belief Propgtion Lgorithm in Nd Pent Penta

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1 IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE Itertive Trust nd Reputtion Mngement Using Belief Propgtion Ermn Aydy, Student Member, IEEE, nd Frmrz Feri, Senior Member, IEEE Abstrct In this pper, we introduce the first ppliction of the belief propgtion lgorithm in the design nd evlution of trust nd reputtion mngement systems. We pproch the reputtion mngement problem s n inference problem nd describe it s computing mrginl lielihood distributions from complicted globl functions of mny vribles. However, we observe tht computing the mrginl probbility functions is computtionlly prohibitive for lrge-scle reputtion systems. Therefore, we propose to utilize the belief propgtion lgorithm to efficiently (in liner complexity) compute these mrginl probbility distributions; resulting fully itertive probbilistic nd belief propgtion-bsed pproch (referred to s BP-ITRM). BP-ITRM models the reputtion system on fctor grph. By using fctor grph, we obtin qulittive representtion of how the consumers (buyers) nd service providers (sellers) re relted on grphicl structure. Further, by using such fctor grph, the globl functions fctor into products of simpler locl functions, ech of which depends on subset of the vribles. Then, we compute the mrginl probbility distribution functions of the vribles representing the reputtion vlues (of the service providers) by messge pssing between nodes in the grph. We show tht BP-ITRM is relible in filtering out mlicious/unrelible reports. We provide detiled evlution of BP-ITRM vi nlysis nd computer simultions. We prove tht BP-ITRM itertively reduces the error in the reputtion vlues of service providers due to the mlicious rters with high probbility. Further, we observe tht this probbility drops suddenly if prticulr frction of mlicious rters is exceeded, which introduces threshold property to the scheme. Furthermore, comprison of BP-ITRM with some well-nown nd commonly used reputtion mngement techniques (e.g., Averging Scheme, Byesin Approch, nd Cluster Filtering) indictes the superiority of the proposed scheme in terms of robustness ginst ttcs (e.g., bllot stuffing, bd mouthing). Finlly, BP-ITRM introduces liner complexity in the number of service providers nd consumers, fr exceeding the efficiency of other schemes. Index Terms Trust nd reputtion mngement, belief propgtion, itertive lgorithms, bd mouthing, bllot stuffing, online services, e-commerce. Ç 1 INTRODUCTION TRUST nd reputtion re crucil requirements for most environments wherein entities prticipte in vrious trnsctions nd protocols mong ech other. In most online service systems, the consumer of the service (e.g., the buyer) hs no choice but to rely on the reputtion of the service provider (e.g., the seller) bsed on the ltter s prior performnce. A reputtion mngement mechnism is promising method to protect the consumer (buyer) of the service by forming some foresight bout the service providers (sellers) before using their services (or purchsing their products). By using reputtion mngement scheme, n individul peer s reputtion cn be formed by the combintion of received reports (rtings). Hence, fter ech trnsction, prty who receives the service or purchses the product (referred to s the rter) provides (to the centrl uthority) its report bout the qulity of the service provided (or the qulity of the product purchsed) for tht trnsction. The centrl uthority collects the reports nd updtes the reputtions of the service providers (sellers). Therefore, the min gol of reputtion mechnism is to determine the service (product), qulities of the. The uthors re with the Deprtment of Electricl nd Computer Engineering, Georgi Institute of Technology, Atlnt 30332, GA. E-mil: eydy@gtech.edu, feri@ece.gtech.edu. Mnuscript received 19 Feb. 2010; revised 25 Aug. 2011; ccepted 14 Dec. 2011; published online 23 Dec Recommended for cceptnce by V. Vrdhrn. For informtion on obtining reprints of this rticle, plese send e-mil to: tdsc@computer.org, nd reference IEEECS Log Number TDSC Digitl Obect Identifier no /TDSC service providers (sellers), nd the trustworthiness of the rters bsed on their reports bout the service qulities. Hence, the success of reputtion scheme depends on the robustness of the mechnism to ccurtely evlute the reputtions of the service providers (sellers) nd the trustworthiness of the rters. Trust nd reputtion mechnisms hve vrious ppliction res from online services to mobile d-hoc networs (MANETs) [1], [2], [3], [4]. Most well-nown commercil websites such s eby, Amzon, Netflix, nd Google use some types of reputtion mechnisms. Hence, it is foreseeble tht the socil web is going to be driven by these reputtion systems. Despite recent dvnces in reputtion systems, there is yet need to develop relible, sclble, nd dependble schemes tht would lso be resilient to vrious wys reputtion system cn be ttced. Moreover, new nd untested pplictions open up new vulnerbilities, nd hence, requiring specific solutions for reputtion systems. As in every security system, trust nd reputtion mngement systems re lso subect to mlicious behviors. Mlicious rters my ttc prticulr service providers (sellers) in order to undermine their reputtions while they help other service providers by boosting their reputtions. Similrly, mlicious service providers (sellers) my provide good service qulities (or sell high-qulity products) for certin customers (buyers) in order to eep their reputtions high while cheting the other customers. Moreover, mlicious rters (or service providers) my collbortively mount sophisticted ttcing strtegies by /12/$31.00 ß 2012 IEEE Published by the IEEE Computer Society

2 376 IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE 2012 exploiting their prior nowledge bout the reputtion mechnism. Hence, building resilient trust nd reputtion mngement system tht is robust ginst mlicious ctivities becomes chllenging issue. In this pper, we introduce the first ppliction of the belief propgtion lgorithm in the design nd evlution of trust nd reputtion mngement systems. In our previous wor, inspired by our erlier wor on itertive decoding of error-control codes in the presence of stopping sets [5], [6], [7], we proposed n lgebric itertive lgorithm [8] for reputtion systems (referred to s ITRM) nd showed the benefit of using itertive lgorithms for trust nd reputtion mngement. Here, we expnd this wor nd introduce fully probbilistic pproch bsed on the belief propgtion lgorithm. Different from our previous wor [8], in this pper, we view the reputtion mngement problem s n inference problem nd describe it s computing mrginl lielihood distributions from complicted globl functions of mny vribles. Further, we utilize the belief propgtion lgorithm to efficiently (in liner complexity) compute these mrginl probbility distributions. The wor is inspired by erlier wor on grph-bsed itertive probbilistic decoding of turbo codes nd low-density prity-chec (LDPC) codes, the most powerful prcticlly decodble error-control codes nown. These decoding lgorithms re shown to perform t error rtes ner wht cn be chieved by the optiml scheme, mximum lielihood decoding, while requiring fr less computtionl complexity (i.e., liner in the length of the code). We believe tht the significnt benefits offered by the itertive probbilistic lgorithms cn be lso tpped in to benefit the field of reputtion systems. In itertive decoding of LDPC, every chec vertex (in the grph representtion of the code) hs some opinion of wht the vlue of ech bit vertex should be. The itertive decoding lgorithm would then nlyze the collection of these opinions to decide, in ech itertion, wht vlue to ssign for the bit vertex under exmintion. Once the vlues of the bit vertices re estimted, in the next itertion, those vlues re used to determine the stisfction of the chec vertex vlues. The