BP-P2P: Belief Propagation-Based Trust and Reputation Management for P2P Networks

BP-PP: Belief Propgtion-Bsed Trust nd Reputtion Mngement for PP Networs Ermn Aydy School of Electricl nd Comp. Eng. Georgi Institute of Technology Atlnt, GA 333, USA Emil: eydy@gtech.edu Frmrz Feri School of Electricl nd Comp. Eng. Georgi Institute of Technology Atlnt, GA 333, USA Emil: feri@ece.gtech.edu Abstrct In this pper, for the first time, we introduce Belief Propgtion (BP)-bsed distributed trust nd reputtion mngement lgorithm. The proposed lgorithm cn be utilized in mny distributed systems from Peer-to-peer (PP) networs to socil nd mesh networs. In this wor, we focus on PP networs nd explore the ppliction of BP-bsed trust nd reputtion mngement in decentrlized environment in the presence of mlicious peers. In typicl PP trust nd reputtion mngement system, fter ech trnsction, the client peer (who receives service) provides its rting bout the qulity of the service provided by the server peer for tht trnsction. In such system, we view the problem of trust nd reputtion mngement s to compute two sets of vribles: 1. the reputtion prmeters of peers bsed on their qulity of service, nd. the trustworthiness prmeters of peers bsed on the rtings they provide fter ech trnsction. We distinguish between these two prmeters s peer might provide high qulity service s server while providing mlicious rtings s client. The proposed scheme, referred to s BP-PP, relies on the BP lgorithm in n ppropritely chosen fctor grph representtion of the PP networ. The reputtion nd trustworthiness prmeters re computed by BPbsed distributed messge pssing lgorithm between the peers on the fctor grph. We provide detiled evlution of BP- PP vi nlysis nd computer simultions. We show tht BP- PP is very robust in computing trustworthiness vlues nd filtering out mlicious rtings. Specificlly, we prove tht BP-PP itertively reduces the error in the reputtion vlues of peers due to the mlicious rtings with high probbility. Further, comprison of BP-PP with some well-nown nd commonly used PP reputtion mngement techniques (e.g., EigenTrust nd Byesin Frmewor) indictes the superiority of the proposed scheme in terms of robustness ginst mlicious behvior. We lso show tht the computtionl complexity of BP-PP grows only linerly with the number of peers nd the communiction overhed of BP-PP is lower thn the well-nown EigenTrust lgorithm. I. INTRODUCTION Peer-to-peer (PP) networs [1] re commonly defined s distributed rchitectures in which the worlod is prtitioned between the peers nd ech peer is eqully privileged. As opposed to trditionl client-server networing (in which certin peers re responsible for providing resources while other peers only consume), every peer plys the role of both client nd server in the PP networs []. In other words, ech peer provides ccess to its resources (e.g., processing power, storge) s server without the need for centrl uthority. PP networs hve especilly becme populr s distributed file shring systems in which peers exchnge files between ech other (such s Gnutell or Npster). Due to their size nd the distributed rchitecture, PP systems re highly vulnerble to ttcs by the mlicious peers. The This mteril is bsed upon wor supported prtilly by the Ntionl Science Foundtion under Grnt No. IIS-1115199, nd gift from the Cisco University Reserch Progrm Fund, n dvised fund of Silicon Vlley Community Foundtion. most common ttc to PP systems is in the form of inecting inuthentic files (or introducing viruses) to the networ. Mlicious behvior in PP networs is minly confronted by utilizing trust nd reputtion mngement systems in which peers (s clients) get to rte the peers (cting s servers) bsed on the qulity of the trnsctions. A trust nd reputtion mngement mechnism is promising method to protect the client peer by forming some foresight bout the server peers before using their resources. Using distributed trust nd reputtion mngement mechnism, reputtion vlues of the servers nd the trustworthiness vlues of the clients (on their rtings) cn be computed by the peers without needing centrl uthority. As result of this, mlicious behvior cn be detected nd honest behvior cn be encourged in the networ. However, this rting mechnism puts the server peers in vulnerble position s the mlicious peers my undermine (or boost) the reputtion vlues of certin peers by providing incorrect rtings. Hence, building resilient trust nd reputtion mngement system tht is robust ginst mlicious ctivities becomes chllenging issue. Despite recent dvnces in trust nd reputtion mngement in PP networs, there is yet need to develop relible, sclble nd dependble schemes tht would lso be resilient to vrious wys distributed trust nd reputtion system cn be ttced. In this pper we introduce the first ppliction of the Belief Propgtion (BP), n itertive probbilistic lgorithm, in the design nd evlution of distributed trust nd reputtion mngement systems. Belief Propgtion [3], [4] is messge pssing lgorithm for performing inference on grphicl models. It is method for computing mrginl distributions of the unobserved nodes conditioned on the observed ones. In our previous wor, we developed the generl theory of BP-bsed reputtion mngement lgorithms for centrlized environments [5]. However, in distributed infrstructure, trust nd reputtion mngement is more complicted thn in the centrlized solutions. Hence, using the bsic theory in our previous wor [5], in this pper, we focus on PP networs nd explore the ppliction of BP-bsed trust nd reputtion mngement in decentrlized environment in the presence of mlicious peers. We note tht the proposed lgorithm cn lso be utilized in vrious distributed systems such s mesh nd socil networs. In mesh networs, the proposed lgorithm cn be used to evlute the types nd behviors of the nodes, while in socil networs, it cn be used to evlute the behviors nd trustworthiness vlues of the peers. We introduce the Belief Propgtion-Bsed Trust nd Reputtion Mngement for PP Networs (herefter referred to s BP-PP). Our obective is to compute both the reputtion vlues of the peers s servers nd their trustworthiness vlues s clients (bsed on the rtings they provide for the servers). We view this problem s n inference problem which involves computing

