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Coalition Formation Games for CollaborativeSpectrum Sensing

Walid Saad, Zhu Han, Tamer Basar, Merouane Debbah, and AreHjørungnes

Abstract—Collaborative Spectrum Sensing (CSS) between sec-ondary users (SUs) in cognitive networks exhibits an inherenttradeoff between minimizing the probability of missing thedetection of the primary user (PU) and maintaining a reasonablefalse alarm probability (e.g., for maintaining a good spectrumutilization). In this paper, we study the impact of this tradeoffon the network structure and the cooperative incentives ofthe SUs that seek to cooperate for improving their detectionperformance. We model the CSS problem as a non-transferablecoalitional game, and we propose distributed algorithms forcoalition formation. First, we construct a distributed coalitionformation (CF) algorithm that allows the SUs to self-organizeinto disjoint coalitions while accounting for the CSS tradeoff.Then, the CF algorithm is complemented with a coalitionalvoting game for enabling distributed coalition formation withdetection probability guarantees (CF-PD) when required bythePU. The CF-PD algorithm allows the SUs to form minimalwinning coalitions (MWCs), i.e., coalitions that achieve the targetdetection probability with minimal costs. For both algorit hms,we study and prove various properties pertaining to networkstructure, adaptation to mobility and stability. Simulati on resultsshow that CF reduces the average probability of miss per SU upto 88.45% relative to the non-cooperative case, while maintaininga desired false alarm. For CF-PD, the results show that up to87.25% of the SUs achieve the required detection probabilitythrough MWCs.

I. I NTRODUCTION

Recently, there has been a surge in wireless services,yielding a huge demand on the radio spectrum. However, thespectrum resources are scarce and most of them have beenalready licensed to existing operators. Various studies haveshown that the actual licensed spectrum remains unoccupiedfor large periods [1]. For efficiently exploiting these spectrumholes,cognitive radiohas been proposed [2]. By monitoringand adapting to the environment, cognitive radios (secondaryusers (SUs)) can share the spectrum with the licensed user(primary user (PU)), operating whenever the PU is not usingthe spectrum. However, deploying and implementing suchflexible SUs faces several challenges [3] at different levelssuch as spectrum access [3]–[5] and spectrum sensing [6]–[15]. Spectrum sensing mainly deals with the stage duringwhich the cognitive users attempt to learn their environment

Copyright (c) 2010 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

W. Saad and A. Hjørungnes are with the UNIK Graduate University Center,University of Oslo, Oslo, Norway, e-mails:{saad,arehj}@unik.no.Z. Han is with Electrical and Computer Engineering Department, Uni-versity of Houston, Houston, Tx, USA, email:zhan2@mail.uh.edu.T. Basar is with the Coordinated Science Laboratory, University of Illi-nois at Urbana Champaign, IL, USA, email:basar1@illinois.edu.M. Debbah is the Alcatel-Lucent chair, SUPELEC, Paris, France e-mail:merouane.debbah@supelec.fr. This research is supported by theResearch Council of Norway through the projects 183311/S10, 176773/S10and 18778/V11, and through a MURI Grant at UIUC. A preliminary versionof this work [50] appeared in the Proceedings of the IEEE Conference onComputer Communications (INFOCOM), April, 2009.

prior to accessing and sharing the spectrum. For instance, theSUs must constantly sense the spectrum in order to detect thepresence of the PU and use the spectrum holes without causingharmful interference to the PU. Hence, efficient and distributedspectrum sensing is of utter importance and constitutes a majorchallenge in cognitive radio networks.

For sensing the PU presence, the SUs must be able to detectits signal using various kinds of detectors such as energydetectors or others [6]. However, the performance of spectrumsensing is significantly affected by the degradation of the PUsignal due to path loss or shadowing (hidden terminal). Ithas been shown that, through collaborative spectrum sens-ing (CSS) among SUs, the effects of this hidden terminal prob-lem can be reduced and the probability of detecting the PU canbe improved [7]–[10], [12]–[15]. In [7], the SUs collaborateby sharing their sensing decisions through a centralized fusioncenter which combines the SUs’ sensing bits using the OR-rule for data fusion. A similar approach is used in [9] usingdifferent decision-combining methods. In [10], it is shownthat, in CSS, soft decisions can have an almost comparableperformance with hard decisions while reducing complexity.The authors in [12] propose an evolutionary game model forCSS in order to inspect the strategies of the SUs and theircontribution to the sensing process. The effect of the sensingtime on the access performance of the SUs in a cognitivenetwork is analyzed in [11]. For improving the performanceof CSS, spatial diversity techniques are presented in [13] as ameans for combatting the error probability due to fading on thereporting channel between the SUs and the central fusion cen-ter. Other interesting performance aspects of spectrum sensingand CSS are studied in [8], [14]–[18]. Existing literaturemainly focused on the performance assessment of CSS in thepresence of a centralized fusion center that combinesall theSUs bits. In practice, the SUs can be at different locationsin the network, and, thus, prefer to form nearby groups forCSS without relying on a centralized entity. Moreover, theSUs can belong to different service providers and need tointeract with each other for CSS, instead of relaying their bitsto a centralized fusion center which may not even exist in anad hocnetwork of SUs. In addition, a centralized approachrequires the deployment and/or re-use of an infrastructureforthe purpose of spectrum sensing, which is quite impracticalas many secondary networks are inherently ad hoc. Further,the use of a centralized spectrum sensing approach can yieldasignificant overhead and complexity, notably in networks witha large number of SUs having varying locations.

The main contribution of this paper is to devise distributedstrategies for CSS between the SUs in a cognitive networkwhich capture the spectrum sensing objectives of these SUs.Afurther contribution of this paper is to inspect how the networkstructure can evolve and change as a result of the cooperative

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incentives of the SUs seeking to improve their spectrumsensing performance, given the inherent tradeoff that existsbetween the CSS gains in terms of reducing the probabilityof missing the detection of the PU and the cooperation costsin terms of maintaining a reasonable false alarm probability(e.g., to maintain an efficient spectrum utilization). We modelthe CSS problem as a non-transferable coalitional game andwe propose a distributed coalition formation (CF) algorithm.Using CF, each SU autonomously decides to form or leavea coalition while maximizing its utility in terms of detectionprobability and accounting for the false alarm cost. We showthat, due to the cost for cooperation, independent disjointcoali-tions will form in the network and a maximum coalition sizeexists for the proposed utility model. For cognitive networkswhere the PU imposes a detection probability level on theSUs, we propose a distributed coalition formation algorithmwith detection probability guarantees (CF-PD) which is builtby overlaying a coalitional voting game on the CF algorithm.The CF-PD algorithm enables the SUs to autonomously takethe decision to form minimal winning coalitions (MWCs), i.e.,coalitions that achieve the target detection probability withminimal costs. Through simulations, we assess the perfor-mance of both algorithms relative to the non-cooperative case,we compare the results with a centralized solution, and weshow how the SUs can self-organize and adapt the networkstructure to environmental changes such as mobility.

Note that, although some existing literature such as [19] and[20] have dealt with the idea of forming clusters for CSS, theobjectives of [19] and [20] are different from the contributionsof our paper. For instance, [19] studies, in the context of thecentralized fusion center based CSS, the possibility of usingclusters for combatting theerrors on the reporting channel.However, the authors do not deal with the idea of distributedCSS and, as clearly stated in [19, Section III], the paper doesnot propose any algorithm for forming the clusters, instead itassumes that the clusters are pre-formed. Moreover, [19] doesnot account for the tradeoff between the probabilities of missand false alarm, which impacts the size and identities of theSUs that will form coalitions or clusters. In this context, ourpaper complements [19] by having a number of different andnovel contributions as discussed in the previous paragraph.

Further, in [20], the authors focus on studying the perfor-mance of CSS in a geographical area, using different detectiontechniques such as energy detection or feature detection. Theidea of clustering in [20] is restricted to dividing, based onthe received signal strength of the PU, a large geographicalarea into a number of defined clusters which are geographicalareas with a large radius, i.e., analogous to the cells of awireless cellular system. Within every cluster or cell, thework in [20] considers that a large number of SUs exist andthese SUs utilize the traditional network-wide fusion centersolution at cell-level. Moreover, in [20], no target probabilityof miss or false alarm constraints are taken into account. Inshort, [20] does not look at distributed strategies enablingthe SUs to, autonomously, form coalitions while optimizingtheir probability of miss given a false alarm constraint. Inessence, our approach complements the work done in [20].For instance, given a cognitive network deployed in a large

geographical area, one can first cluster it geographically intocells using the approach [20] and, then, our proposed coalitionformation approach can be applied by the SUs inside eachcluster or cell.

The rest of this paper is organized as follows: Section IIpresents the system model. In Section III, we present the pro-posed coalitional game and prove different properties while inSection IV we devise a distributed algorithm for autonomouscoalition formation. Section V extends the coalition formationalgorithm to accommodate probability of detection guarantees.In Section VI, we discuss implementation aspects in practicalcognitive networks. Simulation results are presented and ana-lyzed in Section VII. Finally, conclusions are drawn in SectionVIII.

II. SYSTEM MODELConsider a network consisting ofN transmit-receive pairs

of SUs and a single PU withN = {1, . . . , N} denoting theset of all SUs. The SUs and the PU can be either stationaryor mobile. Throughout this paper, we assume that all SUs arehonest nodes with no malicious or cheating behavior. Sincethe focus is on spectrum sensing, we are only interested in thetransmitter part of each of theN SUs. In a non-cooperativeapproach, each of theN SUs continuously senses the spectrumin order to detect the presence of the PU. For detecting the PU,we use energy detectors which are one of the main practicalsignal detectors in cognitive radio networks [7], [9], [13].In such a non-cooperative setting with Rayleigh fading, theprobabilities of miss (i.e. probability of missing the detectionof the PU) and false alarm for an SUi are, respectively, givenby Pm,i andPf,i [7, Eqs. (2), (4)]

Pm,i = 1− e−λ2

m−2∑

n=0

1

n!

(

λ

2

)n

+

(

1 + γi,PU

γi,PU

)m−1

×

[

e− λ

2(1+γi,PU) − e−λ2

m−2∑

n=0

1

n!

(

λγi,PU

2(1 + γi,PU )

)n]

, (1)

Pf,i = Pf =Γ(m, λ

2 )

Γ(m), (2)

where m is the time bandwidth product,λ is the energydetection threshold assumed to be the same for all SUs withoutloss of generality as in [7], [9], [13],Γ(·, ·) is the incompletegamma function andΓ(·) is the gamma function. Moreover,γirepresents the average SNR of the received signal from the PUto SU given byγi,PU =

PPUhPU,i

σ2 with PPU the transmit power ofthe PU,σ2 the Gaussian noise variance, andhPU,i = κ/dµPU,ithe path loss between the PU and SUi, κ is the path lossconstant,µ the path loss exponent, anddPU,i is the distancebetween the PU and SUi.

The expression in (1) averages out the effect of the small-scale fading, and, thus, in the remainder of this paper, weonly deal with large-scale fading effects, e.g., due to pathloss.Further, it is also important to note that the non-cooperativefalse alarm probability expression dependssolely on the de-tection thresholdλ and does not depend on the SU’s location;hence, we dropped the subscripti in (2). Note that we do notaccount for the shadowing effect, however, in future work, (1)can be modified to take into account this effect as shown in[3], [7], [9]. Moreover, an important metric is the probability

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of detection of the PU by a SUi, which is simply defined asPd,i = 1− Pm,i.

In order to minimize the interference on the PU and reducethe probability of miss, the SUs can cooperate by formingcoalitions. Within each coalitionS ⊆ N , an SU designated ascoalition head, collects the sensing bits from the coalition’sSUs and acts as a fusion center in order to make a coalition-based decision on the presence or absence of the PU. Thiscan be seen as having the centralized CSS of [7], [13] appliedat the level of each coalition with the coalition head beingthe fusion center to which all the coalition members report.For combining the sensing bits and making the final detectiondecision, the coalition head uses the decision fusion OR-rulesuch as in [7], [13]. For a coalitionS having coalition headk ∈ S, using CSS, the miss and false alarm probabilities are,respectively, given by [13]

Qm,S =∏

i∈S

[Pm,i(1− Pe,i,k) + (1 − Pm,i)Pe,i,k] , (3)

Qf,S = 1−∏

i∈S

[(1− Pf )(1− Pe,i,k) + PfPe,i,k] , (4)

where Pm,i, Pf are given by (1), (2), andPe,i,k is theprobability of error due to the fading on the reporting channelbetween the SUi of coalition S and the coalition headk.The error over the reporting channel is an important metricthat affects the performance of CSS in terms of probability ofmiss as well as false alarm as demonstrated in [13]. The mainidea is that, whenever an SU needs to report its sensing bit tothe coalition head, the coalition head might receive this bit inerror due to the fading over the wireless channel between theSU and the coalition head. This reporting errorPe,i,k, betweenan SU i and the coalition headk, increases the probabilityof miss and increases the false alarm as shown in in (3)and (4). Inside any coalitionS, assuming BPSK modulationin Rayleigh fading environments, the average probability ofreporting error between SUi ∈ S and the coalition headk ∈ Sis given by [21]

Pe,i,k =1

2

(

1−

√

γi,k1 + γi,k

)

, (5)

where γi,k =Pihi,k

σ2 is the average SNR for bit reportingbetween SUi and the coalition headk insideS with Pi thetransmit power of SUi used for reporting the sensing bit tok andhi,k = κ

dµ

i,k

the path loss between SUi and coalition

headk. Any SU can be chosen as a coalition head within acoalition. However, for the remainder of this paper, we adoptthe following convention without loss of generality.

Convention 1: Within a coalitionS, the SUk ∈ S havingthe lowest non-cooperative probability of missPm,k is chosenas coalition head. Hence, the coalition headk of a coalitionS is given byk ∈ argmin

i∈S

Pm,i with Pm,i given by (1)1.

The motivation behind Convention 1 is that, due to the erroron the reporting channel, whenever an SU needs to report itssensing bit to a coalition head, there is a potential “risk” that

1Note that, if more than one SU in a coalition achieves the minimumprobability of miss, then, the coalition head is picked at random among theset of SUs with minimum probability of miss.

the bit is received in error as per (3)-(5). This risk motivates theadoption of Convention 1 whereby the SU with minimum non-cooperative miss probability, i.e., the SU with the best channeltowards the PU, is used as a coalition head, and, thus, does notneed to send its bit (which is the most reliable in the coalition)to other SUs in the coalition and risk the reception of this bitin error, subsequently, affecting the performance in termsofprobability of miss and false alarm. It must be noted that theanalysis and algorithm discussed in the remainder of this papercan accommodate other coalition head selection approaches(e.g., select the SU closest to the center of a coalition).