contribution of our reserch stems from the observtion tht similr pproch cn be dpted to determine the reputtions of the service providers (sellers) s well s the trustworthiness of the rters. Furthermore, the nlysis of reputtion systems resembles tht of the code design problem. In LDPC, one of the gols is to find the decoding error for fixed set of chec constrints. Similrly, in the reputtion system, our gol is to specify the regions of trust for the set of the system prmeters. A region of trust is the rnge of prmeters for which we cn confidently determine the reputtion vlues within given error bound. We cnowledge, however, tht we hve hrder problem in the cse of reputtion systems s the dversry dynmics is fr more complicted to nlyze thn the ersure chnnel in the coding problem. We introduce the Belief Propgtion-bsed Itertive Trust nd Reputtion Mngement Scheme (BP-ITRM). Belief propgtion [9], [10], [11] is messge pssing lgorithm for performing interfce on grphicl models such s Byesin networs or Mrov rndom fields. It is used for computing mrginl distributions of the unobserved nodes conditioned on the observed ones. Computing mrginl distributions is hrd in generl s it might require summing n exponentilly lrge number of terms. Hence, the belief propgtion lgorithm is usully described in terms of opertions on fctor grphs. The fctor grph representtion of the reputtion systems turned out to be biprtite grph, where the service providers (sellers) nd consumers (buyers) re rrnged s two sets of vrible nd fctor nodes tht re connected vi some edges. The reputtion cn be computed by messge pssing between nodes in the grph. In ech itertion of the lgorithm, ll the vrible nodes (sellers), nd subsequently ll the fctor nodes (buyers), pss new messges to their neighbors until the reputtion vlue converges. We note tht in the rest of this pper, we use the word messge s virtul term. The exchnge of messges re not between the ctul sellers nd buyers; ll messges between the nodes in the grph (i.e., between the vrible nd fctor nodes) re formed by the lgorithm tht is rn in the centrl uthority. We show tht the proposed itertive scheme is relible (in filtering out mlicious/unrelible reports). Further, we prove tht BP- ITRM itertively reduces the error in the reputtion vlues of service providers due to the mlicious rters with high probbility. We observe tht this probbility suddenly drops if the frction of mlicious rters exceeds threshold. Hence, the scheme hs threshold property. The proposed reputtion mngement lgorithm cn be utilized in well-nown online services such s eby or Epinions. In eby, ech seller-buyer pir rte ech other fter trnsction. Thus, BP-ITRM cn be used in eby to compute the reputtion vlues of the sellers nd buyers long with the trustworthiness vlues of the peers in their rtings. Epinions, on the other hnd, is product review site in which users cn rte nd review items. Users cn lso give rtings to the reviews. Hence, the rtings of members on review nd on product re considered seprtely. BP-ITRM cn be utilized in such n environment to compute the reputtions of the reviewers bsed on the rtings given by the users on the reviews. Although we present the proposed lgorithm s centrlized pproch, it cn lso be pplied to decentrlized systems such s d hoc networs nd P2P systems to compute the reputtions of the nodes in the networ. As n exmple, we pplied ITRM, our lgebric but itertive reputtion mngement system, to dely tolernt networs [12] in decentrlized environment. The rest of this pper is orgnized s follows: in the rest of this section, we summrize the relted wor, list the contributions of this wor nd describe the belief propgtion lgorithm. In Section 2, we describe the proposed BP-ITRM in detil. Next, in Section 3, we mthemticlly model nd nlyze BP-ITRM. Further, we support the nlysis vi computer simultions, compre BP-ITRM with the existing nd commonly used trust mngement schemes, nd discuss the computtionl complexity of the proposed scheme. Finlly, in Section 4, we conclude our pper. 1.1 Relted Wor Severl wors in the literture hve focused so fr on building reputtion-mngement mechnisms [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. We my clssify reputtion mechnisms for centrlized systems s 1) globl reputtion systems, where the reputtion of service provider (seller) is bsed on the reports form generl users [26], [27], nd 2) personlized reputtion systems, where the reputtion of service provider (seller) is determined bsed

3 AYDAY AND FEKRI: ITERATIVE TRUST AND REPUTATION MANAGEMENT USING BELIEF PROPAGATION 377 on the reports of group of prticulr users, which my be different in the eyes of different users [28], [29]. We note tht our wor flls under the ctegory of globl reputtion systems. The most fmous nd primitive globl reputtion system is the one tht is used in eby. In eby, ech sellerbuyer pir rte ech other fter trnsction, nd the totl rting of peer is the sum of the individul rtings it received from the other peers. It is shown in [30] tht, even this simple reputtion mechnism provides the sellers with high reputtion to sell their items more thn the other sellers. On the other hnd, since eby s reputtion scheme weights ll individul rtings eqully, the unfir rtings (the ones coming from the unrelible peers) re not filtered, effecting the reputtion vlues of the sellers significntly. Other wellnown web sites such s Amzon, Epinions, nd AllExperts use more dvnced reputtion mechnism thn eby. Their reputtion mechnisms mostly compute the verge (or weighted verge) of the rtings received for product (or peer) to evlute the globl reputtion of product (or peer). Hence, these schemes re vulnerble to collbortive ttcs by mlicious peers. Google s PgeRn lgorithm [26] cn lso be considered s globl reputtion systems. This lgorithm does not require the prticiption of the users to rn the web pges. Bsiclly, the web pge with more bc lins (lins tht point to it) is considered to be more importnt (hs higher rn) thn the one with fewer bc lins. PgeRn lgorithms is lso modified nd used in socil networs for the reputtion of the peers [31], [32]. Use of the Byesin Approch is lso proposed in [27], [33]. In these systems, the posteriori reputtion vlue of peer is computed combining its priori reputtion vlues with the new rtings received for the peer. Further, threshold method is used to determine nd updte the report relibility of the rter peers. Finlly, [29] proposed to use the Cluster Filtering method [34] for reputtion systems to distinguish between the relible nd unrelible rters. We compre our proposed scheme with the existing schemes (in Section 3.3) nd show its superior performnce (i.e., ccurcy nd robustness ginst ttcs). Personlized reputtion systems re lso widely studied for different purposes. In Histos [28], the centrl node (server) eeps ll the rtings between the peers nd genertes grph to clculte the rtings of ech peer for the other peers. However, ech updte of this grph requires lot of computtions. Hence, this scheme hs high-computtionl complexity. The most well-nown method tht is used to build personl reputtions is the Collbortive Filtering [35], [36]. Using this method, the predicted rting of peer i for nother peer (tht i hs not directly rted) is clculted by the min server using memory-bsed lgorithm (such s similrity testing [37]) or model-bsed lgorithm (such s mtrix fctoriztion [38]). However, these types of systems hve cold strt nd dt sprseness problems which cuse them to be vulnerble ginst mlicious behvior. 1.2 Contributions of the Pper The min contributions of our wor re summrized in the following. 1. We introduce the first ppliction of the belief propgtion lgorithm on trust nd reputtion mngement systems. 2. As the core of our trust nd reputtion mngement system, we use the belief propgtion lgorithm which is proven to be powerful tool on decoding of turbo codes nd LDPC codes. Therefore, we introduce grph-bsed trust nd reputtion mngement mechnism tht relies on n ppropritely chosen fctor grph nd computes the reputtion vlues of service providers (sellers) by messge pssing lgorithm. 