Coalitional Games for Distributed Collaborative Spectrum Sensing in Cognitive Radio Networks
Walid Saad, Zhu Han, Merouane Debbah, Are Hjørungnes, Tamer Başar
aa r X i v : . [ c s . G T ] M a y Coalitional Games for Distributed CollaborativeSpectrum Sensing in Cognitive Radio Networks
Walid Saad , Zhu Han , M´erouane Debbah , Are Hjørungnes and Tamer Bas¸ar UNIK - University Graduate Center, University of Oslo, Kjeller, Norway, Email: { saad,arehj } @unik.no Electrical and Computer Engineering Department, University of Houston, Houston, USA, Email: [email protected] Alcatel-Lucent Chair in Flexible Radio, SUP ´ELEC, Gif-sur-Yvette, France, Email: [email protected] Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, USA, Email: [email protected]
Abstract — Collaborative spectrum sensing among secondaryusers (SUs) in cognitive networks is shown to yield a significantperformance improvement. However, there exists an inherenttrade off between the gains in terms of probability of detectionof the primary user (PU) and the costs in terms of false alarmprobability. In this paper, we study the impact of this trade offon the topology and the dynamics of a network of SUs seeking toreduce the interference on the PU through collaborative sensing.Moreover, while existing literature mainly focused on centralizedsolutions for collaborative sensing, we propose distributed collab-oration strategies through game theory. We model the problemas a non-transferable coalitional game, and propose a distributedalgorithm for coalition formation through simple merge and splitrules. Through the proposed algorithm, SUs can autonomouslycollaborate and self-organize into disjoint independent coalitions,while maximizing their detection probability taking into accountthe cooperation costs (in terms of false alarm). We study thestability of the resulting network structure, and show that amaximum number of SUs per formed coalition exists for theproposed utility model. Simulation results show that the proposedalgorithm allows a reduction of up to . of the averagemissing probability per SU (probability of missing the detectionof the PU) relative to the non-cooperative case, while maintaininga certain false alarm level. In addition, through simulations, wecompare the performance of the proposed distributed solutionwith respect to an optimal centralized solution that minimizes theaverage missing probability per SU. Finally, the results also showhow the proposed algorithm autonomously adapts the networktopology to environmental changes such as mobility. I. I
NTRODUCTION
In recent years, there has been an increasing growth inwireless services, yielding a huge demand on the radio spec-trum. However, the spectrum resources are scarce and mostof them have been already licensed to existing operators.Recent studies showed that the actual licensed spectrum re-mains unoccupied for large periods of time [1]. In order toefficiently exploit these spectrum holes, cognitive radio (CR)has been proposed [2]. By monitoring and adapting to theenvironment, CRs (secondary users) can share the spectrumwith the licensed users (primary users), operating wheneverthe primary user (PU) is not using the spectrum. Implementingsuch flexible CRs faces several challenges [3]. For instance,CRs must constantly sense the spectrum in order to detect thepresence of the PU and use the spectrum holes without causing
This work was done during the stay of Walid Saad at the CoordinatedScience Laboratory at the University Of Illinois at Urbana - Champaignand is supported by the Research Council of Norway through the projects183311/S10, 176773/S10 and 18778/V11. harmful interference to the PU. Hence, efficient spectrumsensing constitutes a major challenge in cognitive networks.For sensing the presence of the PU, the secondaryusers (SUs) must be able to detect the signal of the PU. Variouskinds of detectors can be used for spectrum sensing such asmatched filter detectors, energy detectors, cyclostationary de-tectors or wavelet detectors [4]. However, the performance ofspectrum sensing is significantly affected by the degradation ofthe PU signal due to path loss or shadowing (hidden terminal).It has been shown that, through collaboration among SUs,the effects of this hidden terminal problem can be reducedand the probability of detecting the PU can be improved [5]–[7]. For instance, in [5] the SUs collaborate by sharing theirsensing decisions through a centralized fusion center in thenetwork. This centralized entity combines the sensing bitsfrom the SUs using the OR-rule for data fusion and makesa final PU detection decision. A similar centralized approachis used in [6] using different decision-combining methods.The authors in [7] propose spatial diversity techniques forimproving the performance of collaborative spectrum sensingby combatting the error probability due to fading on thereporting channel between the SUs and the central fusioncenter. Existing literature mainly focused on the performanceassessment of collaborative spectrum sensing in the presenceof a centralized fusion center. However, in practice, the SUsmay belong to different service providers and they need tointeract with each other for collaboration without relying ona centralized fusion center. Moreover, a centralized approachcan lead to a significant overhead and increased complexity.The main contribution of this paper is to devise distributedcollaboration strategies for SUs in a cognitive network. An-other major contribution is to study the impact on the networktopology of the inherent trade off that exists between thecollaborative spectrum sensing gains in terms of detectionprobability and the cooperation costs in terms of false alarmprobability. This trade off can be pictured as a trade offbetween reducing the interference on the PU (increasing thedetection probability) while maintaining a good spectrum uti-lization (reducing the false alarm probability). For distributedcollaboration, we model the problem as a non-transferablecoalitional game and we propose a distributed algorithm forcoalition formation based on simple merge and split rules.Through the proposed algorithm, each SU autonomously