It should be clear from (3) and (4) that as the numberof SUs per coalition increases, the probability of miss willdecrease while the probability of false alarm will increase.This is a crucial tradeoff in CSS that can have a major impacton the collaboration strategies of each SU. Thus, our objectiveis to characterize the network structure that will form whenthe SUs collaborate while accounting for this tradeoff. Anexample of the sought network structure is shown in Fig. 1.Finding the optimal coalition structure, such as in Fig. 1,using a centralized approach requires finding the partitionofthe set that minimizes the average probability of miss perSU, while maintaining a false alarm constraint per SU. Forinstance, in the centralized approach, the optimization problemaims at minimizing the overall system’s average probabilityof miss per SU given a constraint on the average false alarmprobability per SU. In other words, denotingB as the set ofall partitions of N , the centralized approach seeks to solvethe following optimization problem

minT ∈B

∑

S∈T |S| ·Qm,S

N, s.t.Qf,S ≤ α ∀ S ∈ T ,

where | · | represents the cardinality of a set operator andSis a coalition belonging to the partitionT . Note that, in thiscentralized optimization problem it is considered that thefalsealarm probability constraint per SU directly maps to a falsealarm probability constraintper coalitionwhich will be moreformally demonstrated in the next section through Property1.

However, it is shown in [22] that finding the optimal coali-tion structure in a centralized manner leads to an optimizationproblem which is NP-complete. This is mainly due to the factthat the number of possible coalition structures (partitions),is given by a value known as the Bell number which growsexponentially withN [23]. Moreover, in a centralized partitionsearch approach, the objective is typically to optimize theoverall system performance while ignoring the individualpreferences and payoffs of the SUs (or coalition of SUs).Hence, for the centralized approach, a particular SU mightbe asked to be part of a coalitionS1 even if this SU, basedon its own payoff, prefers to be part of another coalitionS2. Since the SUs act independently and are typically self-interested, they should be given the choice of making theircoalition formation decisions on their own. Hence, there isa motivation to devise distributed approaches for the SUsto make autonomous decisions, self-organize, and form thecoalitional structure.

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Fig. 1. An illustrative example of coalition formation for collaborativespectrum sensing among SUs.

III. C OLLABORATIVE SPECTRUM SENSING AS

COALITIONAL GAME

A. Game Formulation and Properties

For devising suitable cooperative strategies for CSS amongthe SUs, we refer to coalitional game theory [24]–[26]. Wemodel distributed CSS as a coalitional game with a non-transferable utility which is defined as follows [24]2

Definition 1: A coalitional game with non-transferable util-ity (NTU) is defined by a pair(N , V ) whereN is the setof players andV is a mapping such that for every coalitionS ⊆ N , V (S) is a closed convex subset ofRS that containsthe payoff vectors that players inS can achieve.

In other words, a coalitional game(N , V ) is said to bewith NTU if the value or utility of a coalition cannot bearbitrarily apportioned between the coalition’s players.Hence,the value of any coalitionS can be mapped to a setV of payoffvectors. First, we define the value (utility)v(S) of a coalitionS ⊆ N as a function that captures the tradeoff between theprobability of detection and the probability of false alarm.For this purpose,v(S) must be an increasing function of thedetection probabilityQd,S = 1−Qm,S within coalitionS anda decreasing function of the false alarm probabilityQf,S asfollows.v(S)=Qd,S−C(Qf,S , αS)=(1 −Qm,S)−C(Qf,S, αS), (6)

whereQm,S is the probability of miss for coalitionS given by(3) andC(Qf,S , αS) is a cost function which depends on thefalse alarm probabilityQf,S and on a false alarm constraintαS that S must not exceed when it forms. Without loss ofgenerality, we assumeαS = α, ∀S ⊆ N .

In a coalitional game, one has to distinguish between twodistinct quantities: The value function and the payoff. Givena coalitionS, the value functionv(S) describes the overallutility that the entire coalition S receives when its actingcooperatively. In contrast, for any SUi ∈ S the payoffφi(S)

2This is the general definition of an NTU game as per [24], [25],someliterature may append other properties to the definition which are not usefulfor the game class dealt with in this paper (for more information on the variousdefinitions of an NTU game see [24]–[26]).

describes the utility that SUi receiveswhen acting as part ofS. Hence,φi(S) represents the tradeoff between probabilityof miss and probability of false alarm that SUi achieves whenacting as part of coalitionS. In the proposed CSS game, wehave the following property:

Property 1: The probabilities of miss and false alarm forany SUi ∈ S are given by the probabilities of miss and falsealarm of coalitionS in (3) and (4), respectively. Hence, inthe proposed CSS game, the payoff,φi(S), of any SUi ∈ Sis simply equal to the value of the coalition, i.e.,φi(S) =v(S), ∀ i ∈ S. As a result, the false alarm constraint percoalitionα maps to a false alarm constraint per SU.

This property is an immediate result of the operation of CSSwhereby the SUs belonging to a coalitionS will transmit ornot based on the final coalition head decision after data fusion.Thus, the miss and false alarm probabilities of any SUi ∈ Sare the miss and false alarm probabilities of the coalitionS towhich i belongs as given byQm,S andQf,S in (3) and (4),respectively. As an immediate consequence of this property,we notice that, in the proposed CSS game, the value of acoalition, i.e.,v(S), is not divisible among the SUs since thepayoff of everySU inS is equal tov(S) (and not to a fractionof v(S)). Hence, the proposed CSS game can be modeled as acoalitional game(N , V ) with NTU whereV (S) is asingletonset,

V (S) = {φ(S) ∈ RS | φi(S) = v(S), ∀i ∈ S}, (7)

with v(S) given by (6).(N , V ) is clearly an NTU game sincethe setV (S) is a singleton, and hence closed and convex.

In a coalitional game withno cost, the main interest wouldbe in characterizing the properties and stability of the grandcoalition of all players since it is generally assumed thatthe grand coalition maximizes the utilities of the players[24] (in such a no cost case, concepts such as the core aresuitable solutions and additional properties forV may berequired). However, for the proposed(N , V ) game, althoughCSS improves the detection probability for the SUs, the falsealarm costs limit this gain, and thus, we remark the following:

Remark 1: For the proposed(N , V ) coalitional game, thegrand coalition of all the SUsseldom forms due to thefalse alarm costs resulting from cooperation. Instead, disjointindependent coalitions form in the network.

By inspectingQm,S in (3) and through the results shown in[7], [13] (for centralized CSS) it is clear that, as the numberof SUs in a coalition increases,Qm,S decreases and theperformance improves. Hence, when no cost for collaborationexists, the grand coalition of all SUs is the optimal structurefor maximizing the detection probability. However, when thenumber of SUs in a coalitionS increases, through (3), thefalse alarm probability increases and can exceed the constraintα, hence giving no incentive for large coalitions to form andaffecting the collaboration strategies of the SUs. Therefore,for the proposed CSS model with cost for collaboration, thegrand coalition of all SUs will seldom form due to the falsealarm cost and constraint as accounted for in (6). Due to thisproperty, we deal with a class of coalitional games knownas coalition formation games [25]. Hence, the solution for the

5

proposed(N , V ) CSS coalition formation game is the networkstructure which can characterize the CSS trade off betweengains (probability of miss) and costs (false alarm).

B. Cost Function

Any well designed cost functionC(Qf,S , α) in (6) mustsatisfy several requirements for adequately modeling the falsealarm. On one hand,C(Qf,S , α) must be an increasingfunction of Qf,S with the slope becoming steeper asQf,S

increases. On the other hand,C(Qf,S , α) must impose amaximum tolerable false alarm probabilityα that cannot beexceeded by any SU (as per Property 1, imposing a false alarmconstraint on the coalition maps to a constraint per SU). Awell suited cost function satisfying these requirements isthelogarithmic barrier penalty function given by [27]

C(Qf,S , α) =

−α2 · log

(

1−(

Qf,S

α

)2)

, if Qf,S < α,

+∞, if Qf,S ≥ α,

(8)

where log is the natural logarithm andα is the false alarmconstraint per coalition. The cost function in (8) incurs apenalty which is increasing with the false alarm probability.Moreover, it imposes a maximum false alarm probabilityαper coalition (per SU by Property 1). In addition, as thefalse alarm probability gets closer toα, the collaborationcost increases steeply, requiring a significant improvement indetection probability if the SUs wish to collaborate as per (6).Note that the proposed cost function depends on both distanceand the number of SUs in the coalition, through the false alarmprobability Qf,S (the distance lies within the probability oferror). Hence, the cost for collaboration increases with thenumber of SUs in the coalition as well as when the distancebetween the coalition’s SUs increases.

IV. COALITION FORMATION GAMES IN CSS

A. Distributed Coalition Formation Algorithm (CF)

Coalition formation is a branch of game theory that inves-tigates algorithms for studying the coalitional structures thatform in a network where there is a cost for forming coalitionsand where the grand coalition is not optimal [22], [25],[26], [28]. For constructing a distributed coalition formationalgorithm for CSS, we use the following concepts [25], [28].

Definition 2: A collectionS is the setS = {S1, . . . , Sl}of mutually disjoint coalitionsSi ⊆ N . If the collection spansall the players ofN , i.e.,

⋃lj=1 Sj = N , the collection is a

partition of N .Definition 3: A comparison relation⊲ is defined for

comparing two collectionsR = {R1, . . . , Rl} and S ={S1, . . . , Sm} that are partitions of the same subsetA ⊆ N(same players inR andS). Thus,R⊲S implies that the wayR partitionsA is preferred to the wayS partitionsA basedon criteria defined next.

Various criteria (referred to asorders) can be used ascomparison relations between collections or partitions [25],[28]. Due to the non-transferable nature of the proposed(N , V ) CSS game, an individual value order that compares the

individual payoffs of the players must be used as a comparisonrelation⊲. An adequate individual value order that can be usedin CSS is thePareto order[25], [28]. Denote for a collectionR = {R1, . . . , Rl}, the utility of a playerj in a coalitionRj ∈ R by φj(R) = φj(Rj) = v(Rj) (Property 1); thePareto order is defined as

R⊲ S ⇐⇒ {φj(R) ≥ φj(S) ∀ j ∈ R,S}, (9)

with at least one strict inequality(>) for a playerk.Using the Pareto order as a comparison relation, we propose

a distributed coalition formation algorithm based on two rulescalled “merge” and “split” that allow to modify a partitionTof the SUs setN as follows [28].

Definition 4: Merge Rule - Merge any set of coali-tions{S1, . . . , Sl} where{∪l

j=1Sj}⊲{S1, . . . , Sl}; therefore,{S1, . . . , Sl} → {∪l

j=1Sj}.Definition 5: Split Rule - Split any coalition ∪l

j=1Sj

where {S1, . . . , Sl} ⊲ {∪lj=1Sj}; thus, {∪l

j=1Sj} →{S1, . . . , Sl}.

Using the above rules, multiple coalitions can interact,taking the decision to merge or split based on the comparisonrelation⊲. For the Pareto order, coalitions decide to merge(split) only if at least one SU is able tostrictly improve itsindividual utility through this merge (split) process withoutdecreasing the other SUs’ utilities. Therefore, the merge ruleby Pareto order is a binding agreement among the SUs tooperate together in the sensing phase if it is beneficial for them;and this agreement is partially reversible; i.e., can be reversedonly by an agreement to split. Note that, coalition formationentails algorithms where agreements can be either irreversible,partially reversible, or fully reversible [26]. As the algorithmsbecome more reversible, their complexity and implementationcan increase while their flexibility improves and this motivatesusing a partially reversible algorithm such as the one wepropose which has a balance between implementation andflexibility [26]. The use of these rules for CSS is described indetails further in this section.

As a consequence, for the proposed CSS game, we builda distributed coalition formation (CF) algorithm based onmerge-and-split and it is divided into three phases: Discovery,adaptive coalition formation, and coalition sensing. In thefirst phase, each SU discovers neighboring SUs as well asinformation required for performing coalition formation,e.g.,distance to the PU and the distance of neighboring SUs. Thediscovery phase is discussed in details in Subsection VI-A.Following the discovery phase, adaptive coalition formationbegins and the SUs (or existing coalitions of SUs) interact inorder to assess whether to cooperate for sharing their sensingresults with discovered coalitions. For this purpose, an iterationof sequential merge-and-split rules occurs in the network,whereby each coalition decides to merge (or split) dependingon the utility improvement. In this phase, as time evolvesand SUs (or the PU) move, the SUs can autonomously self-organize and adapt the network’s structure through new merge-and-split iterations with each coalition taking the decision tomerge (or split) subject to satisfying the merge (or split) rulethrough Pareto order (9). In the final coalition sensing phase,once the network partition converges following merge-and-

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TABLE IONE ROUND OF THE PROPOSEDCF ALGORITHM .

Initial State

The network is partitioned byT = {T1, . . . , Tk} (At the beginningof all time T = N = {1, . . . , N} with non-cooperative SUs).Three phases in each round of the CF algorithm

Phase 1 - Discovery:Each SU (or coalition of SUs) discovers its neighboringcoalitions as per Subsection VI-A.

Phase 2 - Adaptive Coalition Formation:In this phase, coalition formation using merge-and-split occurs.

repeata) F = Merge(T ); coalitions inT decide to merge based

on the merge algorithm explained in Subsection IV-A.b) T = Split(F ); coalitions inF decide to split based on

the Pareto order.until merge-and-split converges.

Phase 3 - Coalition Sensing:a) Each SU reports its sensing bit to the coalition head.b) The coalition head of each coalition makes a final decisiononthe absence or presence of the PU using decision fusion OR-rule.c) The SUs in a coalition abide by the final decision made by thecoalition head.

The above phases are repeated periodically throughout the net-work operation. In Phase 2, through distributed merge-and-splitdecisions, the SUs can autonomously adapt the network structureto environmental changes such as mobility.

split, SUs that belong to the same coalition report their localsensing bits to their local coalition head. The coalition headsubsequently uses decision fusion OR-rule [7], [12], [13] tomake a final decision on the presence or the absence of thePU. This decision is then reported by the coalition heads toall the SUs within their respective coalitions. One round ofthe CF algorithm is summarized in Table I and the detailedoperation is as follows:

Each round of the three phases of the CF algorithm of Ta-ble I starts from an initial network partitionT0 = {T1, . . . , Tl}of N . Following neighbor discovery each coalitionTi ∈ T0can acquire the following information (see Subsection VI-Afor details):

• The probabilities of miss of the SUs inTi, i.e., each SUin Ti is aware of its own probability of miss.