3. The proposed itertive lgorithm computes the reputtion vlues of the service providers (sellers) ccurtely (with smll error) in short mount of time in the presence of ttcers. The scheme is lso robust nd efficient methodology for detecting nd filtering out mlicious rtings. Further, the scheme detects the mlicious rters with high ccurcy, nd updtes their trustworthiness ccordingly enforcing them to execute low-grde ttcs to remin undercover. 4. The proposed BP-ITRM significntly outperforms the existing nd commonly used reputtion mngement techniques such s the Averging Scheme, Byesin Approch s in [27] nd [33], nd Cluster Filtering in the presence of ttcers. 1.3 Belief Propgtion Belief propgtion [9], [10], [11] is messge pssing lgorithm for performing interfce on grphicl models (Byesin networs, Mrov rndom fields). It is method for computing mrginl distributions of the unobserved nodes conditioned on the observed ones. Computing mrginl distributions is hrd in generl s it might require summing n exponentilly lrge number of terms. Hence, belief propgtion lgorithm is usully described in terms of opertions on fctor grph. A fctor grph is biprtite grph contining nodes corresponding to vribles nd fctors with edges between them. A fctor grph hs vrible node for ech vrible, fctor node for ech function, nd n edge connecting vrible node to fctor node if nd only if the vrible is n rgument of function corresponding to the fctor node. The mrginl distribution of n unobserved node cn be computed ccurtely using the belief propgtion lgorithm if the fctor grph hs no cycles. However, the lgorithm is still well defined nd often gives good pproximte results even for the fctor grphs with cycles (s it hs been observed in decoding of LDPC codes). Belief propgtion is commonly used in rtificil intelligence nd informtion theory. It hs demonstrted empiricl success in numerous pplictions including LDPC codes, turbo codes, free energy pproximtion, nd stisfibility. In itertive decoding of LDPC, for exmple, every chec vertex (in the grph representtion of the code) hs some opinion of wht the vlue of ech bit vertex should be. The itertive decoding lgorithm would then nlyze the collection of these opinions to decide, in ech itertion, wht vlue to ssign for the bit vertex under exmintion. Once the vlues of the bit vertices re estimted, in the next itertion, those vlues re used to determine the stisfction of the chec-vertex vlues. While the optiml decoding technique of LDPC codes, mximum lielihood (ML)

4 378 IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE 2012 decoding, is n NP problem, belief propgtion lgorithm provides very efficient decoding tht gets close to the bit error rte (BER) performnce of the ML decoding when the code length becomes lrge. In other words, belief propgtion performs t error rtes ner wht cn be chieved by the optiml scheme while requiring fr less computtionl complexity. Here, we propose to exploit such benefits in trust nd reputtion mngement systems. 2 BELIEF PROPAGATION FOR ITERATIVE TRUST AND REPUTATION MANAGEMENT As in every reputtion mngement mechnism, we hve two min gols: 1) computing the service qulity (reputtion) of the peers who provide service (henceforth referred to s Service Providers or SPs) by using the feedbcs from the peers who used the service (referred to s the rters), nd 2) determining the trustworthiness of the rters by nlyzing their feedbc bout SPs. We ssume two different sets in the system: ) the set of service providers, S nd b) the set of service consumers (herefter referred s rters), U. We note tht these two sets re not necessrily disoint. Trnsctions occur between SPs nd rters, nd rters provide feedbcs in the form of rtings bout SPs fter ech trnsction. Let G be the reputtion vlue of SP ( 2 S) nd T i be the rting tht rter i (i 2 U) reports bout SP ( 2 S), whenever trnsction is completed between the two peers. Moreover, let R i denote the trustworthiness of the peer i (i 2 U) s rter. In other words, R i represents the mount of confidence tht the reputtion system hs bout the correctness of ny feedbc/rting provided by rter i. All of these prmeters my evolve with time. However, for simplicity, we omitted time dependencies from the nottion. We ssume there re u rters nd s SPs in the system (i.e., U ¼u nd S ¼s). Let G ¼fG : 2 Sg nd IR ¼ fr i : i 2 Ug be the collection of vribles representing the reputtions of the SPs nd the trustworthiness vlues of the rters, respectively. Further, let T be the s u SP-rter mtrix tht stores the rting vlues (T i ), nd T i be the set of rtings provided by rter i. We consider slotted time throughout this discussion. At ech time-slot (or epoch), the itertive reputtion lgorithm is executed using the input prmeters IR nd T to obtin the reputtion prmeters (e.g., G). After completing its itertions, the BP-ITRM scheme outputs new globl reputtions of the SPs s well s the trustworthiness (IR vlues) of the rters. For simplicity of presenttion, we ssume tht the rting vlues re from the set ¼f0; 1g. The extension in which rting vlues cn te ny rel number cn be developed similrly (we implemented the proposed scheme for both cses nd illustrte its performnce in Section 3.3). 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, pðg T; IRÞ, ech of which is ssocited with vrible G ; the reputtion vlue of SP. Loosely speing, the present Byesin pproches [27], [33] solve for these mrginl distributions seprtely, leding to poor estimtes s they neglect the interply of the entire evidence. In contrst, we formulte the problem by considering the globl function pð G T; IRÞ, which is the oint probbility distribution function of the Fig. 1. Fctor grph between the SPs nd the rters in (3). vribles in G given the rting mtrix nd the trustworthiness vlues of the rters. Then, clerly, ech mrginl probbility function pðg T; IRÞ my be obtined s follows: pðg T; IRÞ¼ X pð G T; IRÞ; ð1þ GnfG g where the nottion GnfG 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 even for binry reputtion vlues. However, we propose to fctorize (1) to locl functions f i using fctor grph nd utilize the belief propgtion lgorithm to clculte the mrginl probbility distributions in liner complexity. A fctor grph is biprtite grph contining two sets of nodes (corresponding to vribles nd fctors) nd edges incident between two sets. Following [10], 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 belief propgtion lgorithm is still well defined nd empiriclly often gives good pproximte results for the fctor grphs with cycles. To describe the reputtion system, we rrnge the collection of the rters nd the SPs together with their ssocited reltions (i.e., the rtings of the SPs by the rters) s biprtite (or fctor) grph, s in Fig. 1. In this representtion, ech rter peer corresponds to fctor node in the grph, shown s squre. Ech SP is represented by vrible node shown s hexgon in the grph. Ech report/rting is represented by n edge from the fctor node to the vrible node. Hence, if rter i (i 2 U) hs report bout SP ( 2 S), we plce n edge with vlue T i from the fctor node i to the vrible node representing SP. We note tht the T i vlue between rter i nd SP is the ggregtion of ll pst nd present rtings between these two peers s described in the following. If ny new rting rrives from rter i bout SP, our scheme updtes the vlue T i by verging the new rting nd the old vlue of the edge multiplied with the fding fctor. The fctor i ðtþ is used to incorporte the fding fctor of the SPs reputtion (service qulity). We use nown fctor i ðtþ ¼ # tt i where # nd t i re the fding prmeter nd the time when the lst trnsction between rter i nd SP occurred, respectively. The prmeter # is chosen to be less thn one to give greter importnce to more recent rtings.