de-cides to form or break a coalition for maximizing its utilityin terms of detection probability while accounting for a falselarm cost. We show that, due to the cost for cooperation,independent disjoint coalitions will form in the network. Westudy the stability of the resulting coalition structure and showthat a maximum coalition size exists for the proposed utilitymodel. Through simulations, we assess the performance ofthe proposed algorithm relative to the non-cooperative case,we compare it with a centralized solution and we showhow the proposed algorithm autonomously adapts the networktopology to environmental changes such as mobility.The rest of this paper is organized as follows: Section IIpresents the system model. In Section III, we present theproposed coalitional game and prove different properties whilein Section IV we devise a distributed algorithm for coalitionformation. Simulation results are presented and analyzed inSection V. Finally, conclusions are drawn in Section VI.II. S
YSTEM M ODEL
Consider a cognitive network consisting of N transmit-receive pairs of SUs and a single PU. The SUs and the PUcan be either stationary and mobile. Since the focus is onspectrum sensing, we are only interested in the transmitterpart of each of the N SUs. The set of all SUs is denoted N = { , . . . , N } . In a non-cooperative approach, each of the N SUs continuously senses the spectrum in order to detectthe presence of the PU. For detecting the PU, we use energydetectors which are one of the main practical signal detectorsin cognitive radio networks [5]–[7]. In such a non-cooperativesetting, assuming Rayleigh fading, the detection probabilityand the false alarm probability of a SU i are, respectively,given by P d,i and P f,i [5, Eqs. (2), (4)] P d,i = e − λ m − X n =0 n ! (cid:18) λ (cid:19) n + (cid:18) γ i,P U ¯ γ i,P U (cid:19) m − × " e − λ ( γi,PU ) − e − λ m − X n =0 n ! (cid:18) λ ¯ γ i,P U γ i,P U ) (cid:19) n , (1) P f,i = P f = Γ( m, λ )Γ( m ) , (2)where m is the time bandwidth product, λ is the energydetection threshold assumed the same for all SUs withoutloss of generality as in [5]–[7], Γ( · , · ) is the incompletegamma function and Γ( · ) is the gamma function. Moreover, ¯ γ i represents the average SNR of the received signal from the PUto SU given by ¯ γ i, PU = P PU h PU ,i σ with P PU the transmit power ofthe PU, σ the Gaussian noise variance and h PU ,i = κ/d µ PU ,i the path loss between the PU and SU i ; κ being the pathloss constant, µ the path loss exponent and d PU ,i the distancebetween the PU and SU i . It is important to note that the non-cooperative false alarm probability expression depends solely on the detection threshold λ and does not depend on the SU’slocation; hence we dropped the subscript i in (2).Moreover, an important metric that we will thoroughly useis the missing probability for a SU i , which is defined as theprobability of missing the detection of a PU and given by [5] P m,i = 1 − P d,i . (3) Fig. 1. An illustrative example of coalition formation for collaborativespectrum sensing among SUs.
For instance, reducing the missing probability directly maps toincreasing the probability of detection and, thus, reducing theinterference on the PU. In order to minimize their missingprobabilities, the SUs will interact for forming coalitionsof collaborating SUs. Within each coalition S ⊆ N = { , . . . , N } , a SU, selected as coalition head , collects thesensing bits from the coalition’s SUs and acts as a fusioncenter in order to make a coalition-based decision on thepresence or absence of the PU. This can be seen as havingthe centralized collaborative sensing of [5], [7] applied atthe level of each coalition with the coalition head being thefusion center to which all the coalition members report. Forcombining the sensing bits and making the final detectiondecision, the coalition head will use the decision fusion OR-rule. Within each coalition we take into account the probabilityof error due to the fading on the reporting channel betweenthe SUs of a coalition and the coalition head [7]. Inside acoalition S , assuming BPSK modulation in Rayleigh fadingenvironments, the probability of reporting error between a SU i ∈ S and the coalition head k ∈ S is given by [8] P e,i,k = 12 − s ¯ γ i,k γ i,k ! , (4)where ¯ γ i,k = P i h i,k σ is the average SNR for bit reportingbetween SU i and the coalition head k inside S with P i thetransmit power of SU i used for reporting the sensing bit to k and h i,k = κd µi,k the path loss between SU i and coalitionhead k . 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 coalition S , the SU k ∈ S havingthe lowest non-cooperative missing probability P m,k is chosenas coalition head . Hence, the coalition head k of a coalition S is given by k = arg min i ∈ S P m,i with P m,i given by (3).The motivation behind Convention 1 is that the SU having thelowest missing probability (best detection probability) within acoalition should not risk sending his local sensing bit over thefading reporting channel; and thus it will serve as a fusionenter for the other SUs in the coalition. By collaborativesensing, the missing and false alarm probabilities of a coalition S having coalition head k are, respectively, given by [7] Q m,S = Y i ∈ S [ P m,i (1 − P e,i,k ) + (1 − P m,i ) P e,i,k ] , (5) Q f,S = 1 − Y i ∈ S [(1 − P f )(1 − P e,i,k ) + P f P e,i,k ] , (6)where P f , P m,i and P e,i,k are respectively given by (2), (3)and (4) for a SU i ∈ S and coalition head k ∈ S .It is clear from (5) and (6) that as the number of SUs percoalition increases, the missing probability will decrease whilethe probability of false alarm will increase. This is a crucialtrade off in collaborative spectrum sensing that can have amajor impact on the collaboration strategies of each SU. Thus,our objective is to derive distributed strategies allowing theSUs to collaborate while accounting for this trade off. Anexample of the sought network structure is shown in Fig. 1.III. C OLLABORATIVE S PECTRUM S ENSING A S C OALITIONAL G AME
In this section, we model the problem of collaborativespectrum sensing as a coalitional game. Then we prove anddiscuss its key properties.