• The existence of other coalitions with whom a potentialcooperation can be done. Hence, each coalitionTi ∈ T0would have a list of all other potential coalitions withwhom a cooperation is possible.

• The distances between the SUs inTi and the SUs in thediscovered coalitions.

Following the acquisition of this information through thediscovery phase, the adaptive coalition formation phase ofthe algorithm begins and consists of the merge and splitprocedures. During the merge procedure, each SU (or coalitionof SUs that would form afterwards) engages in pairwisenegotiations (i.e., one-on-one) with discovered neighbors, overa control channel. One example of such a control channelis the ad hoc temporary control channel [29] that can betemporarily established by each SU (or coalition) for thepurpose of negotiations (these temporary control channelsarewidely used in ad hoc networks [29] and, more recently, incognitive networks [29]–[32]). By signalling over the controlchannels, two coalitions (or SUs) can inform each other of the

identities of their members (e.g., the identity can be defined bythe member’s position in space) and, in the event of a potentialmerge, can decide to join together into a single coalition. Forexample, at the beginning of all time, when all the network isnon-cooperative, following neighbor discovery, each SUi ∈ Ncan, over a control channel, start negotiating, in a pairwisemanner with other discovered SUs. When negotiating with adiscovered coalition or SU, SUi can send its information andan intent to merge over the control channel. Subsequently,depending on the Pareto order as per (9), the other coalitioncan respond with its own information and an approval or denialof the merge.

Essentially, themerge processoccurs as follows: Givenpartition T0, each coalitionTi ∈ T0 negotiates over a controlchannel (e.g., through its coalition head) with the list of neigh-bors for assessing a potential merge. Proceeding sequentiallythrough the list of discovered neighbors, each coalition headk ∈ Ti, attempts to merge with one of discovered neighborsTdisc (at the beginning of all time all the discovered neighborsare non-cooperative SUs) as follows (the sequence of thisprocedure is assumed to be arbitrary depending on whichneighbor was first discovered):

1) Coalition headk ∈ Ti, ∀Ti ∈ T0 signals an intent tomerge withTdisc by sending, over the control channel,the information (probabilities of miss and distance esti-mates) on its own coalition, i.e.,Ti, to one of the SUsin the coalitionTdisc.

2) The SU inTdisc which receives this information, com-putes, in cooperation with the members of its coalitionTdisc, the potential probability of miss that would resultfrom the merge withTi as per (3), and, following thiscomputation, coalitionTdisc would respond tok with oneof the following results:

• If the Pareto order as per (9) is not verified, i.e., themerger would decrease the payoff of one or moreof the coalition members inTi or Tdisc, thenTdisc

responds with a small packet signalling the failureof the merger as well as providing information onits current members. The information on the currentmembers allowsk andTi to get a better idea of theexisting partition for future merge decisions.

• If the Pareto order as per (9) is verified, i.e., coop-eration improves the payoffs of the involved SUs,then Tdisc responds with a small packet signallingthe acceptance of the merger as well as providinginformation on its current members. Subsequently,the two coalitions would merge into a new coalitionTnew∪ Tdisc and they would be aware of each otherfollowing the negotiations phase. Subsequently, thenew coalitionTnew would repeat the above proce-dure, given the union of the lists of neighbors fromTi andTdisc.

3) Coalition Ti continues the above procedure until itmerges or until no further merge possibility exist.

Hence, two coalitions would merge through negotiations overa control channel, and they can inform each others of theidentities of their members through this channel when the

7

merge procedure above is taking place. Note that, althoughwe consider that the order in which the SUs discover theirneighbors (and goes through them for merge) is arbitrary, onecan always define other metrics such as the average distancebetween the members which can be used as a distance betweencoalitions and, subsequently, used to order the discoveredneighbors. In many cases, from the perspective of a coalitionS, any neighbor discovery algorithm is likely to find coalitionswhose members are at a closer distance from the members ofS, before those who are located far away.

Moreover, the aforementioned procedure shows that, ingeneral, the SUs do not need to estimate the probability ofmiss of their neighbors. In contrast, during the pairwise mergesearch, each SU (or coalition) would append the values ofthe probabilities of miss of its members along with the intentto merge message that is sent to one of its neighbors. Inresponse, the neighbor can communicate, along with the mergedecision, the information on the probability of miss of itsmembers3. Certainly, we assume that each SU is aware ofits own probability of miss and the methods that each SU canuse to find itsown probability of miss are discussed in detailsin Subsection VI-A. Note that, once the SUs of two coalitionsexchange their probabilities of miss, they would be able towork out which member is potentially the coalition head asper Convention 1.

Following the merge process, the coalitions are next subjectto split operations, if any is possible. Hence, any coalitionwhich can no longer merge would start to apply the split oper-ation. As the members are part of the same coalition, assessingthe payoffs yielded by a split form is easily performed sincethe members are aware of the identities and characteristicsoftheir partners. An iteration consisting of multiple successivemerge-and-split operations is repeated until it terminates (theconvergence of an algorithm constructed using merge-and-splitrules is guaranteed [28]). It must be stressed that the mergeorsplit decisions are taken in a distributed way without relyingon any centralized entity as each SU or coalition makes itsown merge or split decisions.

The merge-and-split procedure of Phase 2 is performedsequentially for all coalitions in the network, in a certainorder (in practice there is always a fraction of time ofdifference between the action of a coalition/SU or the next one,which dictates an order of operation). The order in which thecoalitions/SUs act is considered arbitrary which is often thecase in a practical cognitive network. As a result, any arbitrarySU (or coalition) can initiate the merge and split procedures.Hence, once the SUs have operated non-cooperatively for awhile and performed the discovery phase, they would start, inan arbitrary order, to perform coalition formation in Phase2through merge-and-split. In general, all SUs will always havean incentive to seek cooperation since it might improvetheir payoff. Nonetheless, the proposed algorithm is alwaysapplicable regardless of the order in which the SUs/coalitionsact, and, thus, a network operator can impose a certain orderfor coalition formation on its SUs if needed (e.g., let the SU

3In case the SUs, using the techniques of Subsection VI-A, areable toestimate the probability of miss of their neighbors, they may be able to reducethe information exchange overhead.

with worst or best performance start and so on).Further, it is interesting to note that, under certain conditions

(see Subsection IV-B and Lemma 1) on the network partition,the order of operation does not affect the final structure thatwill form in the network. Nonetheless, even when differentorders can yield different networks structures, due to thedefinition of the merge and split operation through the Paretoorder in (9), it is always ensured that, any resulting partitionwould improve the payoffs of the SUs as compared to theprevious state. This is due to the fact that any merge orsplit agreement makes sure that at least one SU benefits fromthis operation. Hence, the proposed merge-and-split algorithm,independent of the order, is guaranteed to yield a cooperativenetwork which outperforms (or is at least as good as) the non-cooperative approach.

For handling environmental changes such as mobility (ofthe SUs or the PU) or the joining/leaving of SUs, Phase 2 ofthe proposed algorithm in Table I is repeated periodically overtime. In Phase 2, periodically, as time evolves and SUs (or thePU) move or join/leave, the SUs can take distributed decisionsto adapt the network’s structure through new merge-and-splititerations with each coalition taking the decision to merge(orsplit) subject to satisfying the merge (or split) rule through thePareto order in (9).

For this purpose, one can define a period of timeθ whichspecifies the time elapsed between two consecutive runs ofthe proposed CF algorithm. Consequently, there would bea tradeoff between the number of runs, i.e., overhead forcoalition formation and the adaptation to the dynamics of theenvironment. Hence, for environments that are static or varyingvery slowly,θ would be large. In contrast, for highly varyingenvironments,θ would have a small value, and, thus, enablingadaptation to rapidly changing environments. The value ofθcan be regulated by the SUs depending on their observationson the environment. For example, given an initial value ofθ = θ0, whenever the SUs notice that, during merge-and-split, a lot of changes in the network and the partition haveoccurred, they can decide to reduceθ to a value smaller thanθ0. In contrast, whenever the SUs note that few changes haveoccurred, they can increaseθ to a value aboveθ0. For example,the possible values for the periodθ can be standardized sothat the SUs would know it without the need of a centralizedcontroller. The above procedure would ensure the convergenceof the merge-and-split algorithm. Nonetheless, certainly, weconsider only environments where the changes during a singlerun of the proposed algorithms are small enough so as not todrastically affect the utilities, and, thus, the merge or splitdecisions of the SUs. Note that, performing instantaneouscoalition formation is still a challenging task in game theory(e.g., see [26] and references therein), and can be consideredin future extensions of this work.

It must be also noted that almost no synchronization amongthe SUs is needed to perform coalition formation since theorder in which the merge-and-split operations proceed canbe arbitrary (or synchronized if an operator wishes to doso). In some sense, the SUs would need only some kind ofquasi-synchronization for handshaking and negotiations overthe control channel to communicate their spectral sensing in-

8

formation. The proposed merge-and-split CF process involvesnegotiations that occur in a pairwise manner with coalitions(or SUs) in a small neighboring area and not in the entirenetwork. In addition, as soon as a coalition identifies a mergepossibility, it does not need to go through the remainder ofthe discovered neighbor. This aspect further corroboratesthelittle need for synchronization among the SUs.

In order to show that the SUs need to survey only a specificgeographical area for finding merge partners without lookingat all other possible SUs in the network, we inspect thedifferent merge possibilities between two SUs. In this regards,given two SUs involved in CSS through CF, we have thefollowing theorem (proof is in the appendix):

Theorem 1: Two SUsi and j with non-cooperative prob-abilities of missPm,i andPm,j respectively, such asPm,i ≤Pm,j , can merge into one coalition if the probability of error isbelow a maximum probability of error, i.e.,if Pe,j,i ≤ Pe,j,i,wherePe,j,i is given by the following tight approximation

Pe,j,i ≈Pf (Pf − 2) + α

√

(1 − (1−P 2

f

α2 )ePm,i(1−η)(Pm,j−1)

α2 )

(2Pf − 1)(Pf − 1),

(10)with η = α, if Pm,j ≤

12 andη = 0 otherwise. Consequently,

a corresponding minimum distancedi,j between the two SUscan be easily found by solving (5) forPe,j,i.

From Theorem 1, each SUi can determine merge regions,corresponding to arcs on circles and their respective inter-cepting angles, for forming a coalition with an SUj wherePm,i ≤ Pm,j . For a given detection thresholdλ, one can seethrough (1) that eachPm,j corresponds toone circlecenteredat the PU with a radiusrj hence determining a class of SUshaving the same non-cooperative probability of miss. For eachsuch circle, an SUi maps the conditionPe,j,i ≤ Pe,j,i toan arc on these circles, with a corresponding interceptingangle (angle centered at SUi) and intercepting the mergeregion arc. However, for any two SUsi and j belonging totwo circles of radiiri and rj , the distancedi,j between thetwo SUs (from Theorem 1) must satisfydi,j < (rj − ri),otherwise no merge is possible (the merge region is empty).A detailed analysis of these merge regions, using Theorem 1,is presented in Section VII. Note that, even if no closed formexpression exists for the upper boundPe,j,i, Pe,j,i will alwaysbe bounded by a valuePe,j,i which would be a function ofthe probabilities of miss and false alarm of the two SUs. Thisis a direct consequence of the fact that, whenever the erroron the reporting channel increases, it will incur an increasedprobability of miss and an increased false alarm as per (3) and(4) which, in turn, would limit the gains from cooperation andsets an upper bound on the error level that would be toleratedfor the merge to occur.

An upper bound on the maximum coalition size is imposedby the proposed utility and models as follows:

Theorem 2: For the proposed CSS model, given that theSUs collaborate for improving their payoff given the tradeoffbetween probability of miss and false alarm, any coalitionalstructure resulting from the distributed coalition formationalgorithm will have coalitions limited in size to a maximumof

Mmax =log (1− α)

log (1 − Pf )(11)

SUs.Proof: In the proposed algorithm, the benefit from collab-

oration is limited by the false alarm probability cost modeledby the barrier function (8). A minimum false alarm cost ina coalitionS with coalition headk ∈ S exists whenever thereporting channel is perfect, i.e., exhibiting no error, hence,Pe,i,k = 0 ∀i ∈ S. In this perfect case, the false alarmprobability in a perfect coalitionSp is given by

Qf,Sp= 1−

∏

i∈Sp

(1− Pf ) = 1− (1 − Pf )|Sp|, (12)

where |Sp| is the number of SUs in the perfect coalitionSp. A perfect coalition Sp where the reporting channelsinside are perfect (i.e., SUs are grouped very close to eachother) accommodates the largest number of SUs relative toall other coalitions. Hence, we use this perfect coalition tofind an upper bound on the maximum number of SUs percoalition. For instance, the log barrier function in (8) tendsto infinity whenever the false alarm constraint per coalitionis reached which implies an upper bound on the maximumnumber of SUs per coalition ifQf,Sp

≥ α, yielding by (12),|Sp| ≤

log (1−α)log (1−Pf )

= Mmax.It is interesting to note that the maximum size of a coalition

Mmax depends mainly on two parameters: the false alarmconstraintα and the non-cooperative false alarmPf . Forinstance, larger false alarm constraints allow larger coalitions,as the maximum tolerable cost limit for collaboration isincreased. Moreover, as the non-cooperative false alarmPf

decreases, the possibilities for collaboration are bettersincethe increase of the false alarm due to coalition size becomessmaller as per (4). It must be noted that the dependence ofMmax on Pf yields a direct dependence ofMmax on theenergy detection thresholdλ as per (2). Finally, it is interestingto see that the upper bound on the coalition size does notdepend on the location of the SUs in the network nor on theactual number of SUs in the network. Therefore, deployingmore SUs or moving the SUs in the network for a fixedαandPf does not increase the upper bound on coalition size.

B. Partition StabilityThe result of the proposed algorithm is a network partition

composed of disjoint independent coalitions of SUs. Thestability of this resulting network structure can be investigatedby a defection functionD [28].

Definition 6: A defectionfunction D is a function whichassociates with each partitionT of N a group of collectionsin N . A partition T = {T1, . . . , Tl} of N is D-stable if nogroup of players is interested in leavingT when the playerswho leave can only form the collections allowed byD.