5 AYDAY AND FEKRI: ITERATIVE TRUST AND REPUTATION MANAGEMENT USING BELIEF PROPAGATION 379 Fig. 2. Setup of the scheme. Next, we suppose tht the globl function pð G T; IRÞ fctors into products of severl locl functions, ech hving subset of vribles from G s rguments s follows: pð G T; IRÞ¼ 1 Y f i ðg i ; T i ;R i Þ; ð2þ Z i2 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 rter. As n exmple, the fctor grph in Fig. 1 corresponds to pðg ;G b ;G c T; IRÞ¼ 1 Z f ðg ;G b ;G c ;T ;T b ;T c ;R Þ f m ðg ;G b ;T m ;T mb ;R m Þ f n ðg ;G c ;T n ;T nc ;R n Þ: We note tht using (3) in (1), one cn ttempt to compute the mrginl distributions. However, s discussed before, this cn get computtionlly infesible. Insted, we utilize the belief propgtion lgorithm to clculte the mrginl distributions of the vribles in G. We now introduce the messges between the fctor nd the vrible nodes to compute the mrginl distributions using belief propgtion. We note tht ll the messges re formed by the lgorithm tht is rn in the centrl uthority. To tht end, we choose n rbitrry fctor grph s in Fig. 2 nd describe messge exchnges between rter nd SP. We represent the set of neighbors of the vrible node (SP) nd the fctor node (rter) s N nd N, respectively (neighbors of SP re the set of rters who rted the SP while neighbors of rter re the SPs whom it rted). Further, let ¼ N nfg nd ¼ N nfg. The belief propgtion lgorithm itertively exchnges the probbilistic messges between the fctor nd the vrible nodes in Fig. 2, updting the degree of beliefs on the reputtion vlues of the SPs s well s the confidence of the rters on their rtings (i.e., trustworthiness vlues) t ech step, until convergence. Let G ¼fG : 2 Sg be the collection of vribles representing the vlues of the vrible nodes t the itertion 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 nd, respectively. The messge! ðg Þ denotes the probbility of G ¼, 2f0; 1g, t the th itertion. On the other hnd,! ðg Þ denotes the probbility tht G ¼, for 2f0; 1g, t the th itertion given T nd R. ð3þ Fig. 3. Messge from the fctor node to the vrible node t the th itertion. The messge from the fctor node to the vrible node t the th itertion is formed using the principles of the belief propgtion s! ¼ G X G ð1þ nfg ð1þ g f G ; T ;R ð1þ Y x2 ð1þ x! G ð1þ x ; where G is the set of vrible nodes which re the rguments of the locl function f t the fctor node. This messge trnsfer is illustrted in Fig. 3. Further, R ð1þ (the trustworthiness of rter clculted t the end of ð 1Þth itertion) is vlue between zero nd one nd cn be clculted s follows: R ð1þ ¼ 1 1 N X X i2n x2f0;1g ð4þ T i x ð1þ i! ðxþ: ð5þ The bove eqution cn be interpreted s one minus the verge inconsistency of rter clculted by using the messges it received from ll its neighbors. Using (4) nd the fct tht the reputtion vlues in set G re independent from ech other, it cn be shown tht! ðg Þ/pðG T ;R ð1þ Þ, where p G T ;R ð1þ ¼ "! # R ð1þ þ 1 Rð1Þ T þ 1 Rð1Þ ð1 T Þ 2 2 "! # 1 R ð1þ T þ R ð1þ þ 1 Rð1Þ ð1 T Þ G : G þ This resembles the belief/plusibility concept of the Dempster-Shfer Theory [39], [40]. Given T ¼ 1, R ð1þ cn be viewed s the belief of the th rter tht G is one (t the th itertion). In other words, in the eyes of rter, G is equl to one with probbility R ð1þ. Thus, ð1 R ð1þ Þ corresponds to the uncertinty in the belief of rter. In order to remove this uncertinty nd express pðg T ; Þ s the probbilities tht G is zero nd one, we R ð1þ distribute the uncertinty uniformly between two outcomes (one nd zero). Hence, in the eyes of the th rter, G is equl to one with probbility (R ð1þ þð1 R ð1þ Þ=2), nd zero ð6þ

6 380 IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE 2012 TABLE 1 Nottions nd Definitions Fig. 4. Messge from the vrible node to the fctor node t the th itertion. with probbility (ð1 R ð1þ Þ=2). We note tht similr sttement holds for the cse when T ¼ 0. It is worth noting tht, s opposed to the Dempster-Shfer Theory, we do not combine the beliefs of the rters. Insted, we consider the belief of ech rter individully nd clculte probbilities tht G being one nd zero in the eyes of ech rter s in (6). The bove computtion must be performed for every neighbors of ech fctor nodes. This finishes the first hlf of the th itertion. During the second hlf, the vrible nodes generte their messges () nd send it to their neighbors. Vrible node forms! ðg Þ by multiplying ll informtion it receives from its neighbors excluding the fctor node, s shown in Fig. 4. Hence, the messge from vrible node to the fctor node t the th itertion is given by! G 1 ¼ X Y h2f0;1g i2 i! ðhþ Y i2 i! G : This computtion is repeted for every neighbors of ech vrible node. The lgorithm proceeds to the next itertion in the sme wy s the th itertion. We note tht the itertive lgorithm strts its first itertion by computing ð1þ Þ in (4). However, insted of clculting in (5), the! ðgð1þ ð7þ trustworthiness vlue R from the previous execution of BP-ITRM is used s initil vlues in (6). The itertions stop when ll vribles in G converge. Therefore, t the end of ech itertion, the reputtions re clculted for ech SP. To clculte the reputtion vlue G, we first compute ðg with N, nd then we set G ¼ P 1 Þ using (7) but replcing ðiþ. i¼0 i 3 SECURITY EVALUATION OF BP-ITRM In this section, we mthemticlly model nd nlyze BP- ITRM. Moreover, we support the nlysis vi computer simultions nd compre BP-ITRM with the existing nd commonly used trust mngement schemes. In order to fcilitte future references, frequently used nottions re listed in Tble Attc Models We consider two mor ttcs tht re common for ny trust nd reputtion mngement mechnisms. Further, we ssume tht the ttcers my collude nd collborte with ech other:. Bd mouthing. Mlicious rters collude nd ttc the service providers with the highest reputtion by giving low rtings in order to undermine them. It is lso noted tht in ddition to the mlicious peers, in some pplictions, bd mouthing my be originted by group of selfish peers who ttempt to ween high-reputtion providers in the hope of improving their own chnces s providers.. Bllot stuffing. Mlicious rters collude to increse the reputtion vlue of peers with low reputtions. Just s in bd mouthing, in some pplictions, this could be mounted by group of selfish consumers ttempting to fvor their llies. 