the mrginl probbility distributions of the reputtion vlues from the globl oint probbility distribution function of mny vribles. This problem, however, cnnot be solved directly in lrge-scle system, becuse the number of terms grow exponentilly with the number of peers in the networ. The ey role of the BP lgorithm is tht we cn use it to compute those mrginl distributions in the complexity tht grows only linerly with the number of peers. BP-PP describes the PP networ on fctor grph; llowing the peers to compute the reputtion nd trustworthiness vlues by distributed messge pssing between ech other. The min contributions of our wor re summrized in the following. 1) We introduce the first ppliction of the Belief Propgtion (BP) lgorithm in the design nd evlution of distributed trust nd reputtion mngement systems for PP networs. We introduce grph-bsed mechnism which relies on fctor grph to compute the reputtion of ech peer (s server) nd its trustworthiness vlue (s client) by BP-bsed itertive nd distributed messge pssing lgorithm. ) The proposed distributed lgorithm enbles the peers to compute the reputtion vlues (of other peers) with smll error in the presence of ttcers. Further, it lso llows the peers to obtin the trustworthiness vlues (of other peers) by nlyzing the rtings provided, which enbles them detect nd filter out mlicious rtings effectively. 3) The computtionl complexity of BP-PP is t most liner in the number of peers in the networ, ming it very ttrctive for lrge-scle systems. Further, its communiction overhed is lower thn the well-nown EigenTrust lgorithm which is prticulrly designed for PP networs. 4) The proposed BP-PP outperforms the existing nd commonly used PP reputtion mngement techniques such s the EigenTrust lgorithm [6] nd the Byesin Approch [7] (which is lso proposed s the reputtion mngement system of the well-nown CONFIDANT protocol [8]) in the presence of ttcers. The rest of this pper is orgnized s follows. In the rest of this section, we summrize the relted wor. In Section II, we describe the proposed BP-PP in detil. Next, in Section III, we nlyze BP-PP using mthemticl model for the peers. Further, we evlute BP-PP vi computer simultions nd compre BP-PP with the existing nd commonly used PP reputtion mngement schemes. Finlly, in Section IV, we conclude the pper. A. Relted Wor nd Their Shortcomings Trust nd reputtion mngement systems for PP networs nd online systems received lot of ttention [6], [7], [9] [16]. In [9] nd [1], uthors provide good survey of the wor on the use of trust nd reputtion mngement systems for PP networs. Most proposed PP trust nd reputtion mngement mechnisms utilize the ide tht peer cn monitor others nd obtin direct observtions [11] or peer cn enquire bout the reputtion vlue of nother peer (nd hence, obtin indirect observtions) before using the service provided by tht peer [1], [13]. EigenTrust [6] is one of the most populr reputtion mngement lgorithms for PP networs. However, the EigenTrust lgorithm is constrined by the fct tht trustworthiness of peer (on its feedbc) is equivlent to its reputtion vlue. However, trusting peer s feedbc nd trusting peer s service qulity re two different concepts. As we will discuss in Section III-A, mlicious peer, by providing incorrect or mlicious rtings, my ttc the reputtion mngement system while providing high qulity service. In other words, node my hve high reputtion but low trustworthiness vlue. Thus, equting the two will ffect the performnce of the reputtion mngement. Further, the EigenTrust lgorithm relies on the presence of pre-trusted peers in the networ which is either imprcticl or limiting in most networs. Most importntly, the EigenTrust lgorithm computes the globl reputtion vlues by simple itertive weighted verging mechnism which is vulnerble to collbortive ttcs from the mlicious peers. Use of the Byesin Frmewor is lso proposed in [8]. In the Byesin Frmewor, ech reputtion vlue is computed independent of the other nodes reputtion vlues. However, the rtings provided by the nodes induce probbility distribution on the reputtion vlues. These distributions re correlted becuse they re induced by the overlpping set of nodes. Therefore, ignoring these dependencies could ffect the performnce drmticlly. The strength of BP-PP stems from the fct tht it cptures this correltion in nlyzing the rtings nd computing the trust nd reputtion vlues. Different from the existing schemes, BP- PP lgorithm is grph bsed itertive lgorithm motivted by the previous success on messge pssing techniques nd BP lgorithms on vrious pplictions such s inference nd decoding error correcting codes. II. BELIEF PROPAGATION-BASED TRUST AND REPUTATION MANAGEMENT FOR PP NETWORKS We ssume two different sets in the system: i) the set of servers, S nd ii) the set of clients, U. We further ssume tht every peer in the networ plys the role of both client nd server. In other words, ech peer provides ccess to its resources (e.g., provides files) s server. On the other hnd, ech peer lso uses the resources of other servers s client. Therefore, sets S nd U re not disoint nd ech peer i is represented both in set S (s server) nd in set U (s client). Trnsctions occur between the servers nd clients, nd clients provide feedbcs in the form of rtings bout servers fter ech trnsction (bsed on the service qulity of the trnsction). First, for the simplicity nd clrity of the presenttion, we will describe the fundmentl scheme ssuming tht ech peer computes its own reputtion vlue (s server) nd trustworthiness vlue (s client) vi distributed messge pssing nd report these vlues to other peers when they re queried. However, this fundmentl scheme is not completely secure s mlicious peers my report incorrect vlues for their own reputtion nd trustworthiness vlues upon n inquiry. Then, in Section II-A, we will describe how we me this scheme completely secure by llowing different groups of peers (referred s the score mngers) to compute the reputtion nd trustworthiness vlues of individul peers. Let G be the reputtion vlue of server ( S) nd T i be the rting tht client i (i U) reports bout server ( S), whenever trnsction is completed between the two peers. Moreover, let R i denote the trustworthiness of the peer i (i U) s client. In other words, R i represents the mount of confidence on the correctness of ny rting provided by client i. We ssume there re u clients nd s servers in the system (i.e., U = u nd S = s) 1. Let G = {G : S} nd R = {R i : i U} be the collection of vribles representing the reputtions of the servers nd the trustworthiness vlues of the clients, respectively. Further, let T be the s u serverclient mtrix tht stores the rting vlues (T i ), nd T i be the 1 Since ech peer is both server nd client, u = s in typicl PP networ.

