individuals in the network and theinteractions among them via massive experiments thus theappeals of analytical tractability are neglected Obviously thecomplexity of modeling individual-level details significantlyincreases with the number of individuals and thus the ODEmodel is suitable to act as a quick tool to identify IDD in largecomplex networks

The dissemination of useful information resembles the SIprocess in epidemics since a rational user waits to receivedesired or useful information and then stores it for furtheruse If the information is indeed useful and valid no furthertransaction for this information will be requiredtake placeThus the user experiences a single state transition when(s)he receives useful information from the noninformed state(node can potentially receive data ie susceptible) to theinformed state (ie infected) Several works have providedepidemic-based models for such type of information dissem-ination in the Internet [25] and in wireless networks [19 21]

This is not the case however in maliciousindifferentinformation dissemination Regarding malicious informa-tion spreading users are reluctant to retain the maliciousinformation they receive However malicious informationmight return (if the user is not properly protected) or adver-saries are capable of devising a new malware type or packagefor the malicious information With respect to indifferentinformation even though it is usually of no interest tousers it might further load an already stressed infrastructureand especially in the case of spam its repetitive naturemight eventually become disturbing similarly to malwareConsequently one may consider indifferent information as aspecial case ofmalware with respect to the usersrsquo macroscopicbehavior In our treatment we will employ this observationdue to the fact that we will focus on modeling the macro-scopic behavior of IDD In the following wewill use the termsldquomaliciousrdquo and ldquoindifferent informationrdquo interchangeably

In principle a user is able to recover (ie dispose mali-cious information) and return to the previous state where itis possible to become infected againThis behavior resemblesepidemics where individuals become infected then recoverfrom a disease and become susceptible again to a differentor the same disease (the latter if they do not properly


ldquovaccinaterdquo) Such model is denoted by Susceptible-Infected-Susceptible (SIS) in epidemics [10] Contrary to useful infor-mation we notice that the SIS model defines two potentialtransitions between the two possible states of a user namelyswitch from susceptible to infected and then switch back tothe susceptible state This model has been also adopted in[26] for information dissemination but again it is targetedtowards only a specific type of information and networktopology as the SI epidemic models mentioned above

In summary compared to previous works in this paperwe introduce two analogies between epidemics and infor-mation dissemination We employ the SI model to describethe macroscopic behavior of useful IDD and the SIS modelfor malicious (indifferent) IDD In Sections 4 and 5 weanalyze in detail the proposed IDD models and obtain bothquantitative and qualitative results for IDD in various typesof complex communication networks eventually drawingimportant observations for each network type that can beuseful for future designs and implementations

3 Wireless Complex Networksand Assessment Metrics

In network science (complex network theory) [27 28] anetwork represents a system of interactions and can bemodeled as a graph 119866(119881 119864) consisting of a set 119881 of nodes(ie wireless terminals or Internet users) and a set 119864 ofundirected edges (ie physical channels) The number ofnodes is denoted by 119873 Without loss of generality andin order to focus on the IDD rather than on the wirelesspropagation details two nodes connected by a link are calledneighbors and the number of neighbors for a node is definedas the degree 119896 We define the degree distribution 119875(119896) as theprobability of having 119896 channels for a node and 119896 as the meanvalue of 119896 We assume that there is at most one edge betweenany node pair and no self-loops in 119866(119881 119864)

As it will be shown in the sequel by employing theaforementioned analogy to epidemics IDD can be effectivelydescribed and mathematically analyzed for different typesof complex communication networks and their topologiesEach network type is characterized by different topologicalfeatures and the proposed approach allows in both cases ofuseful and malicious information identifying the impact ofeach topology on the IDD performance and robustness Theevaluation of these models will be based on the followingcritical parameters involved

(i) Degree Distribution It provides a suitable representationof the structure of a network especially for social ones Basedon the degree distribution a network is of homogeneousmixing if the degree distribution of each node is centeredaround 119896 and the degree variance 1205902

119896⩽ 120576 Otherwise it is of

heterogeneous mixing

(ii) Link Connectivity The finite transmission range of awireless node determines its neighborhood Moreover wire-less channel quality affects the success of transmissionsThe transmission range and channel quality jointly affect

Table 1 Complex network classification

Connectivitymixing type

Partiallyconnectible

Equallyconnectible

Unequallyconnectible

Homogeneousmixing

Lattice networkwireless sensor

networkER network Small-world

network

Heterogeneousmixing

Machine-to-machinenetwork

wireless meshSmart Grid

Scale-freenetwork

link connectivity Thus for a network with variable connec-tivity the network is (partially) connectible if each node hasa positive probability to establish an undirected connectionto (partial but not all) any other node in the network Wefurther define a network with (different) same connectionprobability in each node as (un)equally connectible

Additional properties may be identified capturing topo-logical properties of the underlying complex networkinginfrastructures and could be exploited for analyzing IDDin such networks The clustering coefficient (indicative ofthe clusters building up due to social or other types ofinteraction) and the average path length between randomlyselected node pairs are appropriate quantities [1] Based onthe specific metrics the complex networks of interest can beclassified in five main categories cumulatively depicted inTable 1

(i) Homogeneous Mixing and Partially Connectible (HoMPC)In a (regular) lattice with degree 119896 every node connectsto its 119896 nearest neighbors [29] A lattice is an HoMPCnetwork since the degree distribution is a delta function withmagnitude equal to 1 located at 119896 Another example is wirelesssensor networks where sensors are uniformly and randomlydistributed on a plane In this case for grid-based sensornetworks the number of neighbors in the communicationradius of a sensor is fixed which acts exactly as HoMPC

(ii) Homogeneous Mixing and Equally Connectible (HoMEC)Erdos-Renyi (ER) random graphs [30] assume that an edgeis present with probability 119901

119890for119873(119873 minus 1)2 possible edges

The degree distribution of an ER network in relatively largenetworks is

119875 (119896) =

(119890minus119896119890119896119896

119890)

119896

(1)

where 119896119890= 119873119901

119890 It is observed that the degree distribution

is centered around 119896119890 and a node has equal probability

119901119890connecting to any other node in the network Thus an

ER network is an HoMEC network Some complex wirelessnetworks can be effectively modeled by ER graphs especially


when considering the joint cyber-physical system formingfrom a social network overlaying a wireless one [31]

(iii) Homogeneous Mixing and Unequally Connectible(HoMUC) Small-world networks generated by the Watts-Strogatz (WS) model [32] are constructed from a regular ringlattice with 2119895 edges for each node and randomly rewiringeach edge of the lattice with probability 119901

119908such that self-

connections and duplicate edges are excluded The degreedistribution of a small-world network is

119875 (119896)

=

min(119896minus119895119895)

sum

119899=0

(119895

119899)(1 minus 119901

119908)119899

119901119895minus119899

119908

(119901119908119895)119896minus119895minus119899

(119896 minus 119895 minus 119899)119890minus119901119908119895 for 119896 ⩾ 119895

0 otherwise(2)

Since the degree distribution has a pronounced peak at 119896119908=

2119895 it decays exponentially for large 119896 [1] and the connectionprobability is unequal except 119901

119908= 1 (extreme randomness)

we regard small-world as HoMUC networkThis implies thatas 119901119908rarr 1 a small-world network becomes HoMEC Many

social network models are extensions of the WS model sincethe HoMUC property explains well their clustering features[1] Small-world topologies emerge very often in wirelesscomplex networks especially in cyber-physical systems withsocial networks overlaying the physical ones [31]