3.2 Anlytic Evlution We dopted the following models for vrious peers involved in the reputtion 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 qulity of ech service provider remins unchnged during time slots. Moreover, the rting vlues re either 0 or 1 where 1 represents good service qulity. Rtings generted by the nonmlicious rters re distributed uniformly mong the SPs (i.e., their rtings/edges in the grph representtion re distributed uniformly mong SPs). We further ssumed tht the rting vlue r h (provided by the nonmlicious rters) is rndom vrible with Bernoulli distribution, where Prðr h ¼ ^G Þ¼p c nd Prðr h 6¼ ^G Þ¼ð1 p c Þ, nd ^G is the ctul vlue of the globl reputtion of SP. Even though we ssumed binry vlues (0 or 1) for the ctul reputtion vlues of SPs, BP-ITRM lso performs well nd gives ccurte results when the ctul reputtion vlues of the SPs re between 0 nd 1. Indeed in Section 3.3, we implemented BP-ITRM when the rting vlues re from the set f1;...; 5g insted of binry vlues nd illustrted the performnce of the proposed scheme. 1 To the dvntge of mlicious rters, we ssumed tht totl of T time-slots hd pssed since the initiliztion of the system nd frction of the existing rters chnge behvior nd become mlicious fter T time-slots. In other words, mlicious rters behved lie relible rters before mounting their ttcs t the ðt þ 1Þth time-slot. Finlly, we ssumed tht d is rndom vrible with Yule- Simon distribution, which resembles the power-lw distribution used in modeling online systems [41], with the 1. The performnce of BP-ITRM in this nonbinry rting system (in which the rting vlues re from the set f1;...; 5g) lso illustrtes its performnce when the ctul reputtion vlues of the SPs re between 0 nd 1 in the binry rting system. For exmple, reputtion vlue of 4 in the nonbinry rting system stnds for reputtion vlue of 0.8 in the binry rting system.

7 AYDAY AND FEKRI: ITERATIVE TRUST AND REPUTATION MANAGEMENT USING BELIEF PROPAGATION 381 probbility mss function f d ðd; Þ ¼Bðd; þ 1Þ, where B is the Bet function. For modeling the dversry, we mde the following ssumptions. We ssumed tht the mlicious rters initite bd mouthing nd collude while ttcing the SPs (they ttc the SPs who hve the highest reputtion vlues by rting them s r m ¼ 0). Further, the mlicious rters ttc the sme set of SPs t ech time-slot. In other words, we denote by the set of size b in which every victim SP hs one edge from ech of the mlicious rters. We note tht the results we provide in this section re bsed on the thret model described bove. We wish to evlute the performnce for the time-slot ðt þ 1Þ. It is worth noting tht even though we discuss the detils for bd-mouthing ttc, similr counterprt results hold for bllot stuffing nd combintions of bd mouthing nd bllot stuffing s well. -Optiml scheme. The performnce of reputtion scheme is determined by its ccurcy of estimting the globl reputtions of the SPs. We declre reputtion scheme to be -optiml if the men bsolute error (MAE) (G ^G ) is less thn or equl to for every SP. This introduces clss of optiml schemes. Nturlly, we need to nswer the following question: for fixed, wht re the conditions to hve n -optiml scheme? In order to nswer this question we require two conditions to be stisfied: 1) the scheme should itertively reduce the impct of mlicious rters nd decrese the error in the reputtion vlues of the SPs until it converges, nd 2) the error on the G vlue of ech SP should be less thn or equl to once the scheme converges. In the following, we obtined the condition to rrive t the -optiml scheme. Although the discussions of the nlysis re bsed on bdmouthing ttc, the system designed using these criteri will be robust ginst bllot stuffing nd combintions of bd mouthing nd bllot stuffing s well. The bd-mouthing ttc is imed to reduce the globl reputtion vlues of the victim SPs. Hence, G vlue of victim SP should be nondecresing function of itertions. This leds to the first condition on the -optiml scheme. Lemm 1 (Condition 1). The error in the reputtion vlues of the SPs decreses with ech successive itertions (until convergence) if G ð2þ >G ð1þ is stisfied with high probbility for every SP ( 2 S) with ^G ¼ 1. 2 Proof. Let G ð!þ nd G ð!þ1þ be the reputtion vlue of n rbitrry SP with ^G ¼ 1 clculted t the ð!þth nd >G ð!þ ð! þ 1Þth itertions, respectively. G ð!þ1þ following is stisfied t the ð! þ 1Þth itertion: Y 2 U R \N > Y ðwþ1þ 2p c R þ 1 R ðwþ1þ 2p c R ðwþ1þ 2 U R \N þ 1 þ R ðwþ1þ ðwþ 2p c R þ 1 R ðwþ 2p c R ðwþ þ 1 þ R ðwþ ðwþ1þ Y 1 ^R 2 U M \N 1 þ ^R ðwþ1þ ðwþ Y 1 ^R ; 2 U M \N 1 þ ^R ðwþ if the where R ðwþ nd ^R ðwþ re the trustworthiness vlues of relible nd mlicious rter clculted s in (5) t the wth itertion, respectively. 2. The opposite must hold for ny SP with ^G ¼ 0. ð8þ Given G ð!þ >G ð!1þ holds t the!th itertion, we > ^R ðwþ1þ for 2 U M \ N nd R ðwþ1þ for 2 U R \ N. Thus, (8) would hold for the ðw þ 1Þth itertion. On the other hnd, if G ð!