3 set of rtings provided by client i. We consider slotted time throughout this discussion. At ech time-slot (or epoch), BP- PP lgorithm is executed using the input prmeterstnd the present R to obtin the reputtion prmeters nd the updted trustworthiness vlues t ech peer. We note tht ech peer hs only prt of the input prmeters bsed on its previous trnsctions. More specificlly, we ssume tht every peer i nows the rtings it previously provided s client (i.e.,t i ) nd the set of servers for whom it provided these rtings. Moreover, every peer i nows the rtings it previously received from other peers s server nd the set of clients who provided these rtings (similr to [6]). After BP-PP completes its itertions, ech peer computes its new reputtion vlue s server s well s its updted trustworthiness vlue s client. For simplicity of presenttion, we ssume tht the rting vlues re from the set Υ = {,1}. The extension in which rting vlues cn te ny rel number cn be developed similrly. The reputtion mngement problem cn be viewed s finding the mrginl probbility distributions of ech vrible in G, given the observed dt (i.e., evidence). There re s mrginl probbility functions, p(g T,R), ech of which is ssocited with vrible G ; the reputtion vlue of server. We formulte the problem by considering the globl function p(g T, R), which is the oint probbility distribution function of the vribles in G given the rting mtrix nd the trustworthiness vlues of the clients. Then, clerly, ech mrginl probbility function p(g T,R) my be obtined s follows: p(g T,R) = p(g T,R), (1) G\{G } where G\{G } implies ll vribles in G except G. Unfortuntely, the number of terms in (1) grows exponentilly with the number of vribles, ming the computtion infesible for lrge-scle systems. Further, (1) cn only be solved in centrlized environment in which ll the evidence T nd R is vilble t centrl unit. On the other hnd, PP networs re typiclly distributed environments, nd hence, solving (1) t ech peer is not fesible in such networs in which ech peer hs only prt of the evidence. Thus, we propose to fctorize (1) to locl functions f i using fctor grph nd utilize the Belief Propgtion (BP) lgorithm to clculte the mrginl probbility distributions in liner complexity nd in distributed environment. A fctor grph is biprtite grph contining two sets of nodes (corresponding to vribles nd fctors) nd edges incident between two sets. Following [4], we form fctor grph by setting vrible node for ech vrible G, fctor node for ech function f i, nd n edge connecting vrible node to the fctor node i if nd only if G is n rgument of f i. We note tht computing mrginl probbility functions is exct when the fctor grph hs no cycles. However, the BP lgorithm still gives good pproximte results for the fctor grphs with cycles. T m b n c Fig. 1: Setup of the scheme. First, we rrnge the collection of the clients nd the servers together with their ssocited reltions (i.e., the rtings) s fctor grph, s in Fig. 1. In this representtion, ech client corresponds to fctor node shown s squre nd ech server is represented by vrible node shown s hexgon in the grph. Ech rting is represented by n edge from the fctor node to T nc the vrible node. Hence, if client i (i U) hs rting bout server ( S), we plce n edge with vlue T i from the fctor node i to the vrible node representing server. We note tht if ny new rting rrives from client i bout server, the T i vlue is updted by verging the new rting nd the old vlue of the edge multiplied with the fding fctor γ i (t) = ϑ t ti (where ϑ nd t i re the fding prmeter nd the time when the lst trnsction between client i nd server occurred, respectively). Next, we suppose tht the globl function p(g T, R) fctors into products of severl locl functions, ech hving subset of vribles from G s rguments s follows : p(g T,R) = 1 f i (G i,t i,r i ), () Z i U where Z is the normliztion constnt nd G i is subset of G. Hence, in the grph representtion of Fig. 1, ech fctor node is ssocited with locl function nd ech locl function f i represents the probbility distributions of its rguments given the trustworthiness vlue nd the existing rtings of the ssocited client. We now introduce the messges between the fctor nd the vrible nodes (i.e., between the servers nd the clients) to compute the mrginl distributions t ech server using BP. To tht end, we describe the messge exchnge between peer (s client) nd peer (s server) in Fig. 1. We represent the set of neighbors of the vrible node (server) nd the fctor node (client) s N s nd N c, respectively (neighbors of server re the set of clients who rted the server while neighbors of client re the servers whom it rted). Superscripts in the representtion of the neighbors denote whether the neighbors of peer re determined considering the peer s client (c) or s server (s). We note tht neighbors of peer s server (or vrible node) do not need to be the sme s its neighbors s client (or fctor note). We further let Ξ = N s \{} nd = N c \{}. Fctor nd vrible nodes in Fig. 1 itertively exchnge probbilistic messges following the BP lgorithm, updting the degree of beliefs on the reputtion vlues of the servers s well s the trustworthiness vlues of the clients t ech step, until the itertions stop. Let G (ν) = {G (ν) : S} be the collection of vribles representing the vlues of the vrible nodes t the itertionν of the lgorithm. We denote the messges from the vrible nodes to the fctor nodes nd from the fctor nodes to the vrible nodes s µ nd λ, respectively. The messge µ (ν) (G(ν) ) denotes the probbility of G (ν) = l, l {,1}, t the ν th itertion. On the other hnd, λ (ν) (G(ν) ) denotes the probbility tht G (ν) = l, for l {,1}, t the ν th itertion given T nd R. The messge from the fctor node to the vrible node t the ν th itertion is formed using the principles of the BP s λ (ν) (G(ν) ) = f (G (ν),t,r (ν 1) ) µ (ν 1) x (G(ν) x ), G (ν) \{G(ν) } x (3) whereg is the set of vrible nodes which re the rguments of the locl functionf t the fctor node. This messge trnsfer is illustrted in Fig.. Further, R (ν 1) (the trustworthiness of client clculted t the end of (ν 1) th itertion) is vlue between zero nd one nd cn be clculted s follows: R (ν 1) = 1 1 N c i N c x {,1} T i x µ (ν 1) i (x). (4) It is shown tht such fctoriztion eventully gives the mrginl probbility distributions vi the BP lgorithm [4].