Two defection functions are of interest: aDhp functionwhich associates with each partitionT of N the group ofall partitions of N that can form by merging or splittingcoalitions in T and theDc function which associates witheach partitionT of N the group of all collections inN . Twoforms of stability stem from these definitions:Dhp stability anda strongerDc stability. A partitionT is Dhp-stable, if, for thepartitionT , no coalition has an incentive to split or merge. As

9

an immediate result of the definition ofDhp-stability we havethat every partition resulting from our proposed CF algorithmis Dhp-stable. Briefly, a Dhp-stable can be thought of as astate of equilibrium where no coalitions have an incentive topursue coalition formation through merge or split.

With regards toDc stability, aDc-stable partition has thefollowing properties [28]: (i)- In aDc-stable partitionT , noplayers are interested in leavingT to form other collectionsin N (through any operation), (ii)- If it exists, aDc-stablepartition is theuniqueoutcome of anyarbitrary iteration ofmerge-and-split and is aDhp-stable partition, and, (iii)- ADc-stable partitionT is a unique⊲-maximal partition, that is forall partitionsT ′ 6= T of N , T ⊲ T ′. When⊲ is the Paretoorder, theDc-stable partition presents aPareto optimalutilitydistribution.

However, the existence of aDc-stable partition is not alwaysguaranteed [28]. TheDc-stable partitionT = {T1, . . . , Tl} ofthe whole spaceN exists if a partition ofN that verifies thefollowing two necessary and sufficient conditions exists [28]:(i)

1) For each i ∈ {1, . . . , l} and each pair of disjointcoalitions S1 and S2 such that{S1 ∪ S2} ⊆ Ti, wehave{S1 ∪ S2}⊲ {S1, S2}.

2) For the partitionT = {T1, . . . , Tl} a coalitionG ⊂ Nformed of players belonging to differentTi ∈ T is T -incompatible if for noi ∈ {1, . . . , l} we haveG ⊂ Ti.Dc-stability requires that for allT -incompatible coali-tions {G}[T ] ⊲ {G} where{G}[T ] = {G ∩ Ti ∀ i ∈{1, . . . , l}} is the projection of coalitionG on T .

If no partition satisfies these conditions, then noDc-stablepartitions ofN exist. Nevertheless, we have

Lemma 1: For the proposed(N , V ) CSS coalitional game,the proposed CF algorithm converges to the optimalDc-stablepartition, if such a partition exists. Otherwise, the final networkpartition isDhp-stable.

Proof: The proof is immediate due to the fact that, whenit exists, theDc-stable partition is a unique outcome of anymerge-and-split iteration [28] such as any partition resultingfrom our CF algorithm.

For the proposed game, the existence of theDc-stablepartition cannot always be guaranteed although the algorithmwould alwaysconverge to this partition when it exists. Givena partitionT = {T1, . . . , Tl} of SUs, verifying condition (i)for Dc-stability implies that, for any coalitionTi ∈ T , thePareto order as per (9) must be verified for theunion ofevery pair of sub-coalitions ofTi which, in the context ofcollaborative sensing, maps directly into a condition dependenton the distances between the SUs insideTi and their non-cooperative probabilities of miss and false alarm (due to theutility in (6)). For example, ifTi is a coalition of size2,condition (i) implies that Theorem 1 and (10) must be verifiedbetween the two SUs ofTi. Further, ifTi is a coalition of size3, condition (i) implies the following: (a)

1) The distance between every pair of SUs inTi mustsatisfy (10) and Theorem 1.

2) The distances and probabilities of miss and of falsealarm of the SUs inTi must satisfy the merge rule usingthe Pareto order in (9) (in any direction).

Consequently, simple examples of partitions satisfying con-dition (i) of Dc-stability would be a partition composed solelyof coalitions of size2 which verifies (10) and Theorem 1, apartition T composed of coalitions of size3 that verifies (a)and (b), or a partition composed of a mixture of coalitions ofsize2 or 3 satisfying the requirements previously stated. Cer-tainly, for larger coalitions, condition (i) imposes restrictionson the distances and probabilities of miss and false alarm ofthe cooperating SUs analogous to (10), (a), and (b).

The difficulty of guaranteeing aDc-stable partition in thecontext of collaborative sensing stems from the difficulty ofcomputing a closed form solution that characterizes condition(ii). For a partitionT , condition (ii) deals withT -incompatiblecoalitions, which are coalitions composed of players belongingto different coalitionsTi ∈ T . In order to clarify thisrequirement, consider a3-player game and a partitionT ={{1, 2}, {3}}. For this case, theT -incompatible coalitions areG1 = {2, 3}, G2 = {1, 3}, G3 = {1, 2, 3}. For everyGi,condition (ii) requires that{Gi ∩ {1, 2}, Gi ∩ {3}} ⊲ {G}which maps into: (A)

1) {{2}, {3}}⊲{2, 3}which implies that SUs2 and3 mustnot verify (10) as per Theorem 1.

2) {{1}, {3}}⊲{1, 3}which implies that SUs1 and3 mustnot verify (10) as per Theorem 1.

3) {{1, 2}, {3}}⊲{1, 2, 3}which implies that the Pareto or-der must not be satisfied, i.e., forming coalition{1, 2, 3}decreases the payoff of at least one of the SUs.

Consequently, for the3-player example, verifying condition(ii) for the existence of aDc-stable partition depends, mainly,on the distances between the3 SUs and their location withrespect to the PU (through the probability of miss) as perTheorem 1. For larger networks, satisfying condition (ii)would, once again, depend on the distances between the SUsof T -incompatible coalitions, however, the number of suchcoalitions becomes large, and, thus, mapping the requirementinto closed form conditions such as in Theorem 1 wouldbe difficult. Note that, for the3-player example previouslymentioned, in the event where (A)-(C) are satisfied, and, if{1, 2} satisfies (10), then, partitionT = {{1, 2}, {3}} wouldverify conditions (i) and (ii), and, thus, it will be aDc-stablepartition and a unique outcome of the proposed algorithm.

In summary, for verifying condition (i) for existence of theDc-stable partition, the SUs belonging to partitions of eachformed coalition must verify the Pareto order as per (9) usingtheir utility defined in (6). Similarly, for verifying condition (ii)of Dc stability, SUs belonging to allT -incompatible coalitionsin the network must verify the Pareto order. As shown in theprevious examples, as well as through Theorem 1 (for2 SUs),this existence is closely tied to the location of the SUs and thePU (due to the dependence on the individual miss and falsealarm probabilities in the utility expression (6)) which bothcan be random parameters in practical and large networks.Nonetheless, the proposed algorithm will always guaranteeconvergence to this optimalDc-stable partition when it existsas stated in Lemma 1. Whenever aDc-stable partition doesnot exist, the coalition structure resulting from the proposedalgorithm will beDhp-stable as no coalition or SU is able to

10

merge or split any further (can be viewed as a kind of partitionequilibrium).

V. D ISTRIBUTED COALITION FORMATION ALGORITHM

WITH PROBABILITY OF DETECTION GUARANTEES

(CF-PD)

The previously proposed CF algorithm in Section IV allowsthe SUs in a cognitive network to minimize their probabilityofmiss under false alarm constraints. In such a scenario, one cansee the PU rewarding the SUs as much as they improve theirdetection and reduce their interference level, hence, providingan incentive for the SUs to lower their probability of miss asmuch as possible. However, in some cognitive networks, inaddition to the false alarm constraint that the SUs require,thePU (or PU operator) imposes a desired detection probabilityχ that the SUs must meet (i.e., a desired interference level)[3]. The PU introduces such constraints in order to controlthe amount of interference that is allowed by SUs or to limitand control the number of SUs that are allowed to use thespectrum. The presence of such an additional constraint canaffect the behavior of the SUs, their incentives, and theircollaboration strategies. Hence, the coalition formationprocessmust be adequately modified in order to cope with the presenceof such an additional detection probability constraint as theobjective of the SUs is no longer to minimize their probabilityof miss, but rather to achieve the desired probability of missγ = 1 − χ (while meeting the false alarm constraint). Inthis context, during a merge or split process, once a coalitionachieves the probability of missγ, it will have no furtherincentive to pursue the coalition formation process. In fact,in the presence of detection probability constraints, onceacoalition achievesγ increasing the size of the coalition incursan increased false alarm (within the constraintα), as well asextra signalling and sensing time with no extra benefit for thecoalition members.

In order to appropriately capture these new objectives in thecoalition formation process, we introduce, adjunctly withtheproposed(N , V ) coalitional game, asimple voting game[33]which is defined as a coalitional game with utility functionu : 2N → {0, 1}. By introducing this simple voting game inthe coalition formation process, the new objectives of the SUs,i.e., probability of detection guarantees, can be accommodatedas will be seen later in this section. For this purpose, we definethe adjunct utility functionu as follows:

u(S) =

{

1, if Qd,S ≥ χ andQf,S ≤ α,

0, otherwise.(13)

This definition captures the two objectives of the SUs: (i)Satisfying the probability of detectionχ, and, (ii) Maintaininga false alarm level belowα, i.e., keepingQf,S ≤ α. Thus,in (13), u(S) = 1 if the probability of detection satisfiesχand the false alarm constraint isnot violated. For a coalitionS where Qf,S > α, as per (6), the utility goes towardsnegative infinity, and, thus a coalition withQf,S > α cannever form, hence, in this caseu(S) = 0. During coalitionformation, as will be seen later in this section, the SUs canutilize both utilitiesV andu in (7) and (13) in order to make

their cooperative decisions given the objective of achieving thetarget probability of detection while making sure that doing sobenefits them as per (7) while the false alarm constraint is notviolated. Further, within the proposed adjunct simple votinggame framework, we make the following definition:

Definition 7: In the proposed CSS game, a coalitionS issaid to bewinning if u(S) = 1 and losing otherwise. Similarly,a player is winning if it is part of a winning coalition andlosing otherwise.

Consequently, under a desired probability of detection con-straint, the objective is to maximize the number of winningplayers while minimizing the cost in terms of false alarm(given the false alarm constraintα) as well as the signalingoverhead within each formed coalitions. In this regard, thepartition stability concepts introduced in Section IV-B are nolonger adequate to assess the performance of the system sincethe SUs are no longer looking to maximize their utilities butrather to form winning coalitions. Therefore, a more adequatenotion for stability is needed through the following concept.

Definition 8: In a simple voting game(N , u), a coalitionS is said to beminimal winningif the defection ofanyplayerin S renders that coalition losing. In other words, aminimalwinning coalition (MWC)S is a coalition such thatu(S) =1 & u(T ) = 0, ∀ T ⊂ S.From this definition, it follows that, in order to achieve thedetection probabilityχ with minimum overhead, the SUs havean incentive to form MWCs. Using Definition 8 and (13), onecan see that, whenever a group of SUs form an MWC, theyhave no incentive to leave this MWC. In fact, when an SUleaves an MWC, as per Definition 8, this SU becomes losing,i.e., its adjunct utilityu becomes0. Thus, an MWC is highlysuitable to characterize the coalitional structure that formswhen both false alarm and detection probability constraintsare required. Moreover, once the SUs form an MWC, theyhave no further incentive to decrease their probability of miss(which can incur extra false alarm) as they already complywith the requirements set by the PU.

Jointly with the merge and split operations, we define a newoperation on the coalitions as follows:

Definition 9: Adjust Rule - For any coalitionS ⊆ N , theadjust rule is defined as follows:

CS = Adjust(S) =

{

S, if u(S) = 0,

{SMWC, i1, . . . , ik}, if u(S) = 1,

(14)whereSMWC is the MWC generated out of a winning coalitionS by simply excluding all playersi ∈ S such thatu(S) =u(S \ {i}) = 1 and{i1, . . . , ik} is thecollectionof individualSUs ij that were excluded fromS by the adjust operation.The exclusion of the players from a winning coalition in orderto yield an MWC can be done in any order. For the proposedCSS game, we adopt the following convention without anyloss of generality:

Convention 2: The adjust operation generates an MWCfrom a winning coalitionS by excluding the playersi ∈ Ssuch thatu(S) = u(S\{i}) = 1 (note thatu(∅) = 0 by defini-tion of any characteristic function), in asequentialmanner byan increasing order of their non-cooperative probability of miss

11

Pm,i. In other words, the SUs verifyingu(S) = u(S\{i}) = 1and having the smallest non-cooperative probability of missare excluded first and their collection constitutes{i1, . . . , ik}.The motivation behind this convention is that, once and if ex-cluded, SUs with a small non-cooperative probability of misshave higher chances (than SUs with larger miss probabilities)to find other partners with whom to form an MWC.

Having defined the necessary new concepts, the coalitionformation Phase 2 of the proposed CF algorithm in Table Ican be easily modified by introducing the adjust operation. Inthis regard, Phase 2 of the algorithm in Table I is split intotwo phases: Phase 2a and Phase 2b. In Phase 2a, an adjustoperation is applied to every coalition in the network partitionprior to merge-and-split, which results in a partitionT0 thatcan be divided into two collections: a collectionW0 withMWCs only and a collectionL0 with the remaining coalitions.In Phase 2b, adaptive coalition formation through merge-and-split is applied only to partitionL0. Nonetheless, in Phase 2bof the algorithm, the adjust operation needs to be furtherincorporated within merge-and-split. For instance, whenever acoalition forms by merge (or by split), it is subject to an adjustoperation. In the event where a formed coalition is winningand the adjust operation results in an MWC, this MWC nolonger participates in any future merge or split. However, anySUs excluded from the coalition through the adjust operation,continue to participate in merge-and-split. Consequently, theadaptive coalition formation process terminates by resultingin two collections: A collectionWf having only MWCsand another collectionLf with losing coalitions that can nolonger be subject to any merge, split or adjust operations. Thefinal network partition is simplyF = W0 ∪ Wf ∪ Lf . Theconvergence of the CF-PD algorithm to this final network isguaranteed since the presence of the adjust operation (which isan operation internal to each coalition) in Phase 2b does notaffect the convergence of the merge-and-split iterations thatwas assessed in Section IV since this convergence is indepen-dent of the starting partition. This algorithm is referred to asthe distributed coalition formation algorithm with probabilityof detection guarantees (CF-PD) algorithm and is summarizedin Table II. Note that, in the final network, the SUs that belongto the collection of losing coalitionsLf have improved theirsensing performance compared to their non-cooperative state,however, they would need to wait environmental changes, suchas mobility or the arrival of new SUs, in order to becomewinning SUs/coalitions, i.e., meet the required probability ofdetection constraintχ through cooperation.

VI. D ISTRIBUTED IMPLEMENTATION OF CF AND CF-PD

In this section, we discuss and present some of the chal-lenges for the implementation of the proposed CF and CF-PDalgorithms in practical cognitive networks.