þ <G ð!1þ, would get ^R ðwþ R ðwþ we get ^R ðwþ < ^R ðwþ1þ for 2 U M \ N nd R ðwþ1þ <R ðwþ for 2 U R \ N. Hence, (8) is not stisfied t the ðw þ 1Þth itertion. Therefore, if G ð!þ >G ð!1þ holds for some itertion!, then the BP-ITRM lgorithm reduces the error on the globl reputtion vlue (G ) until the itertions stop, nd hence, it is sufficient to stisfy G ð2þ >G ð1þ with high probbility for every SP with ^G ¼ 1 (the set of SPs from which the victims re ten) to gurntee tht BP- ITRM itertively reduces the impct of mlicious rters until it stops. tu As we described in Section 2, itertions of BP-ITRM stop when the G vlues converge for every SP (i.e., do not chnge nymore). The following lemm shows tht BP-ITRM converges to unique solution given Condition 1 is stisfied. Lemm 2. Given Condition 1 holds, G vlue of SP converges to unique solution (G ). Proof. From Lemm 1, BP-ITRM itertively reduces the error in the reputtion vlues of the SPs provided tht Condition 1 is stisfied. Further, given Condition 1 is stisfied, the error in the reputtion vlue of n rbitrry SP stops decresing t the th itertion when G ¼ G ðþ1þ, where the vlue of depends on the frction of mlicious rters. Thus, given tht BP-ITRM stisfies Condition 1, the reputtion vlue of every SP converges to unique vlue. tu Although becuse of the Condition 1, the error in the reputtion vlues of the SPs decrese with successive itertions, it is uncler wht would be the eventul impct of mlicious rters. Hence, in the following, we derive the probbility P for -optimlity. Lemm 3 (Condition 2). Suppose tht the Condition 1 is met. Let be the itertion t which the lgorithm hs converged. Then, BP-ITRM would be n -optiml scheme with probbility P, where P is given in (9) s follows: P ¼ Y Pr 1 2 S Y 2pc R ðþ1þ 2 U R \N Y 2pc R ðþ1þ 2 U R \N þ Y 2pc R ðþ1þ 2 U R \N Y : 1 þ ^R ðþ1þ 2 U M\N þ 1 R ðþ1þ þ 1 R ðþ1þ þ 1 þ R ðþ1þ Y 1 ^R ðþ1þ 2 U M \N Y ð1 ^R ðþ1þ 2 U M \N Þ Proof. Given Condition 1 is stisfied, G vlue of n rbitrry SP (with ^G ¼ 1) increses with itertions. Let BP-ITRM converges t the th itertion. Then, to hve n -optiml scheme, G vlue clculted t the lst itertion of ð9þ

8 382 IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE 2012 Fig. 5. Probbility of BP-ITRM to stisfy Condition 1 versus frction of mlicious rters. BP-ITRM (G ) should result in n error less thn or equl to for every SP. Tht is, the following should hold for every SP 1 G : ð10þ Further, if the scheme continues one more itertion fter convergence, it cn be shown tht G ðþ1þ ¼ G : ð11þ Thus, combining (10) nd (11) leds to (9). We note tht Conditions 1 nd 2 in Lemms 1 nd 3 re to give n insight bout the performnce of the lgorithm prior to the implementtion. Hence, these conditions do not need to be checed t ech execution of BP-ITRM in the rel-life implementtion of the lgorithm. Finlly, the vrition of the probbility of BP-ITRM being n -optiml scheme over time is n importnt fctor ffecting the performnce of the scheme. We observed tht given BP-ITRM stisfies Condition 1 (tht the error in the reputtion vlues of the SPs monotoniclly decreses with itertions), the probbility of BP-ITRM being n -optiml scheme increses with time. This criteri is given by the following lemm: Lemm 4. Let P Tþ1 nd P Tþ2 be the probbilities tht BP-ITRM is -optiml t the ðt þ 1Þth nd ðt þ 2Þth time-slots, respectively. Then, given Condition 1 holds t the ðt þ 1Þth time-slot, we hve P T þ2 >P T þ1. Proof. Due to the fding fctor, the contributions of the pst relible rtings of the mlicious rters to their R i vlues become less dominnt with incresing time. Let R i ðtþ nd ^R i ðtþ be the trustworthiness of relible nd mlicious rter t the T th time-slot, respectively. Then, given tht Condition 1 is stisfied t the ðt þ 1Þth time-slot, it cn be shown tht R i ðt þ 1Þ R i ðtþ nd ^R i ðt þ 1Þ < ^R i ðtþ. Thus, the probbility tht BP-ITRM stisfies Condition 1 increses t the ðt þ 2Þth time-slot. tu In the following exmple, we illustrte the results of our nlyticl evlution. The prmeters we used re U M þu R ¼100, S ¼100, ¼ 1, # ¼ 0:9, T ¼ 50, b ¼ 5 nd p c ¼ 0:8. We note tht there is no motive to select these tu Fig. 6. Probbility tht BP-ITRM is n -optiml scheme versus frction of mlicious rters for different vlues. prmeters. We evluted BP-ITRM with different prmeters nd obtined similr results. BP-ITRM wors properly when the error in the reputtion vlues of the SPs decreses monotoniclly with itertions until convergence. In other words, Condition 1 (in Lemm 1) is fundmentl requirement. In Fig. 5, we illustrted the probbility of BP- ITRM to stisfy Condition 1 versus frction of mlicious rters. We observed tht BP-ITRM stisfies Condition 1 with high probbility for up to 30 percent mlicious rters. Further, we observed threshold phenomenon. Tht is, the probbility of BP-ITRM to stisfy Condition 1 suddenly drops fter exceeding prticulr frction of mlicious rters. Next, in Fig. 6, we illustrted the probbility of BP-ITRM being n -optiml scheme versus frction of mlicious rters for three different vlues. Agin, we observed threshold phenomenon. As the frction of dversry exceeds certin vlue, the probbility of BP-ITRM being n -optiml scheme drops shrply. Moreover, Fig. 7 illustrtes the verge vlues ( v ) for which BP-ITRM is n -optiml scheme with high probbility for different frctions of mlicious rters. We observed tht BP-ITRM provides significntly smll error vlues for up to 30 percent mlicious rters. We note tht these nlyticl results re lso consistent with our simultion results tht re illustrted in the next section. Fig. 7. The verge vlues for which BP-ITRM is n -optiml scheme with high probbility versus frction of mlicious rters.