4 The bove eqution cn be interpreted s one minus the verge inconsistency of client clculted by using the messges it received from ll its neighbors. The bove computtion must be performed for every neighbors of ech fctor nodes. This finishes the first hlf of the ν th itertion. During the second hlf, the vrible nodes (servers) generte their messges (µ) nd send to their neighbors. Vrible node forms µ (ν) (G(ν) ) by multiplying ll informtion it receives from its neighbors excluding the fctor node, s shown in Fig. 3. Hence, the messge from vrible node to the fctor node t the ν th itertion is given by µ (ν) (G(ν) ) = 1 λ (ν) i (h) λ (ν) i (G(ν) ). (5) i Ξ h {,1} i Ξ This computtion is repeted for every neighbors of ech vrible node. The lgorithm proceeds to the next itertion in the sme wy s the ν th itertion. It is worth noting tht since ech peer is both server nd client, t the first hlf of the itertion, ech peer genertes messges s client nd in the second hlf of the itertion, ech peer genertes messges s server. Further, ech peer wits for ll the messges from its neighbors before it cretes its new messge either s client or s server. We note tht the itertive lgorithm strts its first itertion by computing λ (1) (G(1) ) in (3). However, insted of clculting in (4), the trustworthiness vlue R from the previous execution of BP-PP is used s initil vlues in (3). BP-PP stops fter Ψ itertions (which is pre-defined number nd its selection will be discussed in Section III-C). At the end of ech itertion, the reputtions re clculted t ech server. To clculte the reputtion vlue G (ν), ech server computes µ (ν) (G (ν) ) using (5) but replcing Ξ with N s, nd then sets G (ν) = 1 i= iµ(ν) (i). Thus, fter the lst itertion (i.e., Ψ th itertion), ech server peer obtins its updted reputtion vlue nd ech client peer obtins its updted trustworthiness vlue. A. Secure BP-PP As we discussed before, the fundmentl scheme described in Section II is not completely secure since ech peer computes nd reports (upon n inquiry) its own reputtion nd trustworthiness vlues. However, it is cler tht mlicious peer would report its own reputtion nd trustworthiness vlues to other peers incorrectly. Therefore, we propose to use group of rndomly selected peers (referred s the score mngers) to do the messge exchnge, nd hence, compute the reputtion nd trustworthiness vlues on behlf of ech peer s in [6]. It is importnt to note tht the score mngers re not trustworthy peers nd they cn lso be mlicious s will be discussed lter. Similr to [6], to ssign score mngers, we use Distributed Hsh Tble (DHT) [17]. DHTs use hsh function to deterministiclly mp the unique ID of ech peer into the points in logicl coordinte spce. At ny time, the coordinte spce is prtitioned mong the peers in the PP networ such tht every peer covers region in the coordinte spce. Hence, score mnger(s) of n rbitrry peer i is determined by hshing the unique ID of peer i into point in the coordinte spce nd the peer which currently covers this point s prt of its DHT region is ppointed s the score mnger of peer i 3. Thus, ny peer cn esily determine the score mnger(s) of peer i from its unique ID. We ssume tht the DHT cn cope with the dynmics of the networ (e.g., score mngers leving the 3 If peer i hs more thn one score mngers, the unique ID of peer i cn be conctented with n integer before hshing. system) s in [6]. Since it is not the min contribution of this pper, we do not give further detil bout the selection of the score mngers. As we mentioned before, our min contribution is the computtion of trust nd reputtion vlues t the score mngers vi the BP-bsed lgorithm. Next, we show how we modify our fundmentl scheme such tht it cn be executed by score mngers. Using the DHT, ech peer is ssigned with ξ score mngers from the set H = {H 1,H,...,Hξ }. We ssume tht ech score mnger of peer nows: i) neighbors of peer s server (i.e., N s ), nd hence, the score mngers of these neighbors, ii) neighbors of peer s client (i.e., N c ), nd hence, the score mngers of these neighbors, iii) rtings previously provided by peer s client (i.e., T ), iv) rtings previously received by peer s server, nd v) trustworthiness vlue of peer s client computed t the previous execution of the lgorithm. The score mngers in H generte the BP messges on behlf of peer both s server (vrible node) nd s client (fctor node). Further, they compute the reputtion vlue (s server) nd the trustworthiness vlue (s client) of peer bsed on the messges they receive from the score mngers of peer s neighbors. We note tht since ech peer in the networ plys the role of both client nd server; ech score mnger lso hs the sme property. Therefore, t the first hlf of n itertion, ech score mnger genertes messges s client (fctor node) nd in the second hlf of the itertion, the sme score mnger genertes messges s server (vrible node) on behlf of the peer it is responsible for. From now on, for the clrity of presenttion, we refer to score mnger s client score mnger when it genertes messges s client, nd s server score mnger when it genertes messges s server. Thus, different from the fundmentl scheme described in Section II, BP messges re now exchnged between the client score mngers nd the server score mngers. Due to this chnge, the fctor grph in Fig. 1 is lso modified bsed on the score mngers of the peers. As n exmple, we illustrte the chnge in the connectivity of client in Fig. 4 ssuming ξ = 3. As illustrted in the figure, insted of connecting client to the servers it rted (, b nd c), we connect the score mngers of peer to the score mngers of peers, b nd c in the fctor grph. Messges re exchnged between the score mngers of the peers following the principles of the BP lgorithm (s described in Section II) nd the lgorithm stops fter Ψ itertions (selection of Ψ will be