A. Centralized Approach
A centralized approach can be used in order to find theoptimal coalition structure, such as in Fig.1, that allows theSUs to maximize their benefits from collaborative spectrumsensing. For instance, we seek a centralized solution thatminimizes the average missing probability (maximizes theaverage detection probability) per SU subject to a false alarmprobability constraint per SU. In a centralized approach, weassume the existence of a centralized entity in the networkthat is able to gather information on the SUs such as theirindividual missing probabilities or their location. In brief,the centralized entity must be able to know all the requiredparameters for computing the probabilities in (5) and (6) inorder to find the optimal structure. However, prior to derivingsuch an optimal centralized solution, the following propertymust be pinpointed within each coalition.
Property 1:
The missing and false alarm probabilities ofany SU i ∈ S are given by the missing and false alarmprobabilities of the coalition S in (5) and (6), respectively. Proof:
Within each coalition S the SUs report theirsensing bits to the coalition head. In its turn the coalitionhead of S combines the sensing bits using decision fusionand makes a final decision on the presence or absence of thePU. Thus, SUs belonging to a coalition S will transmit or notbased on the final coalition head decision. Consequently, themissing and false alarm probabilities of any SU i ∈ S arethe missing and false alarm probabilities of the coalition S towhich i belongs as given by in (5) and (6), respectively.As a consequence of Property 1 the required false alarmprobability constraint per SU directly maps to a false alarmprobability constraint per coalition . Therefore, denoting B as the set of all partitions of N , the centralized approach seeksto solve the following optimization problem min P∈B P S ∈P | S | · Q m,S N , (7)s.t. Q f,S ≤ α ∀ S ∈ P , where |·| represents the cardinality of a set operator and S is acoalition belonging to the partition P . Clearly, the centralizedoptimization problem seeks to find the optimal partition P ∗ ∈B that minimizes the average missing probability per SU,subject to a false alarm constraint per SU (coalition).However, it is shown in [9] that finding the optimal coalitionstructure for solving an optimization problem such as in (7) isan NP-complete problem. This is mainly due to the fact thatthe number of possible coalition structures (partitions), givenby the Bell number, grows exponentially with the number ofSUs N [9]. Moreover, the complexity increases further dueto the fact that the expressions of Q m,S and Q f,S given by(5) and (6) depend on the optimization parameter P . Forthis purpose, deriving a distributed solution enabling the SUsto benefit from collaborative spectrum sensing with a lowcomplexity is desirable. The above formulated centralizedapproach will be used as a benchmark for the distributedsolution in the simulations, for reasonably small networks. B. Game Formulation and Properties
For the purpose of deriving a distributed algorithm that canminimize the missing probability per SU, we refer to coop-erative game theory [10] which provides a set of analyticaltools suitable for such algorithms. For instance, the proposedcollaborative sensing problem can be modeled as a ( N , v ) coalitional game [10] where N is the set of players (the SUs)and v is the utility function or value of a coalition.The value v ( S ) of a coalition S ⊆ N must capture the tradeoff between the probability of detection and the probabilityof false alarm. For this purpose, v ( S ) must be an increasingfunction of the detection probability Q d,S = 1 − Q m,S withincoalition S and a decreasing function of the false alarmprobability Q f,S . A suitable utility function is given by v ( S ) = Q d,S − C ( Q f,S ) = (1 − Q m,S ) − C ( Q f,S ) , (8)where Q m,S is the missing probability of coalition S givenby (5) and C ( Q f,S ) is a cost function of the false alarmprobability within coalition S given by (6).First of all, we provide the following definition from [10]and subsequently prove an interesting property pertaining tothe proposed game model. Definition 1:
A coalitional game ( N , v ) is said to havea transferable utility if the value v ( S ) can be arbitrarilyapportioned between the coalition’s players. Otherwise, thecoalitional game has a non-transferable utility and each playerwill have its own utility within coalition S . Property 2:
In the proposed collaborative sensing game,the utility of a coalition S is equal to the utility of each SUin the coalition, i.e., v ( S ) = φ i ( S ) , ∀ i ∈ S , where φ i ( S ) denotes the utility of SU i when i belongs to a coalition S .onsequently, the proposed ( N , v ) coalitional game model hasa non-transferable utility. Proof:
The coalition value in the proposed game is givenby (8) and is a function of Q m,S and Q f,S . As per Property 1,the missing probabilities for each SU i ∈ S are also givenby Q m,S and Q f,S and, thus, the utility of each SU i ∈ S is φ i ( S ) = v ( S ) . Hence, the coalition value v ( S ) cannot bearbitrarily apportioned among the users of a coalition; and theproposed coalitional game has non-transferable utility.In general, coalitional game based problems seek to char-acterize the properties and stability of the grand coalitionof all players since it is generally assumed that the grandcoalition maximizes the utilities of the players [10]. In ourcase, although collaborative spectrum sensing improves thedetection probability for the SUs; the cost in terms of falsealarm limits this gain. Therefore, for the proposed ( N , v ) coalitional game we have the following property. Property 3:
For the proposed ( N , v ) coalitional game, thegrand coalition of all the SUs does not always form due tothe collaboration false alarm costs; thus disjoint independentcoalitions will form in the network. Proof:
By inspecting Q m,S in (5) and through the resultsshown in [7] it is clear that as the number of SUs in acoalition increase Q m,S decreases and the performance interms of detection probability improves. Hence, when no costfor collaboration exists, the grand coalition of all SUs is theoptimal structure for maximizing the detection probability.However, when the number of SUs in a coalition S increases,it is shown in [7] through (5) that the false alarm probabilityincreases. Therefore, for the proposed collaborative spectrumsensing model with cost for collaboration, the grand coalitionof all SUs will, in general, not form due to the false alarmcost as taken into consideration in (8).In a nutshell, we have a non-transferable ( N , v ) coalitionalgame and we seek to derive a distributed algorithm for formingcoalitions among SUs. Before deriving such an algorithm, wewill delve into the details of the cost function in (8). C. Cost Function
Any well designed cost function C ( Q f,S ) in (8) must satisfyseveral requirements needed for adequately modeling the falsealarm cost. On one hand, C ( Q f,S ) must be an increasingfunction of Q f,S with the increase slope becoming steeper as Q f,S increases. On the other hand, the cost function C ( Q f,S ) must impose a maximum tolerable false alarm probability, i.e.an upper bound constraint on the false alarm, that cannot beexceeded by any SU in a manner similar to the centralizedproblem in (7) (due to Property 1, imposing a false alarmconstraint on the coalition maps to a constraint per SU).A well suited cost function satisfying the above require-ments is the logarithmic barrier penalty function given by [11] C ( Q f,S ) = − α · log (cid:18) − (cid:16) Q f,S α (cid:17) (cid:19) , if Q f,S < α, + ∞ , if Q f,S ≥ α, (9) where log is the natural logarithm and α is a false alarm con-straint per coalition (per SU). The cost function in (9) allowsto incur a penalty which is increasing with the false alarmprobability. Moreover, it imposes a maximum false alarmprobability per SU. In addition, as the false alarm probabilitygets closer to α the cost for collaboration increases steeply,requiring a significant improvement in detection probability ifthe SUs wish to collaborate as per (8). Also, it is interesting tonote that the proposed cost function depends on both distanceand the number of SUs in the coalition, through the false alarmprobability Q f,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. D ISTRIBUTED C OALITION F ORMATION A LGORITHM
In this section, we propose a distributed coalition formationalgorithm and we discuss its main properties.
A. Coalition Formation Concepts
Coalition formation has been a topic of high interest ingame theory [9], [12]–[14]. The goal is to find algorithms forcharacterizing the coalitional structures that form in a networkwhere the grand coalition is not optimal. For instance, ageneric framework for coalition formation is presented in [13]–[15] whereby coalitions form and break through two simplemerge-and-split rules. This framework can be used to constructa distributed coalition formation algorithm for collaborativesensing, but first, we define the following concepts [13], [14].
Definition 2: A collection of coalitions, denoted S , isdefined as the set S = { S , . . . , S l } of mutually disjointcoalitions S i ⊂ N . If the collection spans all the players of N ; that is S lj =1 S j = N , the collection is a partition of N . Definition 3:
A preference operator or comparison relation ⊲ is defined for comparing two collections R = { R , . . . , R l } and S = { S , . . . , S m } that are partitions of the same subset A ⊆ N (same players in R and S ). Thus, R ⊲ S implies thatthe way R partitions A is preferred to the way S partitions A based on a criterion to be defined next.Various criteria (referred to as orders ) can be used ascomparison relations between collections or partitions [13],[14]. These orders are divided into two main categories:coalition value orders and individual value orders. Coalitionvalue orders compare two collections (or partitions) usingthe value of the coalitions inside these collections such asin the utilitarian order where R ⊲ S implies P li =1 v ( R i ) > P mi =1 v ( S i ) . Individual value orders perform the comparisonusing the actual player utilities and not the coalition value.For such orders, two collections R and S are seen as setsof player utilities of the same length L (number of players).The players’ utilities are either the payoffs after divisionof the value of the coalitions in a collection (transferableutility) or the actual utilities of the players belonging to thecoalitions in a collection (non-transferable utility). Due to thenon-transferable nature of the proposed ( N , v ) collaborativesensing game (Property 2), an individual value order muste used as a comparison relation ⊲ . An important exampleof individual value orders is the Pareto order . Denote for acollection R = { R , . . . , R l } , the utility of a player j ina coalition R j ∈ R by φ j ( R ) = φ j ( R j ) = v ( R j ) (as perProperty 2); hence, the Pareto order is defined as follows R ⊲ S ⇐⇒ { φ j ( R ) ≥ φ j ( S ) ∀ j ∈ R , S} , (10)with at least one strict inequality ( > ) for a player k. Due to the non-transferable nature of the proposed collabora-tive sensing model, the Pareto order is an adequate preferencerelation. Having defined the various concepts, we derive a dis-tributed coalition formation algorithm in the next subsection.