9 AYDAY AND FEKRI: ITERATIVE TRUST AND REPUTATION MANAGEMENT USING BELIEF PROPAGATION Simultions We evluted the performnce of BP-ITRM in the presence of bd mouthing, bllot stuffing, nd combintions of bd mouthing nd bllot stuffing. Here, we provide n evlution of the bd-mouthing ttc only, s similr results hold for bllot stuffing nd combintions of bd mouthing nd bllot stuffing. We compred the performnce of BP-ITRM with three well nown nd commonly used reputtion mngement schemes: 1) The Averging Scheme, 2) Byesin Approch, nd 3) Cluster Filtering. The Averging Scheme is widely used s in eby or Amzon. The Byesin Approch [27], [33] updtes G using Bet distribution. We implemented the Buchegger s Byesin pproch in [27] for the comprison with the devition threshold d ¼ 0:5 nd trustworthiness threshold t ¼ 0:75 3 (for detils refer to [27]). It is worth noting tht since we present evlute BP-ITRM in centrlized setting, Buchegger s wor in [27] nd Whitby s wor in [33] cn be considered s similr. In [27], if rter s rting devites beyond the devition threshold d from the clculted reputtion vlue, its trustworthiness vlue is modified ccordingly. Further, if rter s trustworthiness exceeds definite threshold t, it is detected s mlicious. Similrly, in [33], insted of using the devition threshold, the uthors chec if the clculted reputtion vlue for the SP flls between definite intervl for ech rter s rting distribution. As we will discuss lter, both [27] nd [33] hve the sme problem ginst colluding mlicious rters. Cluster Filtering [29], [34], on the other hnd, performs dissimilrity test mong the rters nd then updtes G using only the relible rters. Finlly, we compred BP-ITRM with our previous wor on itertive trust nd reputtion mngement [8] (referred to s ITRM) to show the benefit of using belief propgtion. We ssumed tht d is rndom vrible with Yule-Simon distribution (with ¼ 1 throughout the simultions) s discussed in Section 3.2. Further, the fding prmeter is set s # ¼ 0:9 4 nd number of rtings, per time-slot, by mlicious rter s b ¼ 5. Let ^G be the ctul vlue of the globl reputtion of SP. Then, we obtined the performnce of BP-ITRM, for ech time-slot, s the men bsolute error (MAE) G ^G, verged over ll the SPs tht re under ttc. We ssumed tht the mlicious rters collude nd ttc the SPs who hve the highest reputtion vlues (ssuming tht the ttcers nows the reputtion vlues) nd received the lowest number of rtings from the relible rters (ssuming tht the ttcers hve this informtion). We note tht this ssumption my not hold in prctice since the ctul vlues of the globl reputtions nd number of rtings received by ech SP my not be vilble to mlicious rters. However, we ssumed tht this informtion is vilble to the mlicious rters to consider the worst cse scenrio. Further, the mlicious rters collude nd ttc the sme set of SPs in ech time-slot (which represents the strongest ttc by the mlicious rters). We further ssumed tht there re U ¼100 rter peers nd S ¼100 SPs. Moreover, totl of T ¼ 50 time-slots hd pssed since the lunch of the system, nd relible reports 3. We note tht these re the sme prmeters used in the originl pper [27]. 4. We note tht for the Averging Scheme, Byesin Approch, nd Cluster Filtering we used the sme fding mechnism s BP-ITRM (discussed in Section 2) nd set the fding prmeter s # ¼ 0:9. Fig. 8. MAE performnce of BP-ITRM versus time when W of the existing rters become mlicious in RepTrp [42]. generted during those time-slots were distributed mong the SPs uniformly. We note tht we strt our observtions t time slot 1 fter the initiliztion period. Initilly, we ssumed tht frction of the existing rters chnge behvior nd become mlicious fter the strt of the system (t time-slot one). The rting vlues re either 0 or 1. Using ll their edges, the mlicious rters collude nd ttc the SPs who hve the highest reputtion vlues nd received the lowest number of rtings from the relible rters, by rting them s r m ¼ 0. We note tht this ttc scenrio lso represents the RepTrp ttc in [42] which is shown to be strong ttc. Since the rtings of the nonmlicious rters devite from the ctul reputtion vlues vi Bernoulli distribution, our ttc scenrio becomes even more severe thn the RepTrp [42]. Further, we ssumed tht the rting r h (provided by the nonmlicious rters) is rndom vrible with Bernoulli distribution, where Prðr h ¼ ^G Þ¼0:8 nd Prðr h 6¼ ^G Þ¼0:2. First, we evluted the MAE performnce of BP-ITRM for U M different frctions of mlicious rters (W ¼ U Mþ U ), t R different time-slots (mesured since the ttc is pplied) in Fig We observed tht the proposed BP-ITRM provides significntly low errors for up to W ¼ 30% mlicious rters. Moreover, MAE t the first time slot is consistent with our nlyticl evlution which ws illustrted in Fig. 7. Next, we observed the chnge in the verge trustworthiness (R i vlues) of mlicious rters with time. Figure 9 illustrtes the drop in the trustworthiness of the mlicious rters with time. We conclude tht the R i vlues of the mlicious rters decrese over time, nd hence, the impct of their mlicious rtings is totlly neutrlized over time. We further observed the verge number of required itertions of BP-ITRM t ech time-slot in Fig. 10. We conclude tht the verge number of itertions for BP-ITRM decreses with time nd decresing frction of mlicious rters. Finlly, we compred the MAE performnce of BP-ITRM with the other schemes. Figure 11 illustrtes the comprison of BP-ITRM with the other schemes for bd mouthing when the frction 5. The plots in Figs. 8, 9, 10, 11, 12, nd 13 re shown from the time-slot the dversry introduced its ttc.