(iv)HeterogeneousMixing and Partially Connectible (HeMPC)Machine-to-machine (M2M) networks wireless mesh net-works or Smart Grid have potentially nonstructured degreedistributions due to finite transmission range heterogeneityin location mobility channel quality variations and time-varying user behavior eventually classifying the underlyingtopology as HeMPC

(v) Heterogeneous Mixing and Unequally Connectible(HeMUC) A power-law distributed network having degreedistribution 119875(119896) sim 119896

minus119903 with 119903 in the range 2 ⩽ 119903 ⩽ 3 isalso called a scale-free network Barabasi and Albert observetwo essential factors of scale-free network growth and pref-erential attachment [33] In this model denoted by BA everynew node connects its119898 edges to existing nodes according tothe preferential attachment rule and 119896

119887= 2119898 The power-

law degree distribution suggests that most nodes have fewneighbors while some ldquosuper nodesrdquo (hubs) tend to have agreat amount of neighbors Thus the power-law distributednetwork is an HeMUC network

Table 1 summarizes the aforementioned complex networkcategories Note that the size of the largest connected com-ponent (ldquogiant componentrdquo) is also an important measureof the effectiveness of the network at information epidemicsor cooperations Regarding this a network is saturated ifthe giant component size approaches the total number ofnodes in the network Otherwise nodesmay be isolated fromthe giant component (nonsaturated network) For instance agiant component emerges almost surely in random graphs ifsum119896119896(119896 minus 2)119875(119896) gt 0 [34]

4 Useful Information DisseminationEpidemic Modeling

As we explained before the macroscopic behavior of usefulinformation dissemination can be described by the SI epi-demic model in which nodes receive the designated dataonce in their lifetime In this section we provide an analyticalapproach for quantifying the process of useful informationdissemination in various types of complex communicationnetworks

41 SI Epidemic SpreadingModel We adopt the SI model [10]in order to describe the IDD of useful information In theanalogy we draw between IDD and epidemics that an unin-fected node in epidemics corresponds to a noninformed nodein IDD that is a node not having received yet useful infor-mation On the other hand an infected node corresponds toa node that has received and stored useful information Suchmodel is appropriate for describing a specific type of usefulinformation spreading for example handbook disseminatedto a population similarly to the SI model describing thedissemination of a single viral threat In the sequel we willuse the terms noninfected and noninformed as well as theinfected and informed terms interchangeably Assume that119873 nodes are initially all susceptible noninformed except fora small number that are infected and contagious (denotedas infectious) These ldquocontagiousrdquo nodes represent the nodesthat generate the useful content to be disseminated Thus bythe previous analogy in this scenario all nodes would like toeventually become infected (ie informed)

We adopt infection parameter 120582 to characterize the rate ofspreading between S-I pairs Considering the volatile natureof wireless channels and MAC information transmissionover a communication link may not be successful Theavailability (or the successful transmission rate) of a link inwireless communication channels is typically modeled as atwo-state Markov chain with on and off states [35] whichis sufficient for the purposes of our study that does notfocus on the impact of the channel propagation effects Fora dynamic topology the set 119881 of nodes is time invariantand the set 119864 of edges varies with time while however thenetwork maintains its basic features The informed nodesadopt a ldquoconsistent broadcastrdquo behavior so that they tendto transmit the useful information to susceptible nodesin contact consistently just as in the spread of biologicalviruses Hence the spreading rate (similar to infection rate)120582 is equivalent to the probability of an available channel(ldquoonrdquo channel state) which is independently determined forall channels such that the long time behavior of channelavailability is equal to 120582

Expressed mathematically if I(119905) and S(119905) are respec-tively the fraction of infected (informed) and susceptible(noninformed) nodes at time 119905 we have

I (119905) + S (119905) = 1

119889I (119905)119889119905

= 120582119896I (119905) S (119905) (3)


0 20 40 60 80 100 120 140 160 180 200

BA modelTraditional SI model

0 100 2000

5

10

15

20(d

eg)

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion

Time

Time

t0

t0

WS model pw = 1

WS model pw = 04

WS model pw = 02WS model pw = 0

Figure 2 IDD in regular lattice (HoMPC 119901119908= 0) ER (HoMEC

119901119908= 1) scale-free (HeMUC) and small-world (HoMUC) networks

for different 119901119908with mean degree equal to 10 120582 = 001 and 119873 =

2500

Then the simple analytic solution obtained is

I (119905) = I (0)I (0) + (1 minus I (0)) 119890minus119896119905

(4)

From (4) it is clear that the informeddensity I(119905) approaches 1as time evolves as expected In this study we assume initiallythere is only one informed node that is I(0) = 1119873 Notethat the saturation of any complex network should serve asone of the important sufficient conditions when adopting theSI model for IDD since the first order differential equation(4) fails to distinguish the saturation of a network In thesequel we demonstrate the proposed SI framework for staticand dynamic saturated complex networks

42 Static Complex Networks The static network is regardedas a time-invariant graph 119866(119881 119864) where the topology isunchanged in time Given that in Figure 2 numericalresults in regular lattice (HoMPC) ER (HoMEC) small-worldWS (HoMUC) and scale-freeBA (HeMUC) networksare presented based on ensemble averages obtained by 100simulations in saturated complex networks Since an ERnetwork can be totally characterized by parameter119873 and 119901

119890

IDD in ER is described through the SI model by adopting119896119890= 119873119901119890 However compared to IDD in ER networks the SI

model amplifies the cumulative informed node fraction withtime as it implicitly assumes the HoMEC property resultingin inaccurate estimation In Section 421 we will address thisdiscrepancy in order to obtain amore accurate IDD SI-basedmodel

We also observe an interesting phenomenon for theIDD in scale-free networks generated by the BA model

The corresponding IDD curve exceeds that of the SI modelin the beginning but at some instance 119905

0 the curve of the SI

model transcendsThis is in accordance with the fact that theinformation has higher possibility to be transmitted to supernodes than the nodes with low degrees at early stages Oncethe information spreading begins the number of informednodes highly increases as super nodes eventually becomeinformed Then the spreading rate decreases as informationdissemination is nowmainly propagated to nodes with lowerdegrees Thus although the SI model is not suitable fordescribing the IDD in scale-free networks it still providesuseful insights for better understanding the IDD in networkswith the HeMUC property Note that as we adopt the factthat homogeneity holds for nodes of the same degree [23] anenhanced model can be derived to accurately match the IDDcurves of HeMUC networks

Regarding small-world networks effects of different re-wiring probabilities 119901

119908are investigated (Figure 2) Ranging

from extreme regularity (119901119908

= 0) to extreme randomness(119901119908= 1) IDD accelerates due to the enhancement of connec-

tivity and the small-world network transforms fromHoMPCnetwork (119901

119908= 0) to HoMUC network (119901

119908isin (0 1)) and

finally HoMEC network (119901119908= 1)

421 Corrected SI Model To overcome the above discusseddiscrepancies between the SI model and the proposedepidemic IDD model in complex networks the emergingldquodegree correlationrdquo problems need to be addressed Thetraditional SI model implicitly assumes that each node isuncorrelated However when a node is informed in a staticnetwork this suggests that at least one of its neighborshas been informed and hence the mean degree has to becorrected accordingly Specifically the average number ofsusceptible neighbors of an infected node is less than 119896 andthus an effective mean degree is proposed to accommodatethis phenomenon This phenomenon should be carefullymodeled otherwise the overestimation problem [6] leadsto significant deviation We take HoMEC network as anexample since it can be characterized by its mean degree 119896The rate of informed nodes will be given by