discussed in Section III-C). We note tht every score mnger wits to receive ll messges from its neighbors before it genertes its new messge. Further, score mngers eep trc of the itertion numbers to both remin loosely synchronized between ech other nd relize when to stop the lgorithm. When client i wnts to use the service of server, it queries the reputtion vlue G from the score mngers of the peer. Similrly, the trustworthiness vlue of peer (s client) cn lso be queried from its score mngers. Once the client i receives ll the computed G vlues from ξ different score mngers in H, it computes the men of the received reputtion vlues to determine the finl reputtion vlue of server. There is one obvious drwbc of using score mngers for BP-PP lgorithm. When mlicious peer becomes the score mnger of relible or mlicious peer, it my crete nd send bogus messges to its neighbors. Therefore, mlicious messges my propgte in the BP lgorithm ffecting the efficiency of the lgorithm. We describe the ttc strtegies of such mlicious score mngers in Section III-A. Further, we evlute BP-PP under this ttc both nlyticlly nd vi simultions

5 λ (v-1) µ b b µ λ m m (v-1) µ c peer s client (v-1) µ d λ n peer s server λi c Fig. : Messge from the fctor node (client) to the vrible node (server) t the ν th itertion. d n Fig. 3: Messge from the vrible node (server) to the fctor node (client) t the ν th itertion. i b c 1 H 1 H H H H H 3 H 3 H Fig. 4: Utilizing score mngers in BP-PP. in Sections III-B nd III-C, respectively. B. Efficient BP-PP In this section, we provide some discussion on the computtionl complexity nd communiction overhed of BP-PP. 1) Computtionl Complexity: One cn show tht the computtionl complexity of BP-PP s mx(o ( ξc ),O ( ξv ) ) per peer in the number of multiplictions, where c nd v re smll numbers representing the verge number of rtings generted by client nd the verge number of rtings received by server. Therefore, the computtionl complexity of BP-PP is t most liner in the number of peers in the networ, ming it very ttrctive for lrge-scle systems. ) Communiction Overhed: In the fundmentl scheme described in Section II, ech client (nd ech server) sends different messges to ech of its neighbors t ech itertion. This introduces extr communiction overhed to the scheme when multiple score mngers re present for ech peer. Therefore, in the following, we modify the messges in (3) nd (5) between the peers (between the score mngers of the peers in the secure version described in Section II-A) to reduce the communiction overhed due to multiple score mngers for ech peer. Before discussing these modifictions in the messges between the score mngers, we first pproximte nd simplify the messge in (3) by ssuming tht the rguments of locl function t fctor node re independent from ech other (to reduce the computtionl complexity t the client peers). Using this ssumption, it cn be shown tht λ (ν) (G(ν) ) p(g (ν) T,R (ν 1) ), where p(g (ν) T,R (ν 1) ) = [ (R (ν 1) [ 1 R (ν 1) + 1 R(ν 1) T +(R (ν 1) H b Hc ] )T + 1 R(ν 1) (1 T ) G (ν) + + 1 R(ν 1) ] )(1 T ) (1 G (ν) ). (6) This resembles the belief/pleusbility concept of the Dempster- Shfer Theory [18], [19]. Given T = 1, R (ν 1) cn be viewed s the belief of the client tht G (ν) is one (t the ν th itertion). In other words, in the eyes of client, G (ν) is equl to one with probbility R (ν 1). Thus, (1 R (ν 1) ) corresponds to the uncertinty in the belief of client. In order to remove this uncertinty nd express p(g (ν) T,R (ν 1) ) s the probbilities tht G (ν) is zero nd one, we distribute the uncertinty uniformly between two outcomes (one nd zero). Hence, in the eyes of the client, G (ν) is equl to one with probbility (R (ν 1) +(1 R (ν 1) )/), nd zero with probbility ((1 R (ν 1) )/). We note tht similr sttement holds for the cse when T =. It is worth noting tht, s opposed to the Dempster-Shfer Theory, we do not combine the beliefs of the clients. Insted, we consider the belief of ech client individully nd clculte probbilities tht G (ν) being one nd zero in the eyes of ech client s in (6). We now describe how we modify the BP messges in (6) nd (5) to reduce the communiction overhed cused by multiple score mngers per ech peer. Let H be the set of score mngers of peer (in the secure version described in Section II-A). In principl, t ech itertion, we let ech score mnger H i in H brodcst single messge to ll of its neighbors insted of sending different messges to ech of its neighbors. Then, ech neighbor of the score mnger H i computes the ctul BP messge in (6) or (5) using the brodcsted messge (from H i ) nd the informtion it lredy possesses. In the following, we discuss the detils of this. Let H N c denote the set of score mngers of neighbors of client. Insted of computing (6) for ll its neighbors seprtely, ech client score mnger in H (in the secure version) only computes nd brodcsts the updted trustworthiness vlue of the client to its neighbors in H N c insted of sending different messges to ech of its neighbors. Since the score mngers in H N c now the rting vlue given by client to the servers they re responsible for, ech score mnger in H N c computes the ctul messge itself. For exmple, since the score mngers of server now T, they compute the messge in (6) by only using the brodcsted R (ν 1) vlue. Similrly, ll messges from server score mnger (H) to its neighbors (in H N s ) my be communicted simultneously vi single brodcst step to decrese the communiction overhed. The messge from H to one of its neighbors i in H N s is formed by multiplying ll the messges received t H excluding the one received from the score mnger i (similr to (5)). Thus, H cn simply brodcst the multipliction of ll received messges to its neighbors, nd llow i (nd ll other neighbors) to deduce the ctul messge from this brodcst.