10 384 IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE 2012 Fig. 9. Chnge in verge trustworthiness of mlicious rters versus time for BP-ITRM when W of the existing rters become mlicious in RepTrp [42]. Fig. 11. MAE performnce of vrious schemes when 30 percent of the existing rters become mlicious in RepTrp [42]. Fig. 10. The verge number of itertions versus time for BP-ITRM when W of the existing rters become mlicious in RepTrp [42]. of mlicious rters (W) is 30 percent. It is cler tht BP- ITRM outperforms ll the other techniques significntly. Next, we simulted the sme ttc scenrio when rtings re integers from the set f1;...; 5g insted of binry vlues. We ssumed tht the rting r h is rndom vrible with folded norml distribution (men ^G nd vrince 0.5), however, it tes only discrete vlues from 1 to 5. Mlicious rters choose SPs from nd rte them s r m ¼ 4. The mlicious rters do not devite very much from the ctul ^G ¼ 5 vlues to remin undercover (while still ttcing) s mny time-slots s possible. We lso tried higher devitions from the ^G vlue nd observed tht the mlicious rters were esily detected by BP-ITRM. Figure 12 illustrtes tht BP-ITRM provides significntly low MAE for up to W ¼ 40% mlicious rters. We then compred the MAE performnce of BP-ITRM with the other schemes in Fig. 13 nd observed tht BP-ITRM outperforms ll the other techniques significntly. In most trust nd reputtion mngement systems, the dversry cuses the most serious dmge by introducing newcomer rters to the system. Since it is not possible for the system to now the trustworthiness of the newcomer rters, the dversry my introduce newcomer rters to the systems nd ttc the SPs using those rters. To study the Fig. 12. MAE performnce of BP-ITRM versus time when W of the existing rters become mlicious nd rting vlues re integers from f1;...; 5g in RepTrp [42]. Fig. 13. MAE performnce of vrious schemes when 30 percent of the existing rters become mlicious nd rting vlues re from f1;...; 5g in RepTrp [42]. effect of newcomer mlicious rters to the reputtion mngement scheme, we introduced 100 more rters s newcomers. Hence, we hd U M þu R ¼200 rters nd S ¼100 SPs in totl. We ssumed tht the rting vlues re either 0 or 1, r h is rndom vrible with Bernoulli distribution s before, nd mlicious rters choose SPs from nd rte them s r m ¼ 0 (this prticulr ttc scenrio

11 AYDAY AND FEKRI: ITERATIVE TRUST AND REPUTATION MANAGEMENT USING BELIEF PROPAGATION 385 Fig. 14. MAE performnce of vrious schemes when 30 percent of the newcomer rters re mlicious. does not represent the RepTrp ttc). We compred the MAE performnce of BP-ITRM with the other schemes for this scenrio in Fig From these simultion results, we conclude tht BP- ITRM significntly outperforms the Averging Scheme, Byesin Approch, nd Cluster Filtering in the presence of ttcers. We identify tht the Byesin Approch performs the worst ginst the RepTrp ttc nd colluding ttcs from mlicious rters. Indeed, both [27] nd [33] hve the sme shortcoming ginst colluding mlicious rters. Both [27] nd [33] first clculte the reputtion vlue of prticulr SP, nd then bsed on the clculted vlue, they dust ech rter s trustworthiness vlue. On the other hnd, when the mlicious rters collude (s in our ttc scenrio), it is liely tht the mority of the rtings to the victim SPs will be from mlicious rters. In this scenrio, the Byesin pproch not only fils to filter the mlicious rtings but it lso punishes the relible rters which rtes the victim SPs. We lso identify tht ITRM (i.e., our lgebric itertive scheme) is the closest in ccurcy to BP- ITRM. This emphsizes the robustness of using itertive messge pssing lgorithms for reputtion mngement. 3.4 Computtionl Complexity In this section, we provide some discussion on the computtionl complexity. It cn be rgued tht the computtionl complexity of BP-ITRM is qudrtic with the number of rters (or SPs) due to the use of the probbility-domin messge pssing lgorithm. This is becuse of multiplictions of probbilities in (7) nd (4). However, this qudrtic computtionl complexity cn be further reduced by using similr techniques developed for messge pssing decoding of LDPC codes (using belief propgtion) for lower complexity. We used log-domin lgorithm in our implementtion, which is often used for LDPC codes [43] to reduce the complexity. Specificlly, ssuming U ¼u rters nd S ¼s SPs in the system, we obtined the computtionl complexity of BP-ITRM s mxðo cu ; O cs Þ in the number of multiplictions, where c is smll constnt number representing the verge number of rtings (reports) per rter. On the other hnd, Cluster Filtering suffers qudrtic complexity versus the number of rters (or SPs). 6. The plot is shown from the time-slot the newcomers re introduced. 4 CONCLUSION In this pper, we introduced the Belief Propgtion-bsed Itertive Trust nd Reputtion Mngement Scheme (BP- ITRM). Our wor is n itertive probbilistic lgorithm motivted by the prior success of messge pssing techniques nd belief propgtion lgorithms on decoding of turbo codes nd low-density prity-chec codes. BP-ITRM relies on grph-bsed representtion of n ppropritely chosen fctor grph for reputtion systems. In this representtion, service providers nd rters re rrnged s two sets of vrible nd fctor nodes tht re connected vi some edges. The reputtion vlues of SPs re computed by messge pssing between nodes in the grph until the convergence. The proposed BP-ITRM is robust mechnism to evlute the qulity of the service of the SPs from the rtings received from the recipients of the service (rters). Moreover, it effectively evlutes the trustworthiness of the rters. We studied BP-ITRM by detiled nlysis nd showed the robustness using computer simultions. We proved tht BP-ITRM itertively reduces the error in the reputtion vlues of SPs due to the mlicious rters with high probbility. Further, we observed tht this probbility demonstrtes threshold property. Tht is, exceeding prticulr frction of mlicious rters reduces the probbility shrply. We lso compred BP- ITRM with some well-nown reputtion mngement schemes nd showed the superiority of our scheme both in terms of robustness nd efficiency. 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Informtion Theory, July Ermn Aydy received the BS degree in electricl nd electronics engineering from the Middle Est Technicl University, Anr, Turey, in He received the MS nd PhD degrees from the School of Electricl nd Computer Engineering (ECE), Georgi Institute of Technology, Atlnt, Georgi, in 2007 nd 2011, respectively. His current reserch interests include wireless networ security, gme theory for wireless networs, trust nd reputtion mngement, nd recommender systems. He is the recipient of 2010 Outstnding Reserch Awrd from the Center of Signl nd Imge Processing (CSIP) t Georgi Tech nd 2011 ECE Grdute Reserch Assistnt (GRA) Excellence Awrd from Georgi Tech. He is student member of the IEEE. Frmrz Feri received the PhD degree from the Georgi Institute of Technology in Since 2000, he hs been with the fculty of the School of Electricl nd Computer Engineering t the Georgi Institute of Technology where he currently holds full professor position. He serves on the editoril bord of the IEEE Trnsctions on Communictions, nd on the Technicl Progrm Committees of severl IEEE conferences. His current reserch interests include the re of communictions nd signl processing, in prticulr coding nd informtion theory, informtion processing for wireless nd sensor networs, nd communiction security. He received the US Ntionl Science Foundtion CAREER Awrd (2001), nd Southern Center for Electricl Engineering Eduction (SCEEE) Reserch Initition Awrd (2003), Outstnding Young fculty Awrd of the School of ECE (2006). He is senior member of the IEEE.. For more informtion on this or ny other computing topic, plese visit our Digitl Librry t