119889I (119905)119889119905

= 120582I (119905) S (119905) = 120582 [119896 minus 119891 (119905)] I (119905) S (119905) (5)

where 119891(119905) is a time-varying function accounting for theaverage number of informed neighbors of an informed nodeBy setting 119891(119905) as a constant a tight upper bound on IDDcan be obtained when 119891(119905) = 1 for HoMEC networks sinceintuitively a node that is infected implies that at least oneof its neighbors is infected Thus we have the followingobservation

Observation 1 For a saturated and HoMEC network giventhe informed rate 120582 and mean degree 119896 parameters thereexists a tight upper bound on IDD at any time instance

By setting appropriate values of 119891(119905) according to thefeatures of different networks the adjusted SImodels success-fully capture the corresponding IDDThis implies that upper


bounds on IDD of HoMUC andHoMPC networks also existTake IDD in sensor network (HoMPC) as an example wheresensors are uniformly and randomly distributed on a planewith node density 120590 the communication radius of a sensor 119877and the radius of the circle containing the informed sensors119903(119905) Obviously 120590120587119903(119905)2 = 119873 sdot I(119905) This is approximatedinto our model by having only the infected nodes that lieon the periphery of an informed circle to communicate withthe susceptible nodes located at a distance of at most 119877outside the infection circle and thus have the potential toinform (infect) them In other words the spatial broadcastingof the information is only contributed from the wavefrontsof informed circles while the infected nodes located in theinterior of the informed circles are not engaged in furtherspatial dissemination Thus we could calculate 119891(119905) as

119891 (119905) = 119896 sdot120590120587[119903 (119905) minus 119877]

2

119873I (119905)cong 119896 sdot (1 minus 119888radic119873I (119905)) (6)

where 119888 = 2radic120590120587119877 and 1205901205871198772 is usually negligible comparedwith119873I(119905) Applying 119891(119905) to the corrected SI model we thusobtain the same result derived in [8] From this example wehave the following observation

Observation 2 For a saturated and homogeneous mixingnetwork the time 119879 needed to inform a fraction 119904 of nodescan be directly obtained from (5) as 119879 = Iminus1(119904) if Iminus1(sdot) exists

As the cumulative informed fraction approaches 1 in asaturated network Observation 2 serves as a more accuratebenchmark to any broadcasting mechanisms in wirelesscomplex networks for evaluating the performance of infor-mation dissemination This can be justified by the fact thatObservation 2 greatly mitigates the biased estimation due todegree correlations

43 Dynamic Complex Networks This section discusses thedynamic case where network topology changes with timewhile maintaining the basic structure and properties Asit will be shown a time-varying topology provides greatchances for information dissemination to the entire net-work and therefore it is suitable for describing compli-cated interactions within large-scale networks with mobilitysupport (eg routing protocols in MANET) Two nodesoriginally disconnected might eventually establish a virtuallink between them due to mobility thus yielding a virtualgiant component Given the condition that a network isoriginally nonsaturated (eg MANET in sparsely populatedarea) mobility may make the network virtually saturatedsince all nodes eventually receive the information due tochange of connectivity in the dynamic sense

Observation 3 A dynamic network is virtually saturated inthe sense that the virtual giant component size approachesthe number of nodes

The mobility of nodes facilitates connectivity in homo-geneous mixing networks originally not saturated andthus dynamic HoMEC HoMPC and HoMUC networks are

0 10 20 30Time

40 50 60 70 800

01

02

03

04

05

06

07

08

09

1

Info

rmed

frac

tion

Simulation N = 2000

Analysis f(t) = 1

Simulation N = 1000

Analysis f(t) = 03

Simulation N = 500

Analysis f(t) = 01

Simulation N = 100

Analysis f(t) = 015

Figure 3 IDD in dynamic MANET with 119877 = 2m 120582 = 1 and 119896 =

251 126 063 and 013 respectively

virtually saturated In the following we focus on and analyzeIDD in such networks where benefits frommobility are moreobvious

Observation 4 The IDD of dynamic homogeneous mixingnetworks can be characterized by the corrected SI model

Due to the virtual saturation property in Observation 3the SI model failing to describe the IDD of nonsaturatednetworks is suitable to characterize the IDD of dynamichomogeneous mixing networks As we define 119891(119905) accordingto the properties of the networks the IDD curves can beaccurately matched by using (5)

To show the significance of these observations and theflexibility of the proposed model Figure 3 illustrates analyticand simulation plots depicting the epidemic routing viabroadcasting in large-scale MANETs Epidemic routing aim-ing at exploring the advantages of path diversitywith concreteanalysis [7] has been regarded as one practical way to achieverouting in dynamicHoMPC [36 37] In the simulation exper-iments we assume that each node has the same transmissionrange 119877 with uniform random deployment in a 100 times 100m2plane with wrap-around condition According to a stationaryand ergodic mobility model such as the Truncated Levy-walk model [38] we set the step length exponent 119904 = 15

and pause time exponent 120593 = 138 which fit the trace-based data of human mobility pattern collected in UCSDand Dartmouth [39] We incorporate the successful packettransmission rate to the spreading rate 120582 Figure 3 shows thatthe MANET is nonsaturated (however virtually saturated)and our model captures the complicated interactions amongnumerous mobile nodes precisely


44 Hybrid Complex Networks When nodes are capable ofcommunicating with each other using multiple heteroge-neous connections a hybrid complex network consistingof multiple complex subnetworks is built via heterogeneouslinks As in the example we mentioned in Section 1 peoplecould exploit mobile smart phones to communicate withindividuals in the address books via traditional phone callsand short messages as well as the individuals in geographicproximity via Wi-Fi [2] or Bluetooth [3] As shown inFigure 4 the IDD in such complicated networks can beinvestigated by separately considering IDD in the socialnetwork constructed by contacts and IDD via broadcastingand then aggregating the results for the combined cyber-physical system

According to the proposed categories we exploit ERnetwork (HoMEC) and sensor network (HoMPC) to respec-tively model the delocalized and broadcasting disseminationpatternsThus the subpopulation function I(119905) = I

119890(119905)+ I119904(119905)

where I119890(119905) and I

119904(119905) are those that have been disseminated

via ER and sensor networks at time 119905 respectively Theaverage degree 119896

119890describing the social relationships between

handsetsmeans the average number of contacts in the addressbook According to (5) the basic differential equation thatdescribes the dynamics of informed subpopulation is

119889I119890(119905)

119889119905= 120582

S (119905) (119896119890minus 1)

119873I (119905) (7)

When an informed node intends to disseminate viabroadcasting it first scans to search the nearby nodes withinits transmission range 119877 and connects to the neighbor so asto determine the susceptible neighbors for propagation Inthis case the average number of neighbors 119896

119904equals 1205881205871198772

The behavior of such spontaneous spreading can be regardedas a ripple centered at the infected source node which growswith time As shown in Figure 4 the spatial spreading of theinformation here is only contributed from the wavefrontsof informed circles while the infected nodes located in theinterior of the informed circles are not engaged in furtherspatial infections

Without loss of generality we assume that a single in-formed circle is generated at time 119905

1by a point source infected

through and kept stretching for 1199052time units Then its

incremental spatial infection at time 1199051+ 1199052is

1198661015840(1199051 1199052) ≜

119889119866 (1199051 1199052)