B. Coalition Formation Algorithm
For autonomous coalition formation in cognitive radio net-works, we propose a distributed algorithm based on two simplerules denoted as “merge” and “split” that allow to modify apartition T of the SUs set N as follows [13]. Definition 4: Merge Rule -
Merge any set of coalitions { S , . . . , S l } where { S lj =1 S j } ⊲ { S , . . . , S l } , therefore, { S , . . . , S l } → { S lj =1 S j } , (each S i is a coalition in T ). Definition 5: Split Rule -
Split any coalition S lj =1 S j where { S , . . . , S l } ⊲ { S lj =1 S j } , thus, { S lj =1 S j } →{ S , . . . , S l } , (each S i is a coalition in T ).Using the above rules, multiple coalitions can merge into alarger coalition if merging yields a preferred collection basedon the selected order ⊲ . Similarly, a coalition would splitinto smaller coalitions if splitting yields a preferred collection.When ⊲ is the Pareto order, coalitions will merge (split) only ifat least one SU is able to strictly improve its individual utilitythrough this merge (split) without decreasing the other SUs’utilities. By using the merge-and-split rules combined withthe Pareto order, a distributed coalition formation algorithmsuited for collaborative spectrum sensing can be constructed.First and foremost, the appeal of forming coalitions usingmerge-and-split stems from the fact that it has been shownin [13] and [14] that any arbitrary iteration of merge-and-splitoperations terminates . Moreover, each merge or split decisioncan be taken in a distributed manner by each individual SU orby each already formed coalition. Subsequently, a merge-and-split coalition algorithm can adequately model the distributedinteractions among the SUs of a cognitive network that areseeking to collaborate in the sensing process.In consequence, for the proposed collaborative sensinggame, we construct a coalition formation algorithm based onmerge-and-split and divided into three phases: local sensing,adaptive coalition formation, and coalition sensing. In the localsensing phase, each individual SU computes its own local PUdetection bit based on the received PU signal. In the adaptivecoalition formation phase, the SUs (or existing coalitions ofSUs) interact in order to assess whether to share their sensingresults with nearby 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 that merging (or splitting) yields. In the final coalition sensing phase, once the network topologyconverges following merge-and-split, SUs that belong to thesame coalition report their local sensing bits to their localcoalition head. The coalition head subsequently uses decisionfusion OR-rule to make a final decision on the presence orthe absence of the PU. This decision is then reported by thecoalition heads to all the SUs within their respective coalitions.Each round of the three phases of the proposed algorithmstarts from an initial network partition T = { T , . . . , T l } of N . During the adaptive coalition formation phase any randomcoalition (individual SU) can start with the merge process. Forimplementation purposes, assume that the coalition T i ∈ T which has the highest utility in the initial partition T starts themerge by attempting to collaborate with a nearby coalition. Onone hand, if merging occurs, a new coalition ˜ T i is formed and,in its turn, coalition ˜ T i will attempt to merge with a nearbySU that can improve its utility. On the other hand, if T i isunable to merge with the firstly discovered partner, it tries tofind other coalitions that have a mutual benefit in merging. Thesearch ends by a final merged coalition T final i composed of T i and one or several of coalitions in its vicinity ( T final i = T i ,if no merge occurred). The algorithm is repeated for theremaining T i ∈ T until all the coalitions have made theirmerge decisions, resulting in a final partition F . Followingthe merge process, the coalitions in the resulting partition F are next subject to split operations, if any is possible.An iteration consisting of multiple successive merge-and-splitoperations is repeated until it terminates. It must be stressedthat the decisions to merge or split can be taken in a distributedway without relying on any centralized entity as each SU orcoalition can make its own decision for merging or splitting.Table I summarizes one round of the proposed algorithm.For handling environmental changes such as mobility or thejoining/leaving of SUs, Phase 2 of the proposed algorithm inTable I is repeated periodically. In Phase 2, periodically, astime evolves and SUs (or the PU) move or join/leave, the SUscan autonomously self-organize and adapt the network’s topol-ogy through new merge-and-split iterations with each coalitiontaking the decision to merge (or split) subject to satisfyingthe merge (or split) rule through Pareto order (10). In otherwords, every period of time θ the SUs assess the possibility ofsplitting into smaller coalitions or merging with new partners.The period θ is smaller in highly mobile environments to allowa more adequate adaptation of the topology. Similarly, everyperiod θ , in the event where the current coalition head of acoalition has moved or is turned off, the coalition membersmay select a new coalition head if needed. The convergenceof this merge-and-split adaptation to environmental changes isalways guaranteed, since, by definition of the merge and splitrules, any iteration of these rules certainly terminates.For the proposed coalition formation algorithm, an upperbound on the maximum coalition size is imposed by theproposed utility and cost models in (8) and (9) as follows: Theorem 1:
For the proposed collaborative sensing model,any coalition structure resulting from the distributed coalition
ABLE IO
NE ROUND OF THE PROPOSED COLLABORATIVE SENSING ALGORITHM
Initial State
The network is partitioned by T = { T , . . . , T k } (At the beginningof all time T = N = { , . . . , N } with non-cooperative SUs). Three phases in each round of the coalition formation algorithm
Phase 1 - Local Sensing:
Each individual SU computes its local PU signal sensing bit.