A. Neighbor Discovery

In the first phase of the CF and CF-PD algorithms, fordiscovering their neighbors, each coalition (or single SU)needs to utilize a neighbor discovery techniques in order todis-cover potential cooperation partners for performing the local

TABLE IIONE ROUND OF THE PROPOSEDCF-PDALGORITHM .

Initial State

The network is partitioned byT = {T1, . . . , Tk} (At the beginningof all time T = N = {1, . . . , N} with non-cooperative SUs).Three phases in each round of the CF-PD algorithm

Phase 1 - Discovery:This phase remains the same as in the CF algorithm of Table I.

Phase 2a - Partition Adjustment:Adjust the initial partition T = {T1, . . . , Tk}

a) CTi = Adjust(Ti), ∀ Ti ∈ T

b) T0 = ∪ki=1

CTi can be divided into 2 collectionsb1) L0 = {L1, . . . , Lp}, Li ∈ T0, u(Li) = 0

b2) W0 = {W1, . . . ,Wq}, Wi ∈ T0, u(Wi) = 1

Phase 2b - Adaptive Coalition Formation with Probability ofDetection Guarantees:

initialize Lf = L0

repeata) {Wf ,Lf} = MergeAndAdjust(Lf ); merge algorithm

with adjust as explained in Section V.b) {Wf ,Lf} = SplitAndAdjust(Lf ); split algorithm with

adjust as explained in Section V.until merge-and-split with adjust terminates.

Final partition isF = W0 ∪Wf ∪ Lf

Phase 3 - Coalition Sensing:This phase remains the same as in the CF algorithm of Table I.

Similar to the CF algorithm, Phase 2 (a and b) of CF-PD isrepeated periodically to allow the SUs to adapt the networkstructure to environmental changes such as mobility.

merge. In other words, each coalition in the network surveysneighboring coalitions within merge range and attempts tomerge according to the Pareto order. For performing merge, theSUs are required to know their own probabilities of miss andtheir distance to the neighbors in order to assess the potentialutility that they can acquire by collaboration which is given by(6). Note that, although, in order to assess (6), the SUs needto also learn the probability of miss of their neighbors, i.e., thedistance between the neighbors and the SU, this informationcan be sent by the neighbors over a control channel duringthe negotiation phase of the merge process. Hence, the SUsdo not necessarily need to estimate this value by themselves.

In Rayleigh fading, the non-cooperative probability of missis mainly a function of the average SNR to the PU, and, hence,the distance to the PU as seen in (1). Consequently, for per-forming coalition formation, the main information that needsto be gathered by the SUs consists of the distances betweenthese neighbors as well as the distance between the PU andthe SU. For finding this information, numerous methods areavailable for the SUs in practical cognitive networks:

1) Information received through control channels in cogni-tive networks such as the recently proposed CognitivePilot Channel (CPC) [34]–[38].

2) Signal-processing based methods [3], [15], [32], [39],[40].

3) Existing ad hoc routing neighbor discovery techniquesapplied in cognitive network [29]–[32].

4) Geo-location techniques [2], [41]–[44].

1) Neighbor discovery through control channels:In orderto maintain a conflict-free environment between the SUs andPUs of a cognitive network, there has been a recent interestin providing, through control channels, assistance to the SUs

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on both the spectrum sensing and access levels by providinguseful information that the SUs can use to improve theirperformance. For example, in [34], a control channel, calledthe Common Spectrum Coordination Channel (CSCC) hasbeen proposed for the announcement of radio and serviceparameters to allow a coordinated coexistence between SUsand PUs. Following a similar concept, the Cognitive PilotChannel (CPC) has been recently proposed [35]–[38] as ameans for providing frequency and geographical informationto cognitive users, assisting them in sensing and accessingthespectrum. Notably, the CPC can assist the SUs to discoverneighboring PUs and SUs in the presence of multiple accesstechnologies and heterogeneous environments. Thus, as ex-plained in [35]–[37], the CPC is a control channel that carriesinformation such as available spectrum opportunities, existingPUs or geographical information on neighboring SUs. A CPCchannel can be deployed in a cognitive network by either abroadcast manner, i.e., a station that broadcasts the CPC toallof its users, or an on-demand manner, i.e., the CPC informationis sent to the SUs upon request. In this paper, the CPC can bedelivered to the SUs by the receivers or a stand alone spectrumbroker which is able to obtain information on the existing SUs,PUs, or other needed CPC details. Following the formation ofa coalition, an SU can also signal to its receiver or to thebroker its membership in a coalition, which is an informationthat can be further appended to the CPC.

Consequently, by listening to the CPC transmission theSUs can discover their own probabilities of miss (i.e., theirdistances to the PU), the presence of neighboring SUs, thedistance to other SUs, and, possibly the coalition membershipsof these SUs. Since various efficient implementations of theCPC or other information control channels for cognitive net-works are ubiquitous, then, using these channels, the SUs canacquire the needed information for forming coalitions throughCF or CF-PD with a reasonable overhead.

2) Signal processing-based methods:In Rayleigh fading,discovering neighboring nodes and finding an estimate ofthe distances to these nodes can be achieved through signalprocessing techniques [40] by computing the average receivedpower, from the neighboring SUs and from the PU. To do so,in the proposed model, prior to any coalition formation, i.e., inPhase 1 of the algorithm, each SU can attempt to compute theaverage received power, while listening to a control channelfor example. Another approach would be for the SUs toestimate this average received power during the operation ofthe network prior to coalition formation. In other words, inagiven initial partitionT0, the SUs start by performing spectrumsensing and, then, use the results in order to access the channel.During spectrum access, the SUs can monitor the transmissionreceived from other SUs, and, then, discover existing SUsin the network as well as the distance to these SUs throughan estimation, using signal processing-based methods, of theaverage received SNR from these SUs. This information onthe neighbors and the distances to them will be useful for theSUs to take adequate coalition formation decisions in Phase2of the algorithm. It must be stressed that the use of receivedpower estimates for neighbor discovery is quite common asshown in [3], [15], [32], [39], [40].

The measured mean received power from the PU (whosetransmission is always monitored by the SUs for cognitiveradio access), can be used by every SU to find its individualprobability of miss. For instance, in this paper, we deal mainlywith the probability of miss averaged over the small-scaleRayleigh fading as given in (1), and which is common inmany cognitive radio literature, e.g. [7]–[10], [12]–[15]andthe references therein. By closely inspecting (1), one wouldsee that, for a given time-bandwidth productm and detectionthresholdλ, the probability of miss of an SU mainly dependson the average received SNR from the PU signal, which, isa function of the path loss, and, thus, the distance betweenthe SU and PU. Typically, an SU needs only to average thereceived power from the PU over several samples to computethis average (with respect to fading) SNR. Therefore, an SUcan use the measured mean received power from the PU toestimate its distance to the PU (or the average SNR), and,subsequently, its individual probability of miss. For a coalitionof SUs, each member can find its own individual probabilityof miss and, consequently, since the SUs of a coalition areaware of the location of each other, they can work out thepotential probability of miss of their coalition as per (3).

Finally, note that, using signal-processing techniques todis-cover neighboring SUs, a coalition of SUsTi ∈ T needs onlyto discover SUs in a small geographical area around it (sinceotherwise no potential merge is possible as demonstrated forexample in Theorem 1 for the 2-SU case) and does not need toknow which coalitions each SU belongs to. The informationon the membership of the SUs into a coalition can be foundout in subsequent pairwise interactions over a control channelbetween the coalitionTi and its discovered neighboring SUs(since each SU knows which coalition it belongs to).

3) Existing ad hoc routing discovery techniques appliedin cognitive networks: In the past decade, techniques forneighbor discovery have been widely spread in the contextof ad hoc routing [29] whereby numerous methods for findingneighboring nodes and information on these nodes are devel-oped. Recently, these approaches have been leveraged in orderto provide neighbor discovery in cognitive radio networks[30]–[32].

While an in-depth study of these neighbor discovery tech-niques is beyond the scope of this paper, we will providea simple example of such techniques which can assist theSUs for the purpose of coalition formation. For instance, onepopular idea for neighbor discovery is the use of methodsbased on the well known on demand distance vector (AODV)protocol of ad hoc routing [32]. In this protocol, each nodeperiodically broadcasts a “Hello” packet, in order to detect itsneighbors. In a cognitive network, several methods based onthis idea can be used by the SUs to discover the PUs or otherSUs. As shown in [32], to do so, each SU can establish anad hoc temporary control information channel, over which itbroadcasts its “Hello” packet which contains the SU’s mobileIP address, location information, frequencies of all ad hoctemporary control channels, exact time when SU will tune toad hoc channels to listen for reception, transmission schedulesof “Hello” messages, time stamp according to its internalclock, coalition membership, or other information. Several

13

techniques for transmitting the “Hello” packets are given in[32]. Upon receiving the “Hello” message, an SU can adjust itsown transmission power according to the distance calculatedfrom the information contained within the broadcasted “Hello”message, and then broadcast a “Hello Ack” message with itsown information. Using this “Hello” and “Hello ACK” proce-dure, the SUs can discover each other [32] and, subsequently,engage in a coalition formation process.

Beyond this idea, numerous other techniques for cognitiveradio neighbor discovery, using similar concepts from ad hocrouting discovery, can be found in [30], [31] (and referencestherein) along with their details. By adopting any of thesetechniques, the neighbor discovery phase of the proposedcoalition formation algorithm can be implemented, allowingthe SUs to gather any needed information for the merge-and-split phase.

4) Geo-location techniques:In some cognitive networks,in order to improve the coexistence between the secondaryand primary networks, the SUs are required to register andconnect, through the Internet, to incumbent databases [41]–[44]. The principal purpose of this requirement is to providea mechanism to inform SUs about their neighboring SUs orPUs [41]. These databases can hold information on the PUs,the SUs, their positions, their capabilities, and so on. Forthepurpose of coalition formation, the databases may also holda field that displays the coalition membership of the SUs, ifany.

The information in these databases is entered either au-tomatically by every SU or PU that enters the network, orby a centralized administrator that controls these databases[41]. In this context, the need for knowing the position of thenodes (SU or PU) prior to filling the database is required.For fixed SUs or PUs, obtaining this information for fillingthe databases is straightforward. However, when the SUs orPUs are mobile devices, two cases can be distinguished. Inan outdoor environment, the SUs or PUs can use techniquesbased on the global positioning service (GPS) in order toobtain their own locations and update the databases. If noGPS is available or the nodes are indoor, finding the positionbecomes more challenging. In this case, the nodes can useadvanced positioning techniques such as reference anchors[41], the signal processing techniques of Subsection VI-A2,or other techniques [42]–[44].

Although the technical details on building the databasesis beyond the scope of this paper (the interested reader isreferred to [41]–[44] and the references therein), nonetheless,for coalition formation, the SUs can access the databases inorder to discover the position of the PU or their neighboringSUs, or even, the coalition memberships of the neighbors. Bydoing so, the SUs would obtain all the necessary informationfor the coalition formation phase of CF or CF-PD.

In summary, neighbor discovery techniques for cognitiveradio networks are abundant. For the proposed coalition for-mation algorithm, any of these techniques can be adoptedin Phase 1 as they all allow the SUs to obtain estimatesof the needed information for the coalition formation phase.Certainly, there is a tradeoff between accuracy and complexityfor each of these techniques, however, for coalition formation,

one can adopt any technique with a reasonable tradeoff (e.g.,the CPC or the signal processing techniques) as long as it canconvey enough information on the neighboring SUs and thePU in the network.

Hence, once these parameters are estimated using either thereceived power estimation, or geo-location techniques, ortheCPC, the coalitions (or individual SUs) can then decide onthe partners, if any, with whom they can cooperate (merge)for spectrum sensing. Moreover, with regards to the splitoperation, each formed coalition can internally make thedecision to split or not depending on whether its memberscan find a split form that is preferred by the Pareto order.

B. Coalition Formation Operations: Merge, Split, and Adjust

The complexity of the coalition formation phase of theproposed CF algorithm lies mainly in the complexity of themerge and split operations. It is important to stress that thecomplexity of the algorithm is independent of whether aDc-stable partition exists or not. For instance, as per Lemma 1and as shown in [28], any implementation of a merge-and-splitalgorithm would reach theDc-stable partition (if it exists) atno extra complexity). Hence, reaching an optimal and stablepartition (when such a partition exists) does not incur extracomplexity. For a given network structure, during one mergeprocess, each coalition attempts to merge with other coalitionsin its vicinity in a pairwise manner. In the worst case scenario,every SU, before finding a suitable merge partner, needs tomake a merge attempt with all the other SUs inN . In thiscase, the first SU requiresN −1 attempts for merge (N beingthe number of SUs), the second requiresN − 2 attempts andso on. The total worst case number of merge attempts will be∑N−1

i=1 i = N(N−1)2 . In practice, the merge process requires a

significantlylower number of attempts since finding a suitablepartner does not always require to go through all the mergeattempts (once a suitable partner is identified the merge willoccur immediately). This complexity is furtherreducedbythe fact that a coalition does not need to attempt to mergewith far away or large coalitions where the false alarm cost islarge relatively to the benefit or where the false alarm is largerthanα (Theorem 1 being an example of such merge regionsfor the two SUs case). Moreover, after the first run of thealgorithm, the initialN non-cooperative SUs will self-organizeinto larger coalitions. Subsequent runs of the algorithms willdeal with a network composed of a number of coalitions thatis much smaller thanN ; reducing the merge attempts percoalition. Finally, once a group of coalitions merges into alarger coalition, the number of merging possibilities for theremaining SUs will decrease as the merge region becomessmaller (due to cooperation costs).

For the split rule, in a worst case scenario, splitting caninvolve finding all the possible partitions of the set formedbythe SUs in a single coalition. For a given coalition, this numberis given by the Bell number which grows exponentially withthe number of SUs in the coalition (here the partition searchis restricted to each coalition and is not over the whole setN ) [23]. In practice, this split operation is restricted to theformed coalitions in the network, and, thus, it will be applied

14

on small sets. Due to the cost function in (6) and (8) , thesize of each coalition resulting from CSS coalition formationis limited by the increasing false alarm cost as well as by theSUs’ locations, thus, the coalitions formed in the proposedCSS game are small. As a result, the split complexity islimited to finding possible partitions for small sets which canbe reasonable in terms of complexity. The split complexity isfurther reduced by the fact that, in most scenarios, a coalitiondoes not need to search for all possible split forms. Forinstance, once a coalition identifies a split structure verifyingthe Pareto order, the SUs in this coalition will split, and thesearch for further split forms is not needed. In summary, thepractical aspects dictated by the cognitive network such asthe increasing false alarm cost, the errors on the reportingchannel, the SUs’ locations and the sequential split searchwill significantly diminish the split complexity. In addition, thecomplexity of both merge and split is also reduced due to thefact that the algorithm is distributed, and hence all decisionsare local to the coalitions.