1198891199052

= 120582S (1199051+ 1199052) sdot (12) 119896

119904

119873119888radic119866 (119905

1 1199052)

(8)

where (12)119896119904accounts for the fact that for an infected node

on a periphery roughly half of neighbors outside the infectioncircle are susceptible The incremental spatial infection attime 119905 of all infection circles is given by

119889I119904(119905)

119889119905= int

119905

0

I1015840119890(120591) 1198661015840(120591 119905 minus 120591) 119889120591 (9)

It means that there are I1015840119890(120591)119889120591 point sources originated at

time 120591 and each contributes 1198661015840(120591 119905 minus 120591) incremental spatialinfection at time 119905

To validate the analytical model we develop experimentsto simulate IDD in a hybrid network among 2000 individualsuniformly deployed in a 50 times 50 plane The constructions ofsocial contact networks and setup of parameters (eg 119896

119890=

6) follow the data set in [24] Figure 5 illustrates analyticand simulation plots depicting the IDD via both delocalizedcommunication and broadcasting in hybrid (HoMEC andHoMPC) complex networks We observe that the curves ofpropagation dynamics closely match our analytical modelwhere limited discrepancy exists mainly due to the fact thatinformation may propagate to individuals who have alreadybeen informed and uncertain boundary conditions could notbe considered in the analysis Comparing with the traditionalSI in Figure 2 the corrected SI could capture the IDD of ERnetwork (HoMEC) more precisely

5 Malicious InformationPropagation Modeling

In this section we adopt and extend an analytical modelwhich is able to capture the behavior of malicious IDD(modeled as SIS epidemics) in wireless multihop networksContrary to the case of useful information regarding mali-ciousindifferent information one is interested in the robust-ness capabilities of the network to sustain such traffic whichin both cases is of no use (even harmful for malware) Theproposed analytical model is based on queuing theory andwe apply it on various types of complex wireless networks asshown cumulatively in Table 1

51 SIS Closed Queuing Network Model In the SIS paradigmsusceptible (noninformed) nodes essentially wait until thearrival ofmalicious information inwhich case they transitionto the infected (informed) state We consider a propagativenetwork where nodes spread further the malicious infor-mation they receive Consequently a node might becomeinfected frommalicious software either froman attacker or analready infected legitimate node This holds for several typesof viruses and worms that have appeared [40 41] Infectionsare assumed to arrive in a nondeterministic fashion Therecovery process (disposing malicious software) is of similarnature but not necessarily of the same waiting behaviorThroughout this work following the current literature [42]we assume that the infection arrival process is Poisson whilethe recovery process of each network node is exponentiallydistributed We denote by 120582

119894the infection arrival rate in link

119894 and by 120583119895the recovery rate of node 119895

Legitimate nodes can be separated at any instance intwo subsets that is infected and susceptible Following theaforementioned modeling approach the operation of thesystem can bemapped to a closed two-queue packet networkas shown in Figure 6 [43] where119873 packets that is networknodes circulate This closed feedback system of queues doesnot model the actual packets exchanged in a network butrather the transition of nodes themselves fromone state to theother and back (SIS behavior) as desired For simplificationpurposes and in order to visually consider the behavior oflegitimate nodes as they become infected and recover one


Infective nodeSusceptible node for broadcasting and delocalized dissemination Susceptible node for delocalized dissemination

Broadcasting dissemination

Informed circle

Delocalized dissemination

Figure 4 The IDD in wireless complex networks (cyber-physical systems) consisting of both long-range and broadcast disseminationpatterns

may map each node to an arbitrary packetcustomer (notthe packets of the specific IDD network but the packets of ahypothetical one with two queues in a closed loop fashion)circulating in this queuing system and thus properly mapqueuing terminology to IDD concepts

At any instance if 119894 nodes are infected then 119873 minus 119894 aresusceptible Both service rates are state-dependent accordingto the number of packets (user nodes) that exist in the cor-responding queue at each time instance Explicit definitionof each queuersquos equivalent service rate that is 120582(119873 minus 119894 + 1)

and 120583(119894) depends on the underlying complex network andemployed infection paradigm Without loss of generality weassume that the lower queue represents the group of infectedlegitimate nodes and denote it as ldquoinfectedrdquo while the upperqueue represents the susceptible nodes and we denote thequeue as ldquosusceptiblerdquo The state-dependent service rate ofthe susceptible queue can be extended to the case of multiple

malicious information sources (ie attackers) in which casethe service rate will have the form 120582(119873minus 119894 +119872)119872 being thenumber of attackers Our analysis here is focused on the caseof a single attacker (ie119872 = 1)

Standard approaches from queuing theory may beemployed to analyze the two queue closed network Thefocus is on the infected queue Its steady state distributiondenoted by 120587(119894) represents the probability that there are 119894packets (nodes) in this queue Using balance equations forthe respective Markov chain the explicit expression for thesteady state distribution can be obtained as

120587 (119894) = 120587 (0) sdot

119894

prod

119895=1

120582 (119873 + 1 minus 119894 + 119895)

120583 (119895) (10)

where 120587(0) is the probability of no infected nodes in the net-work Applying the normalization condition sum

119873

119894=0120587(119894) = 1


0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion

Hybrid simulationDelocalized dissemination simulationBroadcasting dissemination simulationHybrid analysisDelocalized dissemination analysisBroadcasting dissemination analysis

Figure 5 IDD in hybrid (HoMEC and HoMPC) complex networksof propagating information in both delocalized and broadcastfashions where 119896

119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)

Figure 6 Closed queuing model for SIS malicious informationpropagative networks

and appropriately specifying the total infection and recoveryrates (where 120582

119894= 120582 and 120583

119894= 120583 for all 119894 isin 1 2 119873) and by

setting120572minus1 = (120582120587120583)(119877119871)2 while considering a large number

of legitimate nodes the probability of no infected nodes canbe approximated as

120587 (0) cong120572119873

119873sdot 119890minus120572 (11)

and the steady state distribution can be obtained as

120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)

where 1205871015840(119873 minus 119894) is the steady state distribution for the

noninfected queue Using relation (12) the probability of acompletely infected network equals 120587(119873) = 119890

minus119886 It is notedthat the error introduced by the above approximation isnegligible for values of 120572 and 119873 commonly used in practice(in the order of less than 10minus3)

Expression (12) clearly indicates the critical parametersthat affect the behavior of the system Assuming a fixedarea of the network deployment region the number oflegitimate nodes (ie the density of the network) along withthe common transmission radius and the ratio of the linkinfection rate to the node recovery rate are decisive factorsregarding the overall behavior and stability of the system

Based on such model the expected number of infected(informed) nodes (corresponding to the expected number ofpackets in the lower queue) for different types of networksmay be obtained The general expression yielded is

119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)

where 120587(119894) is the steady state distribution of the underlyingMarkov chain and 119888 = (120583120582120587)(119877119871)

2 for an HeMPC net-work (wireless multihop) over a square deployment regionof side 119871 and each node having a transmission radius 119877 and119888 = 120583119873120582119896 for all other types of networks where 119896 is theaverage node degree for each network under discussion (120582

119894=

120582 and 120583119895= 120583 for all 119894 119895without loss of generality)The average

throughput of the susceptible queue 1198641015840[120574] corresponding tothe average rate that nodes receive malicious information canbe computed as

1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890

minus119888) (119873 + 2) minus 119888 minus 119888

2]

(14)