Phase 2 - Adaptive Coalition Formation:
In this phase, coalition formation using merge-and-split occurs. repeat a) F = Merge( T ); coalitions in T decide to mergebased on the merge algorithm explained in Section IV-B.b) T = Split( F ); coalitions in F decide to split based onthe Pareto order. until merge-and-split terminates. 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 decision onthe absence or presence of he 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 throughout the network operation.In Phase 2, through distributed and periodic merge-and-splitdecisions, the SUs can autonomously adapt the network topologyto environmental changes such as mobility. formation algorithm will have coalitions limited in size to amaximum of M max = log (1 − α )log (1 − P f ) SUs.
Proof:
For forming coalitions, the proposed algorithmrequires an improvement in the utility of the SUs throughPareto order. However, the benefit from collaboration is limitedby the false alarm probability cost modeled by the barrierfunction (9). A minimum false alarm cost in a coalition S withcoalition head k ∈ S exists whenever the reporting channel isperfect, i.e., exhibiting no error, hence P e,i,k = 0 ∀ i ∈ S .In this perfect case, the false alarm probability in a perfectcoalition S p is given by Q f,S p = 1 − Y i ∈ S p (1 − P f ) = 1 − (1 − P f ) | S p | , (11)where | S p | is the number of SUs in the perfect coalition S p .A perfect coalition S p where the reporting channels insideare perfect (i.e. SUs are grouped very close to each other)can accommodate the largest number of SUs relative to othercoalitions. Hence, we can use this perfect coalition to find anupper bound on the maximum number of SUs per coalition.For instance, the log barrier function in (9) tends to infinitywhenever the false alarm probability constraint per coalitionis reached which implies an upper bound on the maximumnumber of SUs per coalition if Q f,S p ≥ α, yielding by (11) | S p | ≤ log (1 − α )log (1 − P f ) = M max . (12)It is interesting to note that the maximum size of a coalition M max depends mainly on two parameters: the false alarmconstraint α and the non-cooperative false alarm P f . Forinstance, larger false alarm constraints allow larger coalitions,as the maximum tolerable cost limit for collaboration isincreased. Moreover, as the non-cooperative false alarm P f decreases, the possibilities for collaboration are better sincethe increase of the false alarm due to coalition size becomessmaller as per (6). It must be noted that the dependence of M max on P f yields a direct dependence of M max on theenergy detection threshold λ 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 fixed α and P f does not increase the upper bound on coalition size. C. Stability
The result of the proposed algorithm in Table I is anetwork partition composed of disjoint independent coalitionsof SUs. The stability of this resulting network structure can beinvestigated using the concept of a defection function D [13]. Definition 6: A defection function D is a function whichassociates with each partition T of N a group of collectionsin N . A partition T = { T , . . . , T l } of N is D -stable if nogroup of players is interested in leaving T when the playerswho leave can only form the collections allowed by D .Two important defection functions must be characterized[13]–[15]. First, the D hp ( T ) function (denoted D hp ) whichassociates with each partition T of N the group of allpartitions of N that the players can form through merge-and-split operations applied to T . This function allows anygroup of players to leave the partition T of N through merge-and-split operations to create another partition in N . Second,the D c ( T ) function (denoted D c ) which associates with eachpartition T of N the family of all collections in N . Thisfunction allows any group of players to leave the partition T of N through any operation and create an arbitrary collection in N . Two forms of stability stem from these definitions: D hp stability and a stronger D c stability. A partition T is D hp -stable, if no players in T are interested in leaving T throughmerge-and-split to form other partitions in N ; while a partition T is D c -stable, if no players in T are interested in leaving T through any operation (not necessary merge or split) to formother collections in N .Characterizing any type of D -stability for a partition de-pends on various properties of its coalitions. For instance, apartition T = { T , . . . , T l } is D hp -stable if, for the partition T , no coalition has an incentive to split or merge. As animmediate result of the definition of D hp -stability we have Theorem 2:
Every partition resulting from our proposedcoalition formation algorithm is D hp -stable.Briefly, a D hp -stable can be thought of as a state of equilibriumwhere no coalitions have an incentive to pursue coalitionformation through merge or split. With regards to D c stability,the work in [13]–[15] proved that a D c -stable partition has thefollowing properties:1) If it exists, a D c -stable partition is the unique outcomeof any arbitrary iteration of merge-and-split and is a D hp -stable partition. −2 −1 Number of SUs (N) A v e r age m i ss i ng p r obab ili t y pe r S U Non−cooperative sensingProposed distributed coalition formationCentralized solution
The solution of the centralized approachis mathematically untractable beyond N = 7 SUs
Fig. 2. Average missing probabilities (average over locations of SUs and non-cooperative false alarm range P f ∈ (0 , α ) ) vs. number of SUs.