In CF-PD, the adjust operation can be decided by eachcoalition internally similar to the merge or split operationsas explained in Section VI. Further, for the CF-PD algorithm,typically the merge and split operation will be less complexthan for the CF algorithm in Table I since winning coalitionsdo not participate in the merge and split process. However,for CF-PD, another operation comes into place, which is theadjust operation. The adjust operation islocal to eachwinningcoalition S in the network (adjust has no effect on losingcoalitions) and its complexity grows linearly in|S| (number ofSUs inS). Generally, the coalitions that form in the networkare small, thus, the adjust rule complexity is very small.

C. Coalition Based Sensing

Once the coalitions form, the operation of the networkwithin every coalition (Phase 3 in the CF and CF-PD al-gorithms) is, as briefly prescribed in Sections II and IV-A,systematic. First and foremost, within every coalition each SUprepares asingle sensing bit based on its local observation,and then delivers this bit to the coalition head through thewireless channel. Subsequently, the coalition head can performthe fusion of the bits and broadcast back the result, in a singlebit, to all the coalition members. In this paper, we adopt thecommon OR-rule for decision fusion (hard decisions) that iswidely used in collaborative sensing in cognitive radio [7],[9], [10], [12]–[15]4. The operation of the network in thisphase, i.e., Phase 3, can be used in order to compare thecommunication overhead for collaborative sensing betweenour proposed approach with that of the traditional network-wide fusion center solution such as [7], [9], [10], [12]–[15].The reason is that, the actual collaborative sensing which takesplace in Phase 3 while the coalition formation phase occursonly periodically (seldom in moderately changing environ-ments). For example, in a static network, coalition formation isperformed only once, and subsequently, the SUs can performcollaborative sensing at the level of their coalitions. In this

4Note that, the proposed coalition formation algorithm can be tailored toaccommodate other decision fusion techniques.

regards, in the network-wide fusion center, every single SUinthe network needs to report its sensing bit to the fusion center,which yields up toN bits report whereN is the total numberof SUs. When the network is large and the SUs are spread outfar way from the fusion center, this communication overheadis quite significant. In contrast, in Phase 3 of the proposedalgorithm, in order to perform coalition-based collaborativesensing, the SUs belonging to the same coalition, would needto only do a local exchangeof the sensing bits, at the levelof a coalition, which would have a size much smaller thanN .For instance, as demonstrated in Theorem 2, due to the falsealarm constraint, the size of the coalitions resulting fromtheproposed algorithm is upper bounded. In fact, as will be seenthrough simulations in Section VII, the size of the resultingcoalitions will, in general, be much smaller thanN . Thus,clearly, the bandwidth required for the communications ofthe sensing bits in Phase 3 of our algorithm is quite low, asonly single bits are exchanged between coalition members.In fact, within each coalition, the overhead communicationrequired for performing collaborative sensing will be muchsmaller thanN . Beyond the overhead communication advan-tage, performing collaborative sensing at the level of coalitionswould save power and energy for the SUs since, in general,they would report their bits to a neighboring coalition head.Moreover, the proposed approach would make reporting thesensing bits more reliable since, in each individual formedcoalition, the probability of error over the reporting channelas per (5) would be much smaller than in the case whereall N SUs need to submit their bits to a fusion center. Ina nutshell, performing collaborative sensing at the level ofcoalitions not only reduces the overhead communication but,also, improves reliability and reduces the power consumption(used for sensing) of the SUs. For multiple access within eachcoalition (during sensing bit exchange), well known existingMAC protocols can be readily used, such as simple orthogonalchannels [7], [9], [10], [12]–[15], random access techniques[41], or low temperature handshake solutions such as in [45].For adapting to environmental changes such as mobility, theSUs perform the first two phases of the proposed CF algorithmin Table I periodically over time (see Subsection IV-A).

D. Impact of Potential Estimation Errors

In this paper, as mentioned in Section II, all the SUs areconsidered as honest nodes that do not exhibit any maliciousorcheating behavior. Hence, only estimation errors (e.g., due to alow received SNR from the PU) may have some impact on theformed coalitions, if, once a coalition forms, the probabilityof miss of an SU is not as good as envisioned during thecoalition formation process. Here, all the cooperation decisionsare based on the probability of miss that is averaged over thefading realizations, as per (1). In this context, the probabilityof miss in (1) already captures the possibility of having alow SNR received from the PU, which, in turn, maps intoan estimation error. Therefore, the SUs that are more aptto making estimation errors, i.e., SUs far away from thePU, would exhibit a high non-cooperative probability of missas per (1). In consequence, whenever such SUs report their

15

probabilities of miss to potential merge candidates, thesecandidates would be aware of the high probability of missand would use it in order to find out whether the cooperativeperformance, i.e., as per (3), would satisfy the Pareto order ornot. Hence, the estimation errors due to low SNR, as obtainedby (1), are handled adequately through the Pareto order mergeor split decisions that are based on the utility function in (6).Therefore, these errors do not perturb the convergence of themerge-and-split algorithm.

Nonetheless, following the convergence of merge-and-split,consider a group of SUs that have merged together intoa coalition S. This coalition, in Phase 3 of the algorithm,would start acting cooperatively and performing distributedcollaborative sensing. During this phase, the SUs inS canmonitor their overall performance as opposed to the expectedperformance when the merge agreement has occurred. Inthis case, if it turns out that, when the SUs inS operatecooperatively, the actual probability of miss achieved by one ofthe SUs inS is, due to estimation errors, much higher than thevalue potentially envisioned during coalition formation (e.g.,this can be estimated through the number of collisions thatthe SUs in this coalition had with the PU when they startaccessing the spectrum), then this SU can be excluded by thecoalition. The procedure for excluding these SUs is out of thescope of the present paper; however, one can use approachessimilar to the ones used for identifying malicious SUs (e.g.,onion peeling) such as in [46], [47].

VII. S IMULATION RESULTS AND ANALYSIS

For simulations, the following network is set up: The PUis placed at the center of a3 km ×3 km square area withthe SUs randomly deployed in the area around the PU. Weset the PU transmit powerPPU = 100 mW, the SU transmitpower for reporting the sensing bitsPi = 10 mW, ∀i ∈ Nand the noiseσ2 = −90 dBm. For path loss, we setµ = 3and κ = 1. The maximum false alarm constraint is set toα = 0.1, as recommended by the IEEE 802.22 standard [48].In practical cognitive networks, it is desirable to reduce theaccess delays and the bandwidth needed for sensing [3], [8],hence, the the time bandwidth productm is generally chosento be small [3], [7]–[9], [12], [13]. In the simulations, unlessstated otherwise, we selectm = 5 which is a common valueused widely in all literature [7]–[9], [12], [13]. In this section,all statistical results (when applicable) are averaged over 5000random locations of the SUs as well as a range of energydetection thresholdsλ that do not violate the false alarmconstraint; this in turn, maps into an average over the non-cooperative false alarm rangePf ≤ α (obviously, forPf > αno cooperation is possible).

In Fig. 2, we show a snapshot of the network structureresulting from the proposed CF (dashed line) and CF-PD(solid line) algorithms forN = 16 randomly placed SUs,a non-cooperative false alarmPf = 0.01, and a targetdetection probabilityχ = 95% for CF-PD. We note that, whilethe CF-PD yields10 coalitions which comprise5 SUs thatdecided to remain non-cooperative, the CF algorithm yieldsonly 5 coalitions with no SUs remaining non-cooperative.

−1.5 −1 −0.5 0 0.5 1 1.5

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Fig. 2. A snapshot of the coalitional structure resulting from both CF (indashed line) and CF-PD (solid line) algorithms forN = 16 SUs,Pf = 0.01,and a target detection probabilityχ = 95% for CF-PD.

Due to their proximity to the PU, the non-cooperative detec-tion probabilities of SUs2, 3, 7, 10, and 16 are, respectively,98.8%, 99.8%, 96.6%, 99.1%, and 97.2%, which satisfy therequired target thresholdχ = 95% for these SUs. Thus, inCF-PD, SUs2, 3, 7, 10 and 16 form a singleton MWC eachand have no incentive to cooperate as they already satisfy thedesired performance non-cooperatively. Moreover, coalitions{1, 4}, {8, 12}, and {6, 15} are MWCs, and, thus, do notcooperate any further during CF-PD. In contrast, when theCF algorithm is applied, the SUs seek to minimize theirprobability of miss and, thus, SUs2 and 3 form a coalitionwith SU 9, SUs 10 and 16 form a coalition with SU5,while SU 7 minimizes its probability of miss by mergingwith coalition {1, 4}. Further, for CF, SUs6, 8, 12, and 15minimize their probability of miss by forming a single coali-tion {6, 8, 12, 15}. Finally, we note that for both CF andCF-PD coalition{11, 13, 14} forms, and this coalition is alosing coalition since the achieved probability of detectionby its member SUs is only89.5% (due to the location ofits members). Moreover, this coalition is unable to find anysuitable partners that can help it improve its utility furtherin CF, or meet the target detection threshold in CF-PD. Ina nutshell, Fig. 2 illustrates how, in a practical cognitivenetwork, different coalitions can emerge depending on theincentives and cooperative objectives of the SUs.

Figs. 3 and 4 show, respectively, the average probabilitiesofmiss and the average false alarm probabilities achieved perSUfor different network sizes and for a typical time-bandwidthproduct of m = 5 and a large time-bandwidth product ofm = 1000. In Fig. 3, we show that, form = 5, the proposedCF algorithm yields a significant advantage in the averageprobability of miss reaching up to88.45% reduction (atN =50) relative to the non-cooperative case which is an order ofmagnitude of improvement. This advantage is increasing withthe network sizeN . In this figure, we also show the results ofa centralized exhaustive search solution which minimizes the

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0 5 10 15 20 25 30 35 40 45 50

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Non−cooperative m = 5Proposed CF algorithm m = 5Centralized partition search m = 5Non−cooperative m = 1000Proposed CF algorithm m = 1000Centralized partition search m = 1000

Beyond N=7, the solutionof the centralized approach ismathematically and computationallyintractable

Beyond N=7, the solution of the centralizedapproach is mathematically and computationallyintractable

Fig. 3. Average probability of miss versus number of SUsN for m = 5andm = 1000.

average probability of miss per SU, subject to the false alarmconstraintα. This solution is shown for up toN = 7 since itis mathematically intractable for larger networks.

We note that, for allm, a gap exists between the perfor-mance of the CF algorithm and that of the centralized solution.This gap stems mainly from the individual choices of theSUs when they act in a distributed manner, i.e., selectingtheir partners based on a balance between gains and costs asopposed to a centralized approach that does not capture theseindividual incentives. For CF, these incentives are captured bythe cost function in (8) which increases drastically when thefalse alarm probability is in the vicinity ofα. This increasedcost makes it generally non-beneficial for coalitions with highfalse alarm levels, e.g., false alarm values close toα, tocollaborate in the distributed CF approach as they require alarge probability of miss improvement to compensate the costin their utility (6). As a result, even though the CF algorithmyields a performance gap in terms of miss probability for allm, the individual decisions of the SUs force a false alarmfor the distributed case smaller than that of the centralizedsolution as seen in Fig. 4 at allm. In other words, whenan SU (or coalition) is taking its own decision, it needs toobtain a gain, in terms of probability of miss, that is relativelylarger than the increase in the false alarm (relative to theconstraintα) as captured by the utility and cost function in(6). In contrast, in the centralized approach, the optimizationproblem stated in Section II does not involve the individualutilities or decisions of the SUs, instead it optimizes the overallaverage probability of miss given a constraint on the overallfalse alarm probability. Consequently, the centralized approachwould definitely yield a smaller average probability of miss,but it also yields a false alarm level that is much closerto the constraintα, i.e., a false alarm level that is higherthan in the distributed approach. This result is corroboratedin Fig. 4 where, at allm, the achieved average false alarmby the proposed distributed solution outperforms that of thecentralized solution, however, it is still outperformed bythenon-cooperative case.

In addition, even though a smallm is typical in cognitive

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Beyond N = 7, the solution of the centralized approach is mathematicallyand computationally intractable

Fig. 4. Average false alarm probability versus number of SUsN for m = 5.

networks [7]–[9], [12], [13], a large time-bandwidth productmay be used in some instances when detection at low SNR isrequired. By increasing the observation time (or the bandwidthused for sensing), the SUs can improve their non-cooperativeaverage probability of miss non-cooperatively as shown inFig. 3 form = 1000. However, this gain comes at the expenseof a long delay for spectrum access or a wasted bandwidth.At m = 1000, although for the centralized approach a perfor-mance improvement in terms of probability of miss is possible,we note that for the proposed distributed CF algorithm nocoalitions are formed and the performance is comparable tothe non-cooperative case. This is mainly due to the fact that,at m = 1000, the average non-cooperative probability of missof all SUs is already quite small (around2%), and, hence,when the SUs act in a distributed manner, they find littleimprovement in their utility (6) through cooperation, sincethe gains in probability of miss will be accompanied with alarge increase in the false alarm as shown in Fig. 4 for thecentralized approach atm = 1000. However, by comparingthe results atm = 5 and m = 1000, Fig. 3 shows that,by using our proposed CF algorithm, the SUs can improvetheir detection significantly without the need for a wastedbandwidth or long access delays. For example, for a networkof N = 50 SUs atm = 5, the SUs can achieve a probability ofdetection which is comparable (and slightly outperforms) thenon-cooperative case atm = 1000. Hence, the comparisonbetween the CF results atm = 5 and the non-cooperativeresults atm = 1000 in Fig. 3 demonstrates that, instead ofincreasing the sensing observation time or bandwidth (notablywhen detection at low SNR is needed), the SUs can utilize theproposed CF algorithm to improve their detection performancewhile saving bandwidth and reducing their spectrum accessdelay.

In Fig. 5, we show the average probabilities of miss perSU for different thresholdsλ (range of non-cooperative falsealarmPf ≤ α) for N = 7. Fig. 5 shows that, asPf decreases,the performance advantage of CSS for both the centralizedand distributed CF solutions increases (except for very smallPf where the advantage in terms of miss probability reaches

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Non−cooperative sensingProposed CF algorithmCentralized partition search solution

Fig. 5. Average probabilities of miss per SU versus non-cooperative falsealarm Pf (or energy detection thresholdλ) for a network ofN = 7 SUswith m = 5.

its maximum). The performance gap between centralizedand distributed is once again compensated by a false alarmadvantage for the distributed CF solution as already seen andexplained in Fig. 4 forN = 7. Finally, Fig. 5 shows that asPf goes toα = 0.1, the CSS advantage diminishes as thenetwork tends to the non-cooperative case.