Observing the analytic form of the average number ofinfected nodes for each network type according to thespecific expression of 119888 a major difference between HeMPCnetworks and the rest should be noted More specifically inanHeMPC the spatial dependence among nodes (due to theirmultihop nature) is reflected by the fraction 120587119877

21198712 repre-

senting the coverage percentage of each node with respect tothe whole network area On the contrary in the expression of119888 for the rest of the networks for which the topology is mainlybased on connectivity relations and not spatial dependence asin the multihop case the corresponding quantity expressingthe local neighbor impact is expressed by 119896119873 It is evidentthat for the rest of network types quantity 119896119873 expressessolely the special connectivity properties of the employednetwork through the value of 119896 since for these networks nospatial dependence is expressed in their connectivity graphand thus no such spatial feature has impact on IDD

52 Demonstration Contrary to the SI model in the SISparadigm employed for themalicious information spreading


0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]

Infectionrecovery rate ratio 120582120583

Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03

Infinite lattice k = 4

WS model j = 4

BA model m = 8

Figure 7 Average number of infected users of the legitimatenetwork as a function of 120582120583

average quantities are of interest while in the SI modelinstantaneous quantities were considered as in the long runthe network converges to a pandemic (all nodes are informed)state Particularly the average number of infected legitimatenodes is the most important quantity since malicious infor-mation is of recurrent nature and if one observes the systemmacroscopically nodes oscillate between the S I states

Figure 7 shows the average number of infected nodeswith respect to the infectionrecovery ratio for various typesof complex networks It is evident that as the ratio 120582120583

increases 119864[119894] increases for all types of networks denotinggreater probability of users to get malicious informationfrom a communication link Parameters for each type ofnetworks are in accordancewith the notation employed in theclassification of Section 2

With respect to ad hoc networks (HeMPC) the greatestthe transmission radius of nodes the denser the network andthus the easier it is for themalicious data to spread Especiallyfor ad hoc networks the dependence of the average numberof infected nodes on the number of legitimate nodes is linearas shown in Figure 8 The combination of larger values of 119877and 120582120583 yields the higher number of 119864[119894] for all values oflegitimate nodes while the combination with smaller valuesof119877 and 120582120583 yields the lower values of119864[119894] It should be notedthat the behavior of the spreading dynamics of the networkis more sensitive to changes in the value of 120582120583 rather thanvariations in 119877

Similar to HeMPC networks on average in ER networksthe greatest the probability that two nodes are connected (andthus the denser the network is) the easier it becomes formalicious information to propagate Such property may bealso identified with the rest of the network types leading tothe following

100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]

Legitimate network nodes N

R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075

Figure 8 Average number of infected nodes versus 119873 (numericalresult)

Observation 5 The denser a network becomes irrespectiveof its type the easier for malicious information to spread

For regular ER WS and BA networks with the same 119896the same result would be obtained due to the expression of119888 given before In order to better demonstrate the differentrobustness properties of these network types in Figure 7a different 119896 value has been employed Regarding the dif-ferent types of networks HeMPC (ad hoc) are close to ER(HoMEC) exhibiting that in general randomness aids thespreading of malicious information This is a very usefuloutcome with significant practical value for designing effi-cient countermeasures for malign IDD On the contrarya regular lattice (HoMPC) makes the spread of maliciousinformationmore difficult since each node is only connectedto a typically small number of other nodes and it would takesignificant effort to quickly spread malicious informationthroughout the whole network Similarly WS (HoMUC) andBA (HeMUC) exhibit robustness closer to lattice networks astheir topologies are derived from such regular arrangements[1] Among the latter three categories a scale-free (BA)network may be more prone to spreading than a lattice(as shown in Figure 7) because for the specific networkinstances the given BAnetwork ismore dense than the lattice(the BAhasmean degree 119896 = 16 while the lattice 119896 = 4 for thesame number of network users) The following observationsmay be derived from the above analysis

Observation 6 Topological randomness favors the spreadingof malicious software

Observation 7 Among similar network types the relative(local) density of each topology determines the robustness ofthe system against malicious information spreading


6 Conclusion

In this work we introduced novel epidemic-based modelsfor modeling and understanding information disseminationdynamics (IDD) inwireless complex networks Our approachwas inspired by epidemic approaches developed for thestudy of viruses in social communities Useful informationdissemination was modeled according to the SI epidemicmodel while malicious and indifferent types of propagatedinformation were modeled according to the SIS infectionparadigm We provided analytical approaches for obtainingthe behavior of spreading dynamics in both paradigms andin order to characterize the spreading of information inspecific and diverse types of wireless complex networksNumerical results depicted the effectiveness of the pro-posed approaches for analyzing and utilizing the developedprocesses in the described environment yielding the mostimportant characteristics of complex networks that affectand control the propagation process Useful directions wereidentified for similar studies and developing practical coun-termeasuresinfrastructures

The proposed methodology and the respective analyticalresults could be further exploited for defining more com-plex and application-oriented problems that arise in infor-mation diffusion processes in wireless complex networksFor instance these could be used in order to optimizethe spreading of useful information and designing morerobust networks capable of sustaining large-scale attacks ofmalicious spreading information Depending on the contextof each application more effective dissemination campaignscan be designed and more robust infrastructures developedagainst malicious intentions

References

[1] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[2] H Hu S Myers V Colizza and A Vespignani ldquoWiFi net-works and malware epidemiologyrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 106 no5 pp 1318ndash1323 2009

[3] G Yan H D Flores L Cuellar N Hengartner S EidenbenzandV Vu ldquoBluetooth worm propagation mobility patternmat-tersrdquo inProceedings of the 2ndACMSymposiumon InformationComputer and Communications Security (ASIACCS rsquo07) pp 32ndash44 March 2007

[4] S Staniford V Paxson and N Weaver ldquoHow to own theInternet in your spare timerdquo in Proceedings of the 11th USENIXSecurity Symposium pp 149ndash167 2002

[5] P T Eugster R Guerraoui A-M Kermarrec and L MassoulieldquoEpidemic information dissemination in distributed systemsrdquoComputer vol 37 no 5 pp 60ndash67 2004

[6] C C Zou D Towsley andW Gong ldquoModeling and simulationstudy of the propagation and defense of internet e-mail wormsrdquoIEEE Transactions on Dependable and Secure Computing vol 4no 2 pp 105ndash118 2007

[7] X Zhang G Neglia J Kurose and D Towsley ldquoPerformancemodeling of epidemic routingrdquo Computer Networks vol 51 no10 pp 2867ndash2891 2007

[8] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[9] P-Y Chen and K-C Chen ldquoInformation epidemics in complexnetworks with opportunistic links and dynamic topologyrdquo inProceedings of the 53rd IEEEGlobal Communications Conference(GLOBECOM rsquo10) Miami Fla USA December 2010

[10] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press 2001

[11] E Vynnycky and R G White An Introduction to InfectiousDisease Modelling Oxford University Press 2009

[12] P-Y Chen and K-C Chen ldquoOptimal control of epidemicinformation dissemination in mobile ad hoc networksrdquo inProceedings of the 54th Annual IEEEGlobal TelecommunicationsConference ldquoEnergizing Global Communicationsrdquo (GLOBECOMrsquo11) pp 1ndash5 Houston Tex USA December 2011

[13] M Akdere C C Bilgin O Gerdaneri I Korpeoglu O Ulusoyand U Cetintemel ldquoA comparison of epidemic algorithms inwireless sensor networksrdquo Computer Communications vol 29no 13-14 pp 2450ndash2457 2006