6 Therefore, t ech itertion, server score mnger brodcsts single messge insted of sending different messges to ech of its neighbors. We note tht we used these modified messge formts for the evlution of BP-PP (in Section III). Let Ψ be the totl number of itertions required for single execution of the lgorithm nd ξ be the number of score mngers for ech peer. Then, ech score mnger sends (on the verge) ξψ messges during the execution of the BP-PP lgorithm (Ψ 1 s will be discussed in Section III-C). On the other hnd, in EigenTrust [6] ech score mnger sends (on the verge) mx(o ( ξψc ),O ( ξψv ) ) messges during the lgorithm, where c nd v represent the verge number of rtings generted by client nd the verge number of rtings received by server. Further, Ψ is reported to be (on the verge) 1 for EigenTrust [6]. Therefore, we conclude tht the proposed BP-bsed lgorithm does not introduce significnt communiction overhed to the networ. III. SECURITY EVALUATION In order to fcilitte future references, frequently used nottions re listed in Tble I. S U M U R H i r m d b The set of peers cting s servers The set of mlicious clients The set of non-mlicious clients The set of score mngers of peer i Rting given by mlicious client Totl number of newly generted rtings, per time-slot, per non-mlicious client Totl number of newly generted rtings, per time-slot, per mlicious client TABLE I: Nottions nd definitions. A. Thret Model We minly focus on the mlicious behviors of the clients nd score mngers nd explore their impct on the proposed trust nd reputtion mngement lgorithm. 1) Mlicious Clients: There re two mor ttcs tht re common for ny trust nd reputtion mngement mechnisms: i) Bd-mouthing, in which mlicious clients ttc the servers with the highest reputtion by giving low rtings in order to undermine them, nd ii) Bllot-stuffing, in which mlicious clients try to increse the reputtion vlues of servers with low reputtions. Further, there re opportunities for the mlicious score mngers to ttc specificlly to the BP lgorithm by creting incorrect BP messges. In the following, we describe how we modeled the dversry considering the forementioned threts to evlute BP-PP in the most dverse environment. We ssumed tht the mlicious clients initite bdmouthing 4. Further, ll the mlicious clients collude nd ttc the sme subset Γ of servers in ech time-slot (which represents the strongest ttc), by rting those servers s r m = (ssuming the rting vlues re either or 1). In other words, we denote by Γ the set of size b in which every victim server hs one edge from ech of the mlicious clients (in the fctor grph in Fig. 1). The subset Γ is chosen to include those servers who hve the highest reputtion vlues but received the lowest number of rtings from the non-mlicious clients (ssuming tht the ttcers hve this informtion 5 ). We note tht this ttc scenrio lso represents the RepTrp ttc in [] which is shown to be strong ttc. To the dvntge of mlicious peers, we ssumed tht totl of T time-slots hd pssed 4 Even though we use the bd-mouthing ttc, similr counterprt results hold for bllot-stuffing nd combintions of bd-mouthing nd bllot-stuffing. 5 Although it my pper unrelistic for some pplictions, vilbility of such informtion for the mlicious clients would imply the worst cse scenrio. since the initiliztion of the networ nd frction of the existing peers chnge behvior nd become mlicious fter T time-slots. In other words, mlicious clients behved lie non-mlicious clients nd incresed their trustworthiness vlues before mounting their ttcs t the (T + 1) th time-slot. We will evlute the performnce for the time-slot (T + 1). ) Mlicious Score Mngers: As we discussed in Section II-A score mngers of n rbitrry peer i generte the BP messges on behlf of the peer i (both s server nd s client). Therefore, mlicious score mngers of peer my crete nd send incorrect messges to their neighbors. By doing so, mlicious score mngers specificlly ttc the ccurcy of the BP lgorithm. When mlicious peer is the score mnger of peer i (i.e., H i ) it cretes bogus BP messges depending on the type of peer i s below: If i is non-mlicious peer: When cretes messge s server score mnger on behlf of peeri, it reports n incorrect vlue for the probbility of G (ν) i = l (l {,1}) t every itertion ν (e.g., if normlly G (ν) i = 1 with high probbility, cretes messge reporting tht G (ν) i = with probbility of 1, where is positive number close to zero). When cretes messge s client score mnger on behlf of peer i, it reports low trustworthiness vlue for peer i s client (e.g., reports the trustworthiness vlue of peer i s R (ν) i = σ t every itertion ν, where σ is positive number close to zero). If i is mlicious peer: When cretes messge s server score mnger on behlf of peer i, it reports high vlue for the probbility ofg (ν) i = 1 t every itertion ν to fvor its lly (e.g., cretes messge reporting tht G (ν) i = 1 with probbility of 1, where is positive number close to zero). When cretes messge s client score mnger on behlf of peer i, it reports high trustworthiness vlue for the mlicious peer i s client (e.g., reports the trustworthiness vlue of peer i s R (ν) i = 1 σ t every itertion ν, where σ is positive number close to zero). We note tht since the score mngers of the peers re ssigned vi DHT, we ssume tht mlicious score mngers do not collborte. We considered the bove thret models for both our nlyticl evlution nd simultions. B. Anlyticl Evlution We dopted the following models for vrious peers involved in the PP trust nd reputtion mngement system. We cnowledge tht lthough the models re not inclusive of every scenrio, they re good illustrtions to present our results. We ssumed tht the service qulity of ech server remins unchnged during our evlution. Moreover, the rting vlues re either or 1 where 1 represents good service qulity (e.g., providing uthentic files). Rtings generted by the non-mlicious clients re distributed uniformly mong the servers (i.e., their rtings/edges in the grph representtion re distributed uniformly mong the servers). We wish to evlute the performnce for the time-slot (T +1) t which mlicious peers chnge behvior nd initite their ttcs s discussed in Section III-A.