In Fig. 6 we investigate, through Theorem 1 (withPf =0.01), the merge possibilities between two secondary usersSU1 and SU2, with Pm,1 ≤ Pm,2. The results are shownfor different locations of SU1 when SU2 belongs to circlescentered at the PU and with different radii (each circle radiusrepresents a class of non-cooperative probability of miss).Fig. 6 shows the angle, in degrees, of the sector centeredat SU1 and which intercepts the merge region arc at thevarious circles where SU2 belongs (obviously a larger angleintercepts a larger merge region arc). The angle is obtainedbya simple geometrical computation after finding the minimumdistanced1,2 between SU1 and SU2 using Theorem 1. Firstand foremost, Fig. 6 shows that the approximate solutionobtained by Theorem 1 is almost the same as the exactnumerical (graphical) solution, which corroborates that the ap-proximation in Theorem 1 for the distances is tight. Moreover,Fig. 6 clearly shows that, for the different SU1 locations, themerge region is larger as the radius of SU2’s circle is smaller(i.e. error on the reporting channel is smaller). Further, we notethat as SU1 becomes closer to the PU, the number of circleswhere SU2 can belong and on which a merge region existsbecomes smaller. For instance, when SU1 is at 0.8 km, it canmerge with SU2 moving on circles of radii up to2.3 km. Inother words, when SU2 belongs to a circle with a radius1.5 kmlarger than the radius of the circle to which SU1 belong, thecoalition cannot form. However, when SU1 is at 0.4 km itcan merge with SU2 belonging to circles with radii up to1.5 km, hence, only around1.1 km larger than the radiusof the circle to which SU1 belong. This result shows that,when SU1 has a smaller probability of miss (i.e. better non-cooperative performance), the collaboration possibilities withdistant SUs become smaller, since SU1 is already satisfied withits non-cooperative probability of miss and does not have an

1 1.2 1.3 1.5 1.7 1.8 2 2.3 2.4 2.50

50

100

150

200

250

300

Radius of circle where SU2 belongs (km)

Ang

le in

terc

eptin

g th

e m

erge

reg

ion

arc

betw

een

SU

1 and

SU

2 (de

gree

s)

Approx. via Theorem 1, SU1 at 0.4 km

Exact, SU1 at 0.4 km

Approx. via Theorem 1, SU1 at 0.5 km

Exact, SU1 at 0.5 km

Approx. via Theorem 1, SU1 at 0.8 km

Exact, SU1 at 0.8 km

Fig. 6. Approximate (via Theorem 1) and exact (graphical solution forinequality) angle centered at SU1 and intercepting the merge region arc (forforming a coalition of size2) between SU1 and SU2 (Pm,1 ≤ Pm,2) whenSU2 belongs to circles centered at the PU and having different radii (eachcircle radius represents a different non-cooperative probability of miss) andfor different locations of SU1.

incentive to collaborate with very distant SUs. In summary,Fig. 6 provides interesting insights on how two SUs can merge,depending on their location with respect to the PU.

Fig. 7 shows how the structure of the cognitive networkwith N = 50 SUs evolves and self-adapts, while all theSUs use a basic random walk mobility model whereby thenodes move at a constant speed of of120 km/h in a randomdirection uniformly distributed between0 and2π, over periodsof 5 seconds for a total duration of5 minutes. The proposedCF algorithm is repeated periodically by the SUs everyθ = 5seconds, in order to provide self-adaptation to mobility. Asthe SUs move, the structure of the network changes, with newcoalitions forming and others splitting. The network starts witha non-cooperative structure made up of50 independent SUs. Inthe first step, the network self-organizes in18 coalitions withan average of2.77 SUs per coalition. As time evolves, the SUsself-organize the structure which is changing as new coalitionsform or others split. After the5 minutes have elapsed the finalstructure is made up of20 coalitions with an average of2.5SUs per coalition.

In Fig. 8, we evaluate the performance of the CF-PDalgorithm of Table II by showing the average percentageof winning SUs (average over locations of SUs and non-cooperative false alarm rangePf ≤ α ), i.e., SUs achievinga detection probabilityχ = 95% imposed by the PU (thechoice of χ = 95% conforms with the recommendationof the IEEE 802.22 standard [48]) as a function of thenetwork sizeN . We compare the performance of CF-PDto that of the non-cooperative case as well as the optimalpartition that maximizes the number of winning SUs andwhich is found by a centralized search over all partitionsformed of MWCs. For CF-PD, the percentage of winningSUs increases with the number of SUs since the possibilityof finding cooperating partners increases. In contrast, thenon-cooperative approach presents an almost constant performancewith different network sizes. Cooperation presents a significant

18

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50N

umbe

r of

coa

litio

n

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

8

9

10

Num

ber

of S

Us

per

coal

ition

s

Time (seconds)

Average number of coalitions

Average number of SUs per coalition

Fig. 7. Evolution of the network’s structure over time usingCF for a networkof N = 50 SUs, a constant speed of120 km/h and a non-cooperative falsealarmPf = 0.01.

advantage over the non-cooperative case in terms of averagepercentage of winning SUs, and this advantage increaseswith the network size approximately tripling the number ofwinning SUs (relative to the non-cooperative case) atN = 50.Furthermore, compared to the optimal solution, the proposedCF-PD algorithm achieves a highly comparable performancewith a loss not exceeding3.7 percentage points atN = 7.This shows that, by using the proposed CF-PD algorithm, thenetwork can achieve a performance that is close to optimal.Note that, for networks larger thanN = 7 SUs, finding theoptimal partition through exhaustive search is mathematicallyand computationally intractable. However, as the network sizeincreases CF-PD presents a portion of winning SUs that isclose to100%, reaching up to87.25% at N = 50, hence,as the network grows, the performance gap relatively to theoptimal case will not increase much. Finally, in Fig. 8, we notethat for the proposed algorithm, the non-cooperative case,andthe optimal solution, some SUs (or coalitions) are still losing,i.e., unable to meet theχ = 95% constraint. This is mainly dueto the current locations and channels of these SUs. In orderfor these SUs to meet the desired performance level, they needto wait for environmental changes, such as their own mobility(moving to a better location) or the arrival of new SUs thatcan boost their performance.

In Fig. 9, we show the average percentage of winning SUs(average over locations of SUs and non-cooperative false alarmrange Pf ≤ α ) achieved by CF-PD forN = 50 SUsas the desired probability of missγ = 1 − χ increases(i.e. desired detection probability decreases) within therangerecommended by the IEEE 802.22 [48]. Asγ increases, thenumber of winning SUs increases for both the CF-PD and thenon-cooperative case, as it becomes easier to guarantee thisrequired probability. Fig. 9 shows that the proposed CF-PD al-gorithm guarantees that at least74.1% of the SUs (atγ = 0.01,the most stringent constraint) achieve the desired detectionprobability. Further, at allγ, CF-PD presents a significantadvantage over the non-cooperative case which increases asthe desired detection probability becomes stricter. For instance,at γ = 0.01 (i.e. χ = 99%), CF-PD allows around eight times

0 234567 10 15 20 25 30 40 5020

30

40

50

60

70

80

90

100

Number of SUs (N)

Ave

rage

per

cent

age

of w

inni

ng S

Us

(%)

Non−cooperative sensing Proposed CF−PD algorithm for CSSCentralized partition search solution

The solution of the centralized approach ismathematically and computationally intractable beyond N = 7 SUs

Fig. 8. Average percentage of winning SUs achievingχ = 95% detectionprobability versus number of SUsN .

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

10

20

30

40

50

60

70

80

90

100

Desired probability of miss γ

Ave

rage

per

cent

age

of w

inni

ng S

Us

(%)

Non−cooperative sensingProposed CF−PD algorithm for CSS

Fig. 9. Average percentage of winning SUs versus the desiredprobabilityof missγ = 1− χ for the proposed CF-PD algorithm forN = 50 SUs.

more SUs (relative to the non-cooperative case) to achieve therequired probability of detection. In Fig. 10, for a networkof N = 50 SUs, we evaluate the sizes of the coalitionsresulting from CF and CF-PD (withχ = 95%) and comparethem with the the upper boundMmax derived in Theorem 2.First, as the non-cooperativePf increases, for CF and CF-PD, both the maximum and the average size of the formedcoalitions decrease converging towards the non-cooperativecase asPf reaches the constraintα = 0.1. Through this result,we can clearly see the limitations that the detection-falsealarmprobabilities trade off for CSS imposes on the coalition sizeand network structure. Also, in Fig. 10, we show that, albeitthe upper bound on coalition sizeMmax increases drasticallyasPf becomes smaller, the average maximum coalition sizeachieved by the proposed algorithms does not exceed5 SUsper coalition for CF and3 SUs per coalition for CF-PD. Thisresult shows that the network structure is composed of a largenumber of small coalitions rather than a small number oflarge coalitions, especially for CF-PD, even whenPf is smalland the collaboration possibilities are high. In this context,for CF-PD, the average and average maximum coalition sizes

19

10−1.9

10−1.7

10−1.5

10−1.3

1

2

3

4

5

6

7

8

9

10

Non−cooperative false alarm Pf

Ave

rage

coa

litio

n si

ze (

num

ber

of S

Us)

Upper bound on coalition size (Mmax

)

Average maximum number of SUs for CFActual average number of SUs for CFAverage maximum number of SUs for CF−PDActual average number of SUs for CF−PD

Fig. 10. Maximum and average coalition size versus non-cooperative falsealarm Pf (or energy detection thresholdλ) for CF and CF-PD forN =50 SUs.

are smaller than for CF, since in CF-PD, once a coalitionachieves the detection probabilityχ, no further collaborationis required.

In Fig. 11, for CF and CF-PD (withχ = 95%) we show,over a period of10 minutes, the frequency in terms of merge-and-split and adjust operations per minute for various speedsof the SUs in a mobile cognitive network withN = 50 SUsand Pf = 0.01. As the velocity increases, the frequency ofall operations increases for both CF and CF-PD due to thechanges in the network structure incurred by mobility. TheCF algorithm incurs a higher frequency of merge-and-splitthan CF-PD since, in the CF, the SUs seek to minimize theirprobabilities of miss with no constraints. In contrast, theCF-PD presents a significantly smaller frequency of merge-and-split operations, since the SUs, once they form a winningcoalition, are no longer interested in participating in themerge-and-split operations. However, the CF-PD complements thislower number of merge-and-split operations with a numberof adjust operations, needed in order to form MWCs out ofwinning coalitions. Finally, the frequency of all operations(merge, split and adjust) for CF-PD is comparable to thefrequency of merge-and-split in CF at almost all velocities.However, CF-PD possesses a lower complexity as the adjustrule has a complexity smaller than merge or split.

VIII. C ONCLUSIONS

This paper introduced a novel and distributed model forperforming collaborative sensing in cognitive radio networks.In the proposed model, a network of SUs can interact and formcooperating coalitions for improving their spectrum sensingperformance. We modeled the problem as a coalitional gamewith non-transferable utility and derived distributed algorithmsfor coalition formation. First, we proposed a distributed coali-tion formation (CF) algorithm based on two rules of merge andsplit that enable SUs in a cognitive network to collaborate andmaximize the probability of detecting the PU presence whileaccounting for the costs in terms of false alarm probability.We characterized the network structure resulting from the pro-

0 20 40 60 80 100 1200

10

20

30

40

50

60

SUs velocity (km/h)

Fre

quen

cy o

f ope

ratio

ns p

er m

inut

e

Frequency of merge and split operations for CFFrequency of merge and split operations for CF−PDFrequency of adjust operations for CF−PD

Fig. 11. Frequency of merge-and-split operations per minute (for CF andCF-PD) and adjust operations per minute (for CF-PD) achieved over a periodof 10 minutes for different velocities forN = 50 SUs andPf = 0.01.

posed algorithm as well as studied and proved its various prop-erties. We further complemented the proposed algorithm withan adjunct coalitional voting game for providing distributedcoalition formation with detection probability guarantees (CF-PD) when such detection probabilities are imposed by the PU.Simulation results showed that the CF algorithm reduces theaverage probability of miss per SU up to88.45% compared tothe non-cooperative case while the CF-PD algorithm enablesup to 87.25% of the SUs to achieve the required detectionprobability while forming minimal winning coalitions. Theresults also showed how, through the proposed algorithms,the SUs can self-organize and adapt the network structure toenvironmental changes such as mobility.

APPENDIX

A. Proof of Theorem 1

Consider two SUs having non-cooperative probabilities ofmissPm,i andPm,j , respectively, such thatPm,i ≤ Pm,j . Inthis case, if and when coalitionS = {i, j} forms, then SUiis thecoalition headas per Convention 1 and we have

Qm,S=Pm,i[Pm,j(1 − Pe,j,i)+(1− Pm,j)Pe,j,i], (15)

Qf,S=1− [(1− Pf )[(1 − Pf )(1 − Pe,j,i)+PfPe,j,i]] . (16)

By virtue of Property 1, the payoffs of the SUsi and j ifS forms are given by,φi(S) = φj(S) = v(S). In order forthe merge between SUsi and j to occur (for formingS) wemust haveS ⊲ {{i}, {j}}. Using the Pareto order, and giventhat v({i}) ≥ v({j}), S ⊲ {{i}, {j}} translates intoφi(S) =v(S) > v({i}) ≥ v({j}). After algebraic manipulations andusing the expressions ofv({i}) andv(S) in (6), we obtain

D · Pe,j,i + C

α2≤ log

(

1−

(

A · Pe,j,i +B

α

)2)

, (17)

with Pe,j,i the probability of error between the two SUs, andA=(2Pf − 1)(Pf − 1), B=Pf (2−Pf ), C=Pm,i(Pm,j − 1)+(

α2 · log

(

1−(

Pf

α

)2))

andD=Pm,i(1− 2Pm,j).