[14] S Boyd A Ghosh B Prabhakar and D Shah ldquoRandomizedgossip algorithmsrdquo IEEE Transactions on Information Theoryvol 52 no 6 pp 2508ndash2530 2006

[15] Z J Haas J Y Halpern and L Li ldquoGossip-based ad hocroutingrdquo IEEEACM Transactions on Networking vol 14 no 3pp 479ndash491 2006

[16] D Kempe J Kleinberg and A Demers ldquoSpatial gossip andresource location protocolsrdquo Journal of the ACM vol 51 no 6pp 943ndash967 2004

[17] W Heinzelman J Kulik and H Balakrishnan ldquoAdaptive pro-tocols for information dissemination in wsnrdquo in Proceedings ofthe 5th annual ACMIEEE International Conference on MobileComputing and Networking (MobiCom rsquo99) pp 174ndash185 1999

[18] C Anagnostopoulos S Hadjiefthymiades and E Zervas ldquoAnanalytical model for multi-epidemic information dissemina-tionrdquo Journal of Parallel and Distributed Computing vol 71 no1 pp 87ndash104 2011

[19] C Anagnostopoulos O Sekkas and S HadjiefthymiadesldquoAn adaptive epidemic information dissemination model forwireless sensor networksrdquo Pervasive andMobile Computing vol8 no 5 pp 751ndash763 2012

[20] A-M Kermarrec L Massoulie and A J Ganesh ldquoProbabilisticreliable dissemination in large-scale systemsrdquo IEEE Transac-tions on Parallel and Distributed Systems vol 14 no 3 pp 248ndash258 2003

[21] A Lindgren A Doria and O Schelen ldquoProbabilistic routing inintermittently connected networksrdquo inProceedings of the ServiceAssurance with Partial and Intermittent Resources (SAPIR rsquo04)pp 239ndash254 2004

[22] K A Harras K C Almeroth and E M Belding-Royer ldquoDelaytolerant mobile networks (DTMNs) controlled flooding insparse mobile networksrdquo in Proceedings of the 4th InternationalIFIP-TC6 Networking Conference Networking Technologies Ser-vices and Protocols Performance of Computer and Communi-cation Networks Mobile and Wireless Communications Systems(NETWORKING rsquo05) pp 1180ndash1192 May 2005

[23] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001


[24] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[25] M Vojnovic V Gupta T Karagiannis and C GkantsidisldquoSampling strategies for epidemic-style information dissemina-tionrdquo in Proceedings of the 27th IEEE Communications SocietyConference on Computer Communications (INFOCOM rsquo08) pp2351ndash2359 Phoenix Ariz USA April 2008

[26] F D Sahneh and C M Scoglio ldquoOptimal information dissem-ination in epidemic networksrdquo in Proceedings of the IEEE 51stAnnual Conference onDecision and Control (CDC rsquo12) pp 1657ndash1662 2012

[27] T G Lewis Network Science Theory and Practice John Wileyand Sons 2009

[28] A Barrat M Barthelemy and A Vespignani Dynamical Pro-cesses on Complex Networks Cambridge University Press 2008

[29] G Barrenetxea B Berefull-Lozano and M Vetterli ldquoLatticenetworks capacity limits optimal routing and queueing behav-iorrdquo IEEEACM Transactions on Networking vol 14 no 3 pp492ndash505 2006

[30] P Erdos and A Renyi ldquoOn random graphsrdquo PublicationesMathematicae (Debrecen) vol 6 pp 290ndash297 1959

[31] G Shi and K H Johansson ldquoThe role of persistent graphs inthe agreement seeking of social networksrdquo IEEE Journal onSelected Areas in Communications Special Issue on EmergingTechonlogies in Communications vol 51 no 8 2013

[32] D J Watts and S H Strogatz ldquoCollective dynamics of lsquosmall-worldrsquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[33] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999

[34] MMolloy and B Reed ldquoA critical point for random graphs witha given degree sequencerdquo Random Structures and Algorithmsvol 6 pp 161ndash180 1995

[35] E N Gilbert ldquoCapacity of a burst-noise channelrdquo Bell SystemTechnical Journal vol 39 pp 1253ndash1265 1960

[36] A Vahdat and D Becker ldquoEpidemic routing for partially-connected ad hoc networksrdquo Tech Rep CS-2000-06 DukeUniversity 2000

[37] O Ozkasap Z Genc and E Atsan ldquoEpidemic-based reliableand adaptive multicast for mobile ad hoc networksrdquo ComputerNetworks vol 53 no 9 pp 1409ndash1430 2009

[38] I Rhee M Shin S Hong K Lee and S Chong ldquoOn thelevy-walk nature of human mobilityrdquo in Proceedings of the27th IEEE Communications Society Conference on ComputerCommunications (INFOCOM rsquo08) pp 924ndash932 Phoenix ArizUSA April 2008

[39] S Kim C-H Lee and D Y Eun ldquoSuperdiffusive behavior ofmobile nodes and its impact on routing protocol performancerdquoIEEE Transactions on Mobile Computing vol 9 no 2 pp 288ndash304 2010

[40] S H Sellke N B Shroff and S Bagchi ldquoModeling and auto-mated containment of wormsrdquo IEEE Transactions on Depend-able and Secure Computing vol 5 no 2 pp 71ndash86 2008

[41] M H R Khouzani S Sarkar and E Altman ldquoMaximum dam-age malware attack in mobile wireless networksrdquo in Proceedingsof the IEEE (INFOCOM rsquo10) SanDiego Calif USAMarch 2010

[42] A Ganesh L Massoulie and D Towsley ldquoThe effect of networktopology on the spread of epidemicsrdquo in Proceedings of the 24thAnnual Joint Conference of the IEEE Computer and Communi-cations Societies (INFOCOM rsquo05) pp 1455ndash1466 March 2005

[43] V Karyotis A Kakalis and S Papavassiliou ldquoMalware-prop-agative mobile ad hoc networks asymptotic behavior analysisrdquoJournal of Computer Science and Technology vol 23 no 3 pp389ndash399 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Sensor network

Socialnetwork

Infrastructure network

Ad hoc network

Wireless LAN

Wirednetwork

Satellitenetwork

Vehicular network

Backhaul network

Physical contactSocial contact

Figure 1 Contemporary wireless complex communication network architecture presenting cumulatively all considered types of networksincluding interconnections to wired backhauls

In this work we focus especially on information dis-semination in wireless complex communication networksModern networks consist of various complex subnetworks[1] which merge as a heterogeneous large-scale networkpossibly connecting to a wired infrastructure (Figure 1)Contemporary users equipped with devices having variousradio interfaces can communicate with each other in morecomplicated ways than in the past For example a user maycommunicate with another user via mobile phone over cellu-lar networks while also communicating with a different userin hisher geographic vicinity via WiFi [2] or Bluetooth [3]To evaluate the information dissemination dynamics (IDD)in such heterogeneous complex communication networkswhere communications are affected by both social relationsand physical proximity we establish analytical models withparameters representing IDD for different types of underly-ing communication networks and exploit them to study thedynamics of IDD and obtain means of IDD control