7 The performnce of reputtion mngement mechnism is determined by its ccurcy of estimting the reputtion vlues of the servers. Therefore, we evlute BP-PP in terms of the Men Absolute Error (MAE) ( G Ĝ ) computed t ech non-mlicious score mnger of every server, where Ĝ is the ctul vlue of the reputtion of server. We require two conditions to be stisfied: 1) the scheme should itertively reduce the impct of mlicious peers nd decrese the error in the reputtion vlues of the servers (computed t the nonmlicious score mngers) until the itertions stop, nd ) the error on the G vlue of ech server (computed t the nonmlicious score mngers) should be less thn or equl to ǫ (where ǫ should be smll vlue) fter the lst itertion (i.e., Ψ th itertion). In the following, we obtined the conditions nd probbilities to rrive t such scheme. We note tht lthough the discussions of the nlysis re bsed on RepTrp ttc vi bd-mouthing (s described in Section III-A), the system designed using these criteri will be robust ginst bllot-stuffing nd combintions of bd-mouthing nd bllotstuffing s well. The bd-mouthing ttc is imed to reduce the reputtion vlues of the victim servers. Hence, G vlue of victim server (computed t the non-mlicious score mngers in set H ) should be non-decresing function of itertions. This leds to the below lemm. Lemm 1: The error in the reputtion vlues of the servers decreses with ech successive itertions (until the itertions stop) if G () > G (1) is stisfied with high probbility t the non-mlicious score mngers of peer (H U R ) for every server ( S) with Ĝ = 1 6. Proof: Let G (ω) nd G (ω+1) be the reputtion vlue of n rbitrry server with Ĝ = 1 clculted t the (ω) th nd (ω + 1) th itertions t the non-mlicious score mngers of peer (H U R ), respectively. Further, let H R N denote the s set of score mngers of non-mlicious neighbors of server (i.e., N s U R ) nd H M N denote the set of score mngers of s mlicious neighbors of server (i.e., N s U M ). G (ω+1) G (ω) > if the following is stisfied t the (ω +1) th itertion. H R N s UR H R N s UR R (w+1) +1 1 R (w+1) R (w) +1 1 R (w) ˆR (w) H M N s UR H M N s UR 1 1+ 1 1+ (w+1) ˆR (w+1) ˆR (w) ˆR (w) ˆR >, (7) where R (w) nd re the trustworthiness vlues of relible nd mlicious client clculted t non-mlicious score mnger (s in (4)) t the w th itertion, respectively. Given G (ω) > G (ω 1) holds t the ω th itertion, we would (w) (w+1) get ˆR > ˆR for H M N U s R nd R (w+1) R (w) for H R N U R. Thus, (7) would hold for the(w+1) th itertion. s On the other hnd, if G (ω) < G (ω 1), we get ˆR(w) (w+1) < ˆR for H M N U s R nd R (w+1) < R (w) for H R N U R. s Hence, (7) is not stisfied t the (w+1) th itertion. Therefore, if G (ω) > G (ω 1) holds for some itertion ω t the peers in H U R, then the BP-PP lgorithm reduces the error on the reputtion vlue (G ) until the itertions stop, nd hence, it is sufficient to stisfy G () > G (1) with high probbility 6 The opposite must hold for ny server with ˆ G =. t the non-mlicious score mngers of every server with Ĝ = 1 (the set of servers from which the victims re ten) to gurntee tht BP-PP itertively reduces the impct of mlicious clients t the non-mlicious score mngers until the itertions stop. Although becuse of the Lemm 1, the error in the reputtion vlues of the servers decrese with successive itertions, it is uncler wht would be the eventul impct of the mlicious peers. Once the condition in Lemm 1 is met nd ssuming Ψ be the totl number of itertions required for single execution of the BP-PP lgorithm, the probbility (P ) tht BP-PP provides n MAE tht is less thn ǫ for ech server t every non-mlicious score mngers cn be obtined s in (8). In ˆR (Ψ+1) (8), R (Ψ+1) nd re the trustworthiness vlues of relible nd mlicious client clculted t non-mlicious score mnger, respectively. Further, R(Ψ+1) is the trustworthiness vlue of mlicious client clculted t mlicious score mnger. In the following exmple, we illustrte the results of our nlyticl evlution. The prmeters we used re U M + U R = 1, S = 1,ϑ =.9, T =,b = 1, = σ =.1, ξ = 3, nd Ψ = 1 (selection of Ψ will be discussed in Section III-C). Further, we ssumed tht d is rndom vrible with Yule-Simon distribution, which resembles the power-lw distribution used in modeling PP nd online systems [1], with the probbility mss function f d (d;ρ) = ρb(d,ρ+1) (with ρ = 1), where B is the Bet function. Finlly, we ssumed the thret model described in Section III-A. We note tht we lso evluted BP-PP with different prmeters nd obtined similr results. In Fig. 5, we illustrted the probbility of BP-PP providing MAE tht is less thn ǫ (t ech nonmlicious score mnger) versus frction of mlicious peers for two different ǫ vlues. We observed tht for n cceptble vlue of ǫ (ǫ =.1), BP-PP stisfies MAE < ǫ with high probbility for up to 3% mlicious peers. Moreover, Fig. 6 illustrtes the verge MAE vlues provided by BP-PP (t ech non-mlicious score mnger) with high probbility for different frctions of mlicious peers. We observed tht BP- PP provides significntly smll error vlues for up to 3% mlicious peers. We note tht these nlyticl results re lso consistent with our simultion results tht re illustrted in the next section. C. Simultions We evluted the performnce of BP-PP vi computer simultions (vi MATLAB) nd compred BP-PP with the Byesin reputtion mngement frmewor in [7] (which is lso proposed s the reputtion mngement system of the well-nown CONFIDANT protocol [8]) nd the EigenTrust lgorithm [6] in distributed PP networ environment. We ssumed tht d is rndom vrible with Yule-Simon distribution (with ρ = 1) s discussed in Section III-B. We set T =, b = 1, ρ = 1, U = 1, S = 1, = σ =.1, ξ = 3, nd the fding prmeter s ϑ =.9 7. Further, we ssumed tht rting vlues re from the set Υ = {, 1}. Finlly, we ssumed the thret model described in Section III-A in which there re both mlicious clients nd mlicious score mngers. Let Ĝ be the ctul reputtion vlue of server. We obtined the performnce of BP-PP, t ech time-slot, s the Men Absolute Error (MAE) G Ĝ, verged over the reputtion vlues of ll victim servers (i.e., the servers tht re under ttc) computed t their non-mlicious score mngers. 7 We note tht for the EigenTrust nd the Byesin frmewor we used the sme fding mechnism s BP-PP nd set the fding prmeter s ϑ =.9.