20

The inequality in (17) is a transcendental inequality that ad-mits only a numerical (graphical) solution, with no analyticalexpression [49]. The numerical solution (by solution we implythe range ofPe,j,i that satisfy the inequality) can be derivedby plotting both sides of the equation as a function ofPe,j,i.A sufficient approximate solution (i.e. approximate range ofPe,j,i that satisfy the inequality) for (17) is found by solving

D · Pe,j,i + C

α2≤ log

(

1−

(

A · Pe,j,i +B

α

)2)

, (18)

where Pe,j,i is the value that maximizesD·Pe,j,i+C

α2 (sincethe probability of error is bounded, this maximum is finite)and which directly depends on the sign ofD. By algebraicmanipulation, (18) yields a quadratic inequality inPe,j,i

A2P 2e,j,i + (2AB)Pe,j,i +B2 + α2(e(

C+Pe,j,iD

α2 ) − 1) ≤ 0,(19)

with discriminant∆ = A2α2(1−e(C+Pe,j,iD

α2 )). We distinguishtwo cases depending on the sign ofD=Pm,i(1− 2Pm,j).

Case 1: D ≥ 0, that isPm,j ≤12 . First, we note that: (i)Pf

which depends solely on the detection thresholdλ is generallychosen relatively small compared to1 (a typical value is0.01),and (ii) the inequality (17) admits a solution if and only ifA·Pe,j,i+B < α (otherwise the right hand side of (17) goes to−∞). A direct result of (ii) is that any solutionPe,j,i for (17)cannot exceedα−B

A. By using (i), we have thatA ≈ 1 andB ≈

0, hence, any solutionPe,j,i can be approximately bounded byPe,j,i ≤ α, and thus, the probability of errorPe,j,i = α. In thiscase,∆ is always positive. After manipulation(1−e(

C+αD

α2 )) >0 yields

Pm,i(1−Pm,j)+Pm,i(2Pm,j − 1)>α2 log (1−P 2f

α2). (20)

The RHS is always negative (barrier function [27]). The LHScan be shown to be always positive since otherwise, we needPm,i(1−Pm,j)+αPm,i(2Pm,j−1) < 0 which yields two casesPmj

> α−12α−1 ≥ 1 if α < 0.5 or Pmj

< α−12α−1 ≤ 0 if α > 0.5

that are both impossible for0 ≤ Pm,j ≤ 1 (probability ofmiss). Hence, whenPm,j ≤

12 , (18) admits two roots

Pe,j,i =(−B + α

√

(1− e(C+αD

α2 )))

A,

P′

e,j,i =(−B − α

√

(1− e(C+αD

α2 )))

A. (21)

Or P′

e,j,i < 0 (A > 0 due to (i)). Consequently, thesolution of (19) isPe,j,i ≤ Pe,j,i where Pe,j,i is given by(after algebraic manipulation and substitutingA,B,C, andDfor their values)

Pe,j,i =Pf (Pf − 2) + α

√

(1− (1−P 2

f

α2 )ePm,i(1−α)(Pm,j−1)

α2 )

(2Pf − 1)(Pf − 1).

(22)The solutionPe,j,i ≤ Pe,j,i with Pe,j,i in (22) provides a

sufficient analytical approximation for the solution (range ofprobabilities) of the initial transcendental inequality in (17)whenPm,j ≤

12 .

Case 2: D < 0, that isPm,j >12 . In this case, we can easily

see thatPe,1,2 = 0, and it is easily seen that the discriminant

∆ is positive. By following a reasoning analogous to Case 1,we find that the sufficient analytical approximation isPe,j,i ≤Pe,j,i with Pe,j,i given by

Pe,j,i =Pf (Pf − 2) + α

√

(1− (1−P 2

f

α2 )ePm,i(Pm,j−1)

α2 )

(2Pf − 1)(Pf − 1).

(23)We cam combine these two cases into one solution, that is

Pe,j,i ≤ Pe,j,i with

Pe,j,i=Pf (Pf − 2)+α

√

(1− (1−P 2

f

α2 )ePm,i(1−η)(Pm,j−1)

α2 )

(2Pf − 1)(Pf − 1),

(24)with η = α, if Pm,j ≤

12 andη = 0 otherwise.

REFERENCES

[1] Federal Communications Commission, “Spectrum policy task forcereport,” Report ET Docket no. 02-135, Nov. 2002.

[2] S. Haykin, “Cognitive radio: brain-empowered wirelesscommunica-tions,” IEEE J. Select. Areas Commun., vol. 23, pp. 201–220, Feb. 2005.

[3] D. Niyato, E. Hossain, and Z. Han,Dynamic spectrum access incognitive radio networks. Cambridge, UK: Cambridge University Press,in print 2009.

[4] M. Dohler, S. Ghorashi, M. Ghozzi, M. Arndt, F. Said, and A. Aghvami,“Opportunistic scheduling using cognitive radio,”Elsevier Science Jour-nal, Special Issue on Cognitive Radio, vol. 7, pp. 805–815, Sep. 2006.

[5] D. Niyato and E. Hossain, “Cognitive radio for next generation wirelessnetworks: An approach to opportunistic channel selection in IEEE802.11-based wireless mesh,”IEEE Trans. Wireless Commun., vol. 16,no. 1, pp. 46–54, Feb. 2009.

[6] D. Cabric, M. S. Mishra, and R. W. Brodersen, “Implementation issuesin spectrum sensing for cognitive radios,” inProc. Asilomar Conferenceon Signals, Systems, and Computers, Pacific Grove, CA, USA, Nov.2004, pp. 772–776.

[7] A. Ghasemi and E. S. Sousa, “Collaborative spectrum sensing foropportunistic access in fading environments,” inIEEE Symp. NewFrontiers in Dynamic Spectrum Access Networks, Baltimore, USA, Nov.2005, pp. 131–136.

[8] M. Ghozzi, M. Dohler, F. Marx, and J. Palico, “Cognitive radio: Methodsfor detection of free bands,”Elsevier Science Journal, Special Issue onCognitive Radio, vol. 7, pp. 794–805, Sep. 2006.

[9] E. Visotsky, S. Kuffner, and R. Peterson, “On collaborative detectionof TV transmissions in support of dynamic spectrum sensing,” in IEEESymp. New Frontiers in Dynamic Spectrum Access Networks, Baltimore,USA, Nov. 2005, pp. 338–356.

[10] S. M. Mishra, A. Sahai, and R. Brodersen, “Cooperative sensing amongcognitive radios,” in Proc. Int. Conf. on Communications, Istanbul,Turkey, Jun. 2006, pp. 1658–1663.

[11] S. Huang, X. Liu, and Z. Ding, “Optimal transmission strategies fordynamic spectrum access in cognitive radio networks,”IEEE Trans.Mobile Computing, vol. 8, pp. 1636–1648, Dec. 2009.

[12] B. Wang, K. J. Liu, and T. Clancy, “Evolutionary cooperative spectrumsensing game: how to collaborate?”IEEE Trans. Communications,vol. 58, no. 3, pp. 890–900, Mar. 2010.

[13] W. Zhang and K. B. Letaief, “Cooperative spectrum sensing withtransmit and relay diversity in cognitive networks,”IEEE Trans. WirelessCommun., vol. 7, no. 12, pp. 4761–4766, Dec. 2008.

[14] C. Sun, W. Zhang, and K. B. Letaief, “Cooperative spectrum sensingfor cognitive radios under bandwidth constraint,” inProc. IEEE WirelessCommunications and Networking Conf., Hong Kong, Mar. 2007.

[15] L. S. Cardoso, M. Debbah, P. Bianchi, and J. Najim, “Cooperative spec-trum sensing using random matrix theory,” inin Proc. of InternationalSymposium on Wireless Pervasive Computing, Santorini, Greece, May2008.

[16] F. Moghimi, A. Nasri, and R. Schober, “Lp-norm spectrumsensing forcognitive radio networks impaired by non-gaussian noise.”in Proc. IEEEGlobal Communication Conference, Hawaii, USA, Dec. 2009.

[17] E. Peh and Y. Liang, “Optimization for cooperative sensing in cognitiveradio networks,” inProc. IEEE Wireless Communications and Network-ing Conf., Hong Kong, Mar. 2007, pp. 27–32.

[18] Y. Liang, Y. Zeng, E. Peh, and A. T. Hoang;, “Sensing-throughputtradeoff for cognitive radio networks,”IEEE Trans. Wireless Commun.,vol. 7, no. 4, pp. 1326–1337, Apr. 2008.

[19] C. Sun, W. Zhang, and K. B. Letaief, “Cluster-based cooperativespectrum sensing in cognitive radio systems,” inProc. Int. Conf. onCommunications, Glasgow, Scotland, Jun. 2007, pp. 2511–2515.

[20] H. Kim and K. G. Shin, “In-band spectrum sensing in cognitive radionetworks: energy detection or feature detection?” inProc. of 14th ACMInt. Conf. on Mobile Computing and Networking, San Francisco, CA,USA, Sep. 2008, pp. 14–25.

21

[21] J. Proakis,Digital Communications, 4th edition. New York: McGraw-Hill, 2001.

[22] T. Sandholm, K. Larson, M. Anderson, O. Shehory, and F. Tohme,“Coalition structure generation with worst case guarantees,” ArtificalIntelligence, vol. 10, pp. 209–238, Jul. 1999.

[23] L. Lovasz,Combinatorial Problems and Exercises 2nd edition. Ams-terdam, the Netherlands: North-Holland, Aug. 1993.

[24] R. B. Myerson,Game Theory, Analysis of Conflict. Cambridge, MA,USA: Harvard University Press, Sep. 1991.

[25] W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Basar,“Coalitiongame theory for communication networks: A tutorial,”IEEE SignalProcessing Mag., Special issue on Game Theory in Signal Processingand Communications, vol. 26, no. 5, pp. 77–97, Sep. 2009.

[26] D. Ray, A Game-Theoretic Perspective on Coalition Formation. NewYork, USA: Oxford University Press, Jan. 2007.

[27] S. Boyd and L. Vandenberghe,Convex Optimization. New York, USA:Cambridge University Press, Sep. 2004.

[28] K. Apt and A. Witzel, “A generic approach to coalition formation,” inProc. of the Int. Workshop on Computational Social Choice (COMSOC),Amsterdam, the Netherlands, Dec. 2006.

[29] C. S. R. Murthy and B. Manoj,Ad Hoc Wireless Networks: Architecturesand Protocols. New Jersey, NY, USA: Prentice Hall, Jun. 2004.

[30] C. J. L. Arachchige, S. Venkatesan, and N. Mittal, “An asynchronousneighbor discovery algorithm for cognitive radio networks,” in Proc.IEEE Symp. on New Frontiers in Dynamic Spectrum Access Networks(DySPAN), Chicago, IL, USA, Oct. 2008, pp. 1–5.

[31] S. Krishnamurthy, N. Mittal, and R. Chandrasekaran, “Neighbor dis-covery in multi-receiver cognitive radio networks,”Int. Journal ofComputers and Applications, Jun. 2008.

[32] Z. Damljanovic, “Cognitive radio access discovery strategies,” inProc.of 6th Int. Symp. on Communication Systems, Networks, and DigitalSignal Processing, Graz, Austria, Jul. 2008, pp. 251–255.

[33] G. Owen,Game Theory, 3rd edition. London, UK: Academic Press,Oct. 1995.

[34] D. Raychaudhuri and X. Jing, “A spectrum etiquette protocol for efficientcoordination of radio devices in unlicensed bands,” inProc. of IEEEInt. Symp. on Personal, Indoor and Mobile Radio Communications(PIMRC), Beijing, China, Sep. 2003.

[35] P. Houz, S. B. Jemaa, and P. Cordier, “Common pilot channel fornetwork selection,” inProc. of IEEE Vehicular Technology - Spring,Melbourne, Australia, May 2006.

[36] M. Filo, A. Hossain, A. R. Biswas, and R. Piesiewicz, “Cognitive pilotchannel: Enabler for radio systems coexistence,” inProc. of Second Int.Workshop on Cognitive Radio and Advanced Spectrum Management,Aalborg, Denmark, May 2009.

[37] O. Sallent, J. Perez-Romero, R. Agusti, and P. Cordier,“Cognitivepilot channel enabling spectrum awareness,” inProc. Int. Conf. onCommunications, Workshop on, Dresden, Germany, Jun. 2009.

[38] J. Perez-Romero, O. Salient, R. Agusti, and L. Giupponi, “A novel on-demand cognitive pilot channel enabling dynamic spectrum allocation,”in Proc. of IEEE Int. Symp. on New Frontiers in Dynamic SpectrumAccess Networks (DySPAN), Dublin, Ireland, Apr. 2007.

[39] M. Maida, J. Najim, P. Bianchi, and M. Debbah, “Performance analysisof some eigen-based hypothesis tests for collaborative sensing,” in Proc.of IEEE Workshop on Statistical Signal Processing, Cardiff, UK, Aug.2009.

[40] J. Proakis,Digital Communications. McGraw-Hill, 4th edition, Aug.2000.

[41] S. Shellhammer, A. Sadek, and W. Zhang, “Technical challenges forcognitive radio in TV white space spectrum,” inInformation Theoryand Applications Workshop (ITA), San Diego, USA, Feb. 2009.

[42] T. Fuji and Y. Suzuki, “Ad hoc cognitive radio - development tofrequency sharing system by using multi-hop network,” inProc. of IEEEDySPAN.

[43] M. Hoyhtya, A. Hekkala, and A. Mammela, “Spectrum awareness: tech-niques and challenges for active spectrum sensing,”Springer CognitiveNetworks, vol. 3, pp. 353–372, Apr. 2007.

[44] M. Siebert, M. Lott, M. Schinnenburg, and S. Gbbels, “Hybrid informa-tion system,” inProc. of IEEE Vehicular Technology Conference (VTC- Spring), Milan, Italy, May 2004.

[45] B. Shreshta, D. Niyato, Z. Han, and E. Hossain, “Wireless access invehicular environments using bit torrent and bargaining,”in Proc. IEEEGlobal Communication Conference, New Orleans, USA, Dec. 2008.

[46] H. Li and Z. Han, “Catching attacker(s) for collaborative spectrumsensing in cognitive radio systems: A universal approach,”in Proc. ofIEEE DySPAN, Singapore, Apr. 2010.

[47] ——, “Blind dogfight in spectrum: Combating primary useremulationattacks in cognitive radio systems with unknown channel statistics,” inProc. Int. Conf. on Communications, Cape Town, South Africa, May2010.

[48] IEEE 802.22, “Cognitive wireless regional area network - functionalrequirements,” 802.22-06/0089r3, Tech. Rep., Jun. 2006.

[49] V. L. Zaguskin, Handbook of numerical methods for the solution ofalgebraic and transcendental equations. Oxford, UK: Pergamon Press,Oct. 1961.

[50] W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Basar,“Coalitionalgames for distributed collaborative spectrum sensing in cognitive radionetworks,” inProc. of IEEE Int. Conference on Computer Communica-tions (INFOCOM), Rio de Janeiro, Brazil, Apr. 2009.