Substantial works [4ndash9] have employed an analogy ofIDD in a single network as infectious spreading diseasesby using ordinary differential equation (ODE) in the fieldof epidemics [10 11] which could act as a quick referenceto efficiently gather approximate knowledge of informationdissemination speed and status with various settings ofaverage node degrees and attain further control [12] in thosenetworks A more thorough summary of such protocolscan be found in [13] However IDD are further complicated

in wireless complex communication networks due to thepresence of dynamic behaviors regarding the variability oftopology and user behavior Randomized approaches basedon both stochastic processes and epidemic models are thosecharacterized as epidemic routing [14ndash16] However all suchworks mainly focus on specific types of information and net-work topology Our work serves as one of the early attemptsto systematically analyze IDD in a generic manner and allowassessing the dissemination and robustness performance inboth current and future wireless complex networks anddifferent types of information

The main contribution of our work is in realizing thatdifferent dynamics are governing the propagation of differenttypes of information and provide appropriatemodels for eachcase Specifically with regard to useful data the objective is tospread such information to as many users as possible so thatuseful information reaches potentially all nodes of a networkOn the contrary in the event of maliciousindifferent infor-mation spreading the objective is to study the robustnessof various network types against harmful or indifferentbut nevertheless network-stressing content To address theabove in this paper we propose two different paradigms thatdescribe the behavior of systems a Susceptible Infected (SI)for the first and Susceptible-Infected-Susceptible (SIS) forthe second and third types of information Both paradigmswere inspired by drawing analogies to the field of epidemics[10] We then propose and analyze two specific approaches

























Equallyconnectible


Homogeneousmixing



network

Heterogeneousmixing



Scale-freenetwork







119875 (119896) =

(119890minus119896119890119896119896

119890)

119896

(1)

where 119896119890= 119873119901










119875 (119896)

=

min(119896minus119895119895)

sum

119899=0

(119895

119899)(1 minus 119901

119908)119899

119901119895minus119899

119908

(119901119908119895)119896minus119895minus119899


0 otherwise(2)









119887= 2119898 The power-








I (119905) + S (119905) = 1

119889I (119905)119889119905

= 120582119896I (119905) S (119905) (3)


0 20 40 60 80 100 120 140 160 180 200


0 100 2000

5

10

15

20(d

eg)

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion

Time

Time

t0

t0

WS model pw = 1

WS model pw = 04





2500


I (119905) = I (0)I (0) + (1 minus I (0)) 119890minus119896119905

(4)



119890













119908isin (0 1)) and



119889I (119905)119889119905

= 120582I (119905) S (119905) = 120582 [119896 minus 119891 (119905)] I (119905) S (119905) (5)






119891 (119905) = 119896 sdot120590120587[119903 (119905) minus 119877]

2








0 10 20 30Time

40 50 60 70 800

01

02

03

04

05

06

07

08

09

1

Info

rmed

frac

tion

Simulation N = 2000

Analysis f(t) = 1

Simulation N = 1000

Analysis f(t) = 03

Simulation N = 500

Analysis f(t) = 01

Simulation N = 100

Analysis f(t) = 015











119890(119905)+ I119904(119905)

where I119890(119905) and I





119889I119890(119905)

119889119905= 120582

S (119905) (119896119890minus 1)

119873I (119905) (7)


119904equals 1205881205871198772






1198661015840(1199051 1199052) ≜

119889119866 (1199051 1199052)

1198891199052

= 120582S (1199051+ 1199052) sdot (12) 119896

119904

119873119888radic119866 (119905

1 1199052)

(8)



119889I119904(119905)

119889119905= int

119905

0

I1015840119890(120591) 1198661015840(120591 119905 minus 120591) 119889120591 (9)




119890=











Informed circle








120587 (119894) = 120587 (0) sdot

119894

prod

119895=1

120582 (119873 + 1 minus 119894 + 119895)

120583 (119895) (10)


119873

119894=0120587(119894) = 1


0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion



119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)



119894= 120582 and 120583




120587 (0) cong120572119873

119873sdot 119890minus120572 (11)


120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)






119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)



119894=



1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890


2]

(14)


21198712 repre-




0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Equallyconnectible


Homogeneousmixing



network

Heterogeneousmixing



Scale-freenetwork







119875 (119896) =

(119890minus119896119890119896119896

119890)

119896

(1)

where 119896119890= 119873119901










119875 (119896)

=

min(119896minus119895119895)

sum

119899=0

(119895

119899)(1 minus 119901

119908)119899

119901119895minus119899

119908

(119901119908119895)119896minus119895minus119899


0 otherwise(2)









119887= 2119898 The power-








I (119905) + S (119905) = 1

119889I (119905)119889119905

= 120582119896I (119905) S (119905) (3)


0 20 40 60 80 100 120 140 160 180 200


0 100 2000

5

10

15

20(d

eg)

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion

Time

Time

t0

t0

WS model pw = 1

WS model pw = 04





2500


I (119905) = I (0)I (0) + (1 minus I (0)) 119890minus119896119905

(4)



119890













119908isin (0 1)) and



119889I (119905)119889119905

= 120582I (119905) S (119905) = 120582 [119896 minus 119891 (119905)] I (119905) S (119905) (5)






119891 (119905) = 119896 sdot120590120587[119903 (119905) minus 119877]

2








0 10 20 30Time

40 50 60 70 800

01

02

03

04

05

06

07

08

09

1

Info

rmed

frac

tion

Simulation N = 2000

Analysis f(t) = 1

Simulation N = 1000

Analysis f(t) = 03

Simulation N = 500

Analysis f(t) = 01

Simulation N = 100

Analysis f(t) = 015











119890(119905)+ I119904(119905)

where I119890(119905) and I





119889I119890(119905)

119889119905= 120582

S (119905) (119896119890minus 1)

119873I (119905) (7)


119904equals 1205881205871198772






1198661015840(1199051 1199052) ≜

119889119866 (1199051 1199052)

1198891199052

= 120582S (1199051+ 1199052) sdot (12) 119896

119904

119873119888radic119866 (119905

1 1199052)

(8)



119889I119904(119905)

119889119905= int

119905

0

I1015840119890(120591) 1198661015840(120591 119905 minus 120591) 119889120591 (9)




119890=











Informed circle








120587 (119894) = 120587 (0) sdot

119894

prod

119895=1

120582 (119873 + 1 minus 119894 + 119895)

120583 (119895) (10)


119873

119894=0120587(119894) = 1


0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion



119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)



119894= 120582 and 120583




120587 (0) cong120572119873

119873sdot 119890minus120572 (11)


120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)






119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)



119894=



1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890


2]

(14)


21198712 repre-




0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119875 (119896)

=

min(119896minus119895119895)

sum

119899=0

(119895

119899)(1 minus 119901

119908)119899

119901119895minus119899

119908

(119901119908119895)119896minus119895minus119899


0 otherwise(2)









119887= 2119898 The power-








I (119905) + S (119905) = 1

119889I (119905)119889119905

= 120582119896I (119905) S (119905) (3)


0 20 40 60 80 100 120 140 160 180 200


0 100 2000

5

10

15

20(d

eg)

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion

Time

Time

t0

t0

WS model pw = 1

WS model pw = 04





2500


I (119905) = I (0)I (0) + (1 minus I (0)) 119890minus119896119905

(4)



119890













119908isin (0 1)) and



119889I (119905)119889119905

= 120582I (119905) S (119905) = 120582 [119896 minus 119891 (119905)] I (119905) S (119905) (5)






119891 (119905) = 119896 sdot120590120587[119903 (119905) minus 119877]