2) A D c -stable partition T is a unique ⊲ -maximal partition,that is for all partitions T ′ = T of N , T ⊲ T ′ . In the casewhere ⊲ represents the Pareto order, this implies thatthe D c -stable partition T is the partition that presents a Pareto optimal utility distribution for all the players.However, the existence of a D c -stable partition is not alwaysguaranteed [13]. The D c -stable partition T = { T , . . . , T l } ofthe whole space N exists if a partition of N that verifies thefollowing two necessary and sufficient conditions exists [13]:1) For each i ∈ { , . . . , l } and each pair of disjoint coalitions S and S such that { S ∪ S } ⊆ T i we have { S ∪ S } ⊲ { S , S } .2) For the partition T = { T , . . . , T l } a coalition G ⊂N formed of players belonging to different T i ∈ T is T -incompatible if for no i ∈ { , . . . , l } we have G ⊂ T i . D c -stability requires that for all T -incompatiblecoalitions { G } [ T ] ⊲ { G } where { G } [ T ] = { G ∩ T i ∀ i ∈{ , . . . , l }} is the projection of coalition G on T .If no partition of N can satisfy these conditions, then no D c -stable partitions of N exists. Nevertheless, we have Lemma 1:
For the proposed ( N , v ) collaborative sensingcoalitional game, the proposed algorithm of Table I convergesto the optimal D c -stable partition, if such a partition exists.Otherwise, the proposed algorithm yields a final networkpartition that is D hp -stable. Proof:
The proof is an immediate consequence of The-orem 2 and the fact that the D c -stable partition is a uniqueoutcome of any arbitrary merge-and-split iteration which isthe case with any partition resulting from our algorithm.Moreover, for the proposed game, the existence of the D c -stable partition cannot be always guaranteed. For instance,for verifying the first condition for existence of the D c -stablepartition, the SUs belonging to partitions of each coalitionsmust verify the Pareto order through their utility given by(8). Similarly, for verifying the second condition of D c sta-bility, SUs belonging to all T -incompatible coalitions in thenetwork must verify the Pareto order. Consequently, finding a
0 1 2 3 4 5 6 7 10 15 20 25 3010 −2 −1 Number of SUs (N) A v e r age f a l s e a l a r m p r obab ili t y pe r S U Non−cooperative sensingProposed distributed coalition formationCentralized solution
The solution of the centralized approachis mathematically untractable beyondN = 7 SUs
Fig. 3. Average false alarm probabilities (average over locations of SUs andnon-cooperative false alarm range P f ∈ (0 , α ) ) vs. number of SUs. geometrical closed-form condition for the existence of such apartition is not feasible as it depends on the location of theSUs and the PU through the individual missing and false alarmprobabilities in the utility expression (8). Hence, the existenceof the D c -stable partition is closely tied to the location of theSUs and the PU which both can be random parameters inpractical networks. However, the proposed algorithm will al-ways guarantee convergence to this optimal D c -stable partitionwhen it exists as stated in Lemma 1. Whenever a D c -stablepartition does not exist, the coalition structure resulting fromthe proposed algorithm will be D hp -stable (no coalition or SUis able to merge or split any further).V. S IMULATION R ESULTS AND A NALYSIS
For simulations, the following network is set up: The PUis placed at the origin of a km × km square with theSUs randomly deployed in the area around the PU. We setthe time bandwidth product m = 5 [5]–[7], the PU transmitpower P PU = 100 mW, the SU transmit power for reportingthe sensing bit P i = 10 mW ∀ i ∈ N and the noise level σ = − dBm. For path loss, we set µ = 3 and κ = 1 .The maximum false alarm constraint is set to α = 0 . , asrecommended by the IEEE 802.22 standard [16].In Figs. 2 and 3 we show, respectively, the average missingprobabilities and the average false alarm probabilities achievedper SU for different network sizes. These probabilities areaveraged over random locations of the SUs as well as a rangeof energy detection thresholds λ that do not violate the falsealarm constraint; this in turn, maps into an average over thenon-cooperative false alarm range P f ∈ (0 , α ) (obviously, for P f > α no cooperation is possible). In Fig. 2, we show thatthe proposed algorithm yields a significant improvement in theaverage missing probability reaching up to . reduction(at N = 30 ) compared to the non-cooperative case. Thisadvantage is increasing with the network size N . However,there exists a gap in the performance of the proposed algorithmand that of the optimal centralized solution. This gap stemsmainly from the fact that the log barrier function used inthe distributed algorithm (9) increases the cost drastically −3 −2 −1 −2 −1 Non−cooperative false alarm P f A v e r age m i ss i ng p r obab ili t y pe r S U Non−cooperative sensingProposed distributed coalition formationCentralized solution
Fig. 4. Average missing probabilities per SU vs. non-cooperative false alarm P f (or energy detection threshold λ ) for N = 7 SUs. when the false alarm probability is in the vicinity of α . Thisincreased cost makes it harder for coalitions with false alarmlevels close to α to collaborate in the distributed approachas they require a large missing probability improvement tocompensate the cost in their utility (8) so that a Pareto ordermerge or split becomes possible. However, albeit the proposedcost function yields a performance gap in terms of missingprobability, it forces a false alarm for the distributed casesmaller than that of the centralized solution as seen in Fig. 3.For instance, Fig. 3 shows that the achieved average falsealarm by the proposed distributed solution outperforms thatof the centralized solution but is still outperformed by thenon-cooperative case. Thus, while the centralized solutionachieves a better missing probability; the proposed distributedalgorithm compensates this performance gap through the av-erage achieved false alarm. In summary, Figs. 2 and 3 clearlyshow the performance trade off that exists between the gainsachieved by collaborative spectrum sensing in terms of averagemissing probability and the corresponding cost in terms