8 P = SPr { ǫ 1 (R (Ψ+1) H R N s UR 1 +1) H M N s UR +1) H R N s UR (R (Ψ+1) (1 (Ψ+1) ˆR ) H M N s UM (Ψ+1) (1 ˆR ) (1 H M N s UR H M N s UM (1 (Ψ+1) R )+ H R N s UR (1 R (Ψ+1) H M N s UR ) R (Ψ+1) ) (1+ (Ψ+1) ˆR ) H M N s UM (1+ R (Ψ+1) ) } (8) Pr(MAE < ε).8.6.4. ε =.1 ε =.1 5 1 15 5 3 35 % mlicious peers Fig. 5: Probbility of MAE being less thn ǫ versus frction of mlicious peers. For the Byesin frmewor [7], we used the prmeters from the originl wor [7] (devition thresholdd =.5 nd trustworthiness threshold t =.75). Further, in fvor of the Byesin frmewor, we ssumed tht ech peer hve ccess to the server-client mtrix T. Therefore, we observed the reputtion vlues computed t ll non-mlicious peers nd we verged the MAE over the reputtion vlues of the victim servers. For the EigenTrust, we implemented the distributed lgorithm described in [6] (with ξ = 3 score mngers for ech peer s in BP-PP) nd observed the reputtion vlues computed t the non-mlicious score mngers s we did for BP-PP. We note tht we did not ssume the existence of the pre-trusted peers for ny schemes. First, we determined the totl number of itertions (Ψ) required for the BP-PP lgorithm. Thus, in Fig. 7 8, we observed the verge number of required itertions for BP- PP to converge t ech peer (i.e., computed reputtion vlues stop chnging) for different frctions of mlicious peers (W = U M U M + U R ), t different time-slots (mesured since the ttc is pplied). We conclude tht the verge number of itertions for convergence is lwys less thn 1 nd it decreses with time nd decresing frction of mlicious peers. Thus, we used Ψ = 1 for the remining of this section. Then, we evluted the MAE performnce of BP-PP for different frctions of mlicious peers (W ), t different time-slots in Fig. 8. We observed tht BP-PP provides significntly low errors for up to bout W = 5% mlicious peers. Next, we observed the chnge in the verge trustworthiness (R i vlues) of mlicious clients computed t the non-mlicious score mngers. Figure 9 illustrtes the drop in the trustworthiness of the mlicious clients with time. We conclude tht the R i vlues of the mlicious clients (computed t non-mlicious score mngers) decrese over time, nd hence, the impct of their mlicious rtings is neutrlized over time. Finlly, we compred the MAE performnce of BP-PP with the Byesin Frmewor nd the EigenTrust lgorithm. Figure 1 illustrtes the comprison of BP-PP with these schemes for different frctions of mlicious peers t the first time-slot the ttc is pplied. It is cler tht 8 The plots in Figs. 7, 8 nd 9 re shown from the time-slot the dversry introduced its ttc. Averge(log 1 MAE) 5 1 15 5 5 1 15 5 3 35 % mlicious peers Fig. 6: The verge MAE versus frction of mlicious peers. BP-PP outperforms the Byesin Frmewor nd EigenTrust significntly. We note tht t lter time-slots, BP-PP still eeps its superiority over the other schemes. From this comprison, we conclude tht in EigenTrust, even the non-mlicious score mngers compute the reputtion vlues with lrge MAE in the presence of the ttcers. Further, when the mlicious nodes collbortively ttc, Byesin Frmewor results in high MAE in the reputtion vlues of the servers. Finlly, we observed the impct of ξ (the number of score mngers for ech peer) to the performnce of BP-PP under the sme ttc scenrio (in which there re both mlicious clients nd mlicious score mngers s described in Section III-A). In Fig. 11, we illustrted MAE performnce of BP-PP for different vlues of ξ nd for different frctions of mlicious peers t the first time-slot the ttc is pplied. As expected, for smll vlues of ξ (i.e., ξ = 1 nd ξ = ), BP-PP provides higher MAE vlues since the probbility tht ll score mngers of victim client being mlicious increses with decresing vlues of ξ. Next, we simulted the sme ttc scenrio when rtings re integers from the set Υ = {1,...,5} insted of binry vlues 9. Mlicious clients choose the victim servers from Γ nd rte them s r m = 4. The mlicious clients do not devite very much from the ctul Ĝ = 5 vlues to remin undercover s mny time-slots s possible. We lso tried higher devitions from the Ĝ vlue nd observed tht the mlicious clients were esily detected by BP-PP. We compred the MAE performnce of BP-PP with the other schemes t the first timeslot the ttc is pplied in Fig. 1 nd observed tht BP-PP outperforms ll the other techniques. We observed tht BP-PP provides significntly low MAE for up to W = 5% mlicious clients. We further observed tht the Byesin Frmewor performs better thn EigenTrust for this scenrio. IV. CONCLUSION In this pper, we introduced the first ppliction of the Belief Propgtion lgorithm in the design nd evlution of distributed trust nd reputtion mngement systems for 9 For the ttc ginst the BP lgorithm, we ssumed similr scenrio to the binry cse s discussed in Section III-A.

9 Number of itertions 8 7 6 5 4 3 W=5% W=1% W=15% W=% W=5% W=3% 4 6 8 1 time slot Fig. 7: The verge number of itertions versus time for BP-PP to converge when W of the existing peers become mlicious. MAE.6.5.4.3..1 BP PP EigenTrust Byesin Frmewor 5 1 15 5 3 % mlicious peers Fig. 1: MAE performnce of vrious schemes when different frctions of the existing peers become mlicious t the first time-slot the ttc is pplied. PP networs. We presented the generl protocol for Belief Propgtion-Bsed Trust nd Reputtion Mngement for PP Networs (BP-PP). BP-PP is grph-bsed system in which the reputtion nd trustworthiness vlue of ech peer is computed by distributed messge pssing mong the peers in the grph. We studied BP-PP in detiled nlysis nd computer simultions. We showed tht proposed BP-PP is robust mechnism to evlute the reputtion vlues of the peers from the received rtings. Moreover, it effectively evlutes the trustworthiness of the peers (in the relibility of their rtings). We lso compred BP-PP with some well-nown PP reputtion mngement schemes nd showed the superiority of the proposed scheme in terms of its robustness ginst mlicious behvior. Finlly, we showed tht the computtionl complexity of the proposed scheme grows only linerly with the number of peers in the networ while its communiction overhed is lower tht the well-nown EigenTrust lgorithm. REFERENCES [1] B. Fn, J. C. S. Lui, nd D.-M. Chiu, The design trde-offs of bittorrentlie file shring protocols, IEEE/ACM Trnsctions on Networing, vol. 17, pp. 365 376, April 9. [] M. Adler, R. Kumr, K. Ross, D. Rubenstein, T. Suel, nd D. D. Yo, Optiml peer selection for pp downloding nd streming, In Proceedings of IEEE INFOCOM 5, 5. [3] J. Perl, Probbilistic Resoning in Intelligent Systems: Networs of Plusible Inference. Morgn Kufmnn Publishers, Inc., 1988. [4] F. Kschischng, B. Frey, nd H. A. Loeliger, Fctor grphs nd the sumproduct lgorithm, IEEE Trnsctions on Informtion Theory, vol. 47, pp. 498 519, Feb. 1. 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