2








0 10 20 30Time

40 50 60 70 800

01

02

03

04

05

06

07

08

09

1

Info

rmed

frac

tion

Simulation N = 2000

Analysis f(t) = 1

Simulation N = 1000

Analysis f(t) = 03

Simulation N = 500

Analysis f(t) = 01

Simulation N = 100

Analysis f(t) = 015











119890(119905)+ I119904(119905)

where I119890(119905) and I





119889I119890(119905)

119889119905= 120582

S (119905) (119896119890minus 1)

119873I (119905) (7)


119904equals 1205881205871198772






1198661015840(1199051 1199052) ≜

119889119866 (1199051 1199052)

1198891199052

= 120582S (1199051+ 1199052) sdot (12) 119896

119904

119873119888radic119866 (119905

1 1199052)

(8)



119889I119904(119905)

119889119905= int

119905

0

I1015840119890(120591) 1198661015840(120591 119905 minus 120591) 119889120591 (9)




119890=











Informed circle








120587 (119894) = 120587 (0) sdot

119894

prod

119895=1

120582 (119873 + 1 minus 119894 + 119895)

120583 (119895) (10)


119873

119894=0120587(119894) = 1


0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion



119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)



119894= 120582 and 120583




120587 (0) cong120572119873

119873sdot 119890minus120572 (11)


120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)






119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)



119894=



1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890


2]

(14)


21198712 repre-




0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120 140 160 180 200


0 100 2000

5

10

15

20(d

eg)

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion

Time

Time

t0

t0

WS model pw = 1

WS model pw = 04





2500


I (119905) = I (0)I (0) + (1 minus I (0)) 119890minus119896119905

(4)



119890













119908isin (0 1)) and



119889I (119905)119889119905

= 120582I (119905) S (119905) = 120582 [119896 minus 119891 (119905)] I (119905) S (119905) (5)






119891 (119905) = 119896 sdot120590120587[119903 (119905) minus 119877]

2








0 10 20 30Time

40 50 60 70 800

01

02

03

04

05

06

07

08

09

1

Info

rmed

frac

tion

Simulation N = 2000

Analysis f(t) = 1

Simulation N = 1000

Analysis f(t) = 03

Simulation N = 500

Analysis f(t) = 01

Simulation N = 100

Analysis f(t) = 015











119890(119905)+ I119904(119905)

where I119890(119905) and I





119889I119890(119905)

119889119905= 120582

S (119905) (119896119890minus 1)

119873I (119905) (7)


119904equals 1205881205871198772






1198661015840(1199051 1199052) ≜

119889119866 (1199051 1199052)

1198891199052

= 120582S (1199051+ 1199052) sdot (12) 119896

119904

119873119888radic119866 (119905

1 1199052)

(8)



119889I119904(119905)

119889119905= int

119905

0

I1015840119890(120591) 1198661015840(120591 119905 minus 120591) 119889120591 (9)




119890=











Informed circle








120587 (119894) = 120587 (0) sdot

119894

prod

119895=1

120582 (119873 + 1 minus 119894 + 119895)

120583 (119895) (10)


119873

119894=0120587(119894) = 1


0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion



119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)



119894= 120582 and 120583




120587 (0) cong120572119873

119873sdot 119890minus120572 (11)


120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)






119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)



119894=



1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890


2]

(14)


21198712 repre-




0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (119905) = 119896 sdot120590120587[119903 (119905) minus 119877]

2








0 10 20 30Time

40 50 60 70 800

01

02

03

04

05

06

07

08

09

1

Info

rmed

frac

tion

Simulation N = 2000

Analysis f(t) = 1

Simulation N = 1000

Analysis f(t) = 03

Simulation N = 500

Analysis f(t) = 01

Simulation N = 100

Analysis f(t) = 015











119890(119905)+ I119904(119905)

where I119890(119905) and I





119889I119890(119905)

119889119905= 120582

S (119905) (119896119890minus 1)

119873I (119905) (7)


119904equals 1205881205871198772






1198661015840(1199051 1199052) ≜

119889119866 (1199051 1199052)

1198891199052

= 120582S (1199051+ 1199052) sdot (12) 119896

119904

119873119888radic119866 (119905

1 1199052)

(8)



119889I119904(119905)

119889119905= int

119905

0

I1015840119890(120591) 1198661015840(120591 119905 minus 120591) 119889120591 (9)




119890=











Informed circle








120587 (119894) = 120587 (0) sdot

119894

prod

119895=1

120582 (119873 + 1 minus 119894 + 119895)

120583 (119895) (10)


119873

119894=0120587(119894) = 1


0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion



119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)



119894= 120582 and 120583




120587 (0) cong120572119873

119873sdot 119890minus120572 (11)


120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)






119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)



119894=



1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890


2]

(14)


21198712 repre-




0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119890(119905)+ I119904(119905)

where I119890(119905) and I





119889I119890(119905)

119889119905= 120582

S (119905) (119896119890minus 1)

119873I (119905) (7)


119904equals 1205881205871198772






1198661015840(1199051 1199052) ≜

119889119866 (1199051 1199052)

1198891199052

= 120582S (1199051+ 1199052) sdot (12) 119896

119904

119873119888radic119866 (119905

1 1199052)

(8)



119889I119904(119905)

119889119905= int

119905

0

I1015840119890(120591) 1198661015840(120591 119905 minus 120591) 119889120591 (9)




119890=











Informed circle








120587 (119894) = 120587 (0) sdot

119894

prod

119895=1

120582 (119873 + 1 minus 119894 + 119895)

120583 (119895) (10)


119873

119894=0120587(119894) = 1


0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion



119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)



119894= 120582 and 120583




120587 (0) cong120572119873

119873sdot 119890minus120572 (11)


120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)






119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)



119894=



1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890


2]

(14)


21198712 repre-




0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Informed circle








120587 (119894) = 120587 (0) sdot

119894

prod

119895=1

120582 (119873 + 1 minus 119894 + 119895)

120583 (119895) (10)


119873

119894=0120587(119894) = 1


0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion



119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)



119894= 120582 and 120583




120587 (0) cong120572119873

119873sdot 119890minus120572 (11)


120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)






119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)



119894=



1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890


2]

(14)


21198712 repre-




0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30Time

40 50 60

1

09

08

07

06

05

04

03

02

01

0

Info

rmed

frac

tion



119890= 6 119896

119887= 3 and 120582 = 005

Nminus i

i

120582(N minus i + 1)

N packets

120583(i)



119894= 120582 and 120583




120587 (0) cong120572119873

119873sdot 119890minus120572 (11)


120587 (119894) =120572119873minus119894

(119873 minus 119894)sdot 119890minus120572

= 1205871015840(119873 minus 119894) (12)






119864 [119894] =

119873

sum

119894=1

119894 sdot 120587 (119894) = 119873 minus 119888 (13)



119894=



1198641015840[120574] =

119873

sum

119894=1

120582 (119894) sdot 1205871015840(119894)

=120582119896

119873[119888 (119873 + 1) minus (1 minus 119890


2]

(14)


21198712 repre-




0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 025 05 075 1 125 15 175 20

20

40

60

80

100

120

140

160

180

200

Expe

cted

infe

ctio

nE[i]


Ad hocAd hoc

R = 200mR = 300m

ER model p = 07

ER model p = 03


WS model j = 4

BA model m = 8







100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Expe

cted

num

ber o

f inf

ecte

d no

desE

[i]


R = 200 120582120583 = 025

R = 200 120582120583 = 075

R = 300 120582120583 = 025

R = 300 120582120583 = 075







6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6 Conclusion



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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