ofaverage false alarm probability.In Fig. 4, we show the average missing probabilities perSU for different energy detection thresholds λ expressed bythe feasible range of non-cooperative false alarm probabilities P f ∈ (0 , α ) for N = 7 . In this figure, we show that asthe non-cooperative P f decreases the performance advantageof collaborative spectrum sensing for both the centralizedand distributed solutions increases (except for very small P f where the advantage in terms of missing probability reachesits maximum). The performance gap between centralizedand distributed is once again compensated by a false alarmadvantage for the distributed solution as already seen andexplained in Fig. 3 for N = 7 . Finally, in this figure, it mustbe noted that as P f approaches α = 0 . , the advantage forcollaborative spectrum sensing diminishes drastically as thenetwork converges towards the non-cooperative case.In Fig. 5, we show a snapshot of the network structureresulting from the proposed distributed algorithm (dashed −1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 Primary User12 34 56 7Position in x (km) P o s i t i on i n y ( k m ) Fig. 5. Final coalition structure from both distributed (dashed line) andcentralized (solid line) collaborative spectrum sensing for N = 7 SUs. line) as well as the centralized approach (solid line) for N = 7 randomly placed SUs and a non-cooperative falsealarm P f = 0 . . We notice that the structures resultingfrom both approaches are almost comparable, with nearby SUsforming collaborative coalitions for improving their missingprobabilities. However, for the distributed solution, SU 4 ispart of coalition S = { , , , } while for the centralizedapproach SU 4 is part of coalition { , , } . This difference inthe network structure is due to the fact that, in the distributedcase, SU 4 acts selfishly while aiming at improving its ownutility. In fact, by merging with { , } SU 4 achieves a utilityof φ ( { , } ) = 0 . with a missing probability of . whereas by merging with { , , } SU 4 achieves a utilityof φ ( { , , , } ) = 0 . with a missing probability of . . Thus, when acting autonomously in a distributedmanner, SU 4 prefers to merge with { , , } rather than with { , } regardless of the optimal structure for the network as awhole. In brief, Fig. 5 shows how the cognitive network struc-tures itself for both centralized and distributed approaches.Furthermore, in Fig. 6 we show how our distributed algo-rithm in Table I handles mobility during Phase 2 (adaptivecoalition formation). For this purpose, after the network struc-ture in Fig. 5 has formed, we allow SU 1 to move horizontallyalong the positive x-axis while other SUs are immobile. InFig. 6, at the beginning, the utilities of SUs { , , , } aresimilar since they belong to the same coalition. These utilitiesdecrease as SU 1 distances itself from { , , } . After moving . km SUs { , } split from coalition { , , , } by Paretoorder as φ ( { , } ) = 0 . > φ ( { , , , } ) = 0 . , φ ( { , } ) = 0 . > φ ( { , , , } ) = 0 . , φ ( { , } ) =0 . > φ ( { , , , } ) = 0 . and φ ( { , } ) = 0 . >φ ( { , , , } = 0 . (this small advantage from splittingincreases as SU 1 moves further). As SU 1 distances itselffurther from the PU, its utility and that of its partner SU 6decrease. Subsequently, as SU 1 moves . km it finds itbeneficial to split from { , } and merge with SU 7. Throughthis merge, SU 1 and SU 7 improve their utilities. Meanwhile, S U u t ili t y SU 7SU 1 (mobile)SU 2SU 4SU 6
SU 1 splits from {1,6} andforms coalition {1,7} whileSU 6 merges with {2,4} andforms {2,4,6}Coalition {1,2,4,6} splits into two coalitions{2,4} and {1,6} as SU 1 moves
Fig. 6. Self-adaptation of the network’s topology to mobility through merge-and-split as SU 1 moves horizontally on the positive x-axis.
SU 6 rejoins SUs { , } forming a 3-SU coalition { , , } while increasing the utilities of all three SUs. In a nutshell,this figure illustrates how adaptive coalition formation throughmerge and split operates in a mobile cognitive radio network.Similar results can be seen whenever all SUs are mobile oreven the PU is mobile but they are omitted for space limitation.In Fig. 7, for a network of N = 30 SUs, we evaluate thesizes of the coalitions resulting from our distributed algorithmand compare them with the the upper bound M max derived inTheorem 1. First and foremost, as the non-cooperative P f in-creases, both the maximum and the average size of the formedcoalitions decrease converging towards the non-cooperativecase as P f reaches the constraint α = 0 . . Through thisresult, we can clearly see the limitations that the detection-false alarm probabilities trade off for collaborative sensingimposes on the coalition size and network topology. Also, inFig. 7, we show that, albeit the upper bound on coalition size M max increases drastically as P f becomes smaller, the averagemaximum coalition size achieved by the proposed algorithmdoes not exceed SUs per coalition for the given networkwith N = 30 . This result shows that, in general, the networktopology is composed of a large number of small coalitionsrather than a small number of large coalitions, even when P f is small and the collaboration possibilities are high.VI. C ONCLUSIONS
In this paper, we proposed a novel distributed algorithm forcollaborative spectrum sensing in cognitive radio networks.We modeled the collaborative sensing problem as a coali-tional game with non-transferable utility and we derived adistributed algorithm for coalition formation. The proposedcoalition formation algorithm is based on two simple rulesof merge-and-split that enable SUs in a cognitive networkto cooperate for improving their detection probability whiletaking into account the cost in terms of false alarm probability.We characterized the network structure resulting from theproposed algorithm, studied its stability and showed that amaximum number of SUs per coalition exists for the proposed −1.8 −1.6 −1.4 −1.2 f A v e r age c oa li t i on s i z e ( nu m be r o f S U s ) Upper bound on number of SUs per coalition (M max )Average maximum number of SUs per coalitionActual average number of SUs per coalition
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