Application of Belief Propagation to Trust and Reputation Management

Transcription

1 Appliction of Belief Propgtion to Trust nd Reputtion Mngement Ermn Aydy School of Electricl nd Comp. Eng. Georgi Institute of Technology Atlnt, GA 333, USA Emil: Frmrz Feri School of Electricl nd Comp. Eng. Georgi Institute of Technology Atlnt, GA 333, USA Emil: Abstrct This pper introduces the first ppliction of Belief Propgtion (BP) in reputtion systems. We view the reputtion mngement s n inference problem, nd hence, describe the reputtion mngement problem s computing mrginl lielihood distributions from complicted globl functions of mny vribles. However, we observe tht computing the mrginl probbility functions of the reputtion vribles is computtionlly prohibitive for lrge scle reputtion systems. Therefore, we propose to utilize the BP lgorithm to efficiently (i.e., in liner complexity) compute these mrginl probbility distributions; leding to fully itertive probbilistic nd BP-bsed pproch (referred to s BP-ITRM). BP-ITRM describes the reputtion system on fctor grph, using which we cn obtin qulittive representtion of how the service providers (sellers) nd consumers (buyers) re relted. Further, by using such grph representtion, we compute the mrginl probbility distribution functions of the vribles representing the globl reputtion vlues vi n itertive messge pssing lgorithm. We show tht BP-ITRM significntly outperforms the well-nown nd commonly used reputtion mngement schemes such s the Averging Scheme, Byesin Approch nd Cluster Filtering in the presence of ttcers. Further, its complexity is liner in the number of service providers nd consumers, fr exceeding the efficiency of other schemes. I. INTRODUCTION Trust nd Reputtion re crucil requirements for most environments wherein entities prticipte in vrious trnsctions nd protocols mong ech other. The recipient of the service often hs insufficient informtion bout the service qulity of the service provider before the trnsction. Hence, the service recipient should te prior ris before receiving the ctul service. This ris puts the recipient into n unprotected position since he hs no opportunity to try the service before he receives it. This problem gives rise to the use of reputtion systems in which reputtions re determined by rules tht evlute the evidence generted by the pst behvior of n entity within protocol. Hence, fter ech trnsction, prty who receives the service (referred to s the rter) provides (to the centrl uthority) its report bout the qulity of the service provided for tht trnsction. The centrl uthority collects the reports nd updtes the reputtions of the service providers. Reputtion systems hve found widespred doption in online communities, web services, d-hoc networs, PP computing, nd e-commerce. Reputtion mngement systems re subect to vrious mnipultions, lunched by the mlicious prticipnts. Thus, the success of reputtion scheme depends on the robustness of the mechnism to ccurtely evlute the service providers service qulities (i.e., reputtions) nd the trustworthiness of the rters bsed on their reports bout the service providers. Further, the emergence of lrge-scle online services nd networs clls for efficient nd sclble lgorithms to solve for the reputtion mngement problem. Hence, there is need to develop relible, sclble nd dependble reputtion schemes tht would lso be resilient to vrious wys reputtion system cn be ttced. In this wor, for the first time, we view the reputtion mngement problem s n inference problem nd describe the reputtion mngement problem s computing mrginl lielihood distributions from complicted globl functions of mny vribles. To solve this problem whose complexity grows exponentilly, we resort to use Belief Propgtion (BP) [1] whose computtionl efficiency (i.e., liner in the number of service providers or consumers) is driven by exploring the wy in which the globl functions fctors into product of simpler locl functions. Thus, we introduce the Belief Propgtion bsed Itertive Trust nd Reputtion Mngement scheme (BP-ITRM). The wor is inspired by erlier wor on grphbsed itertive probbilistic decoding of low-density pritychec codes [], the most powerful prcticlly decodble errorcontrol codes nown. We believe tht the significnt benefits offered by the BP lgorithms cn be tpped in to benefit the field of reputtion systems. In BP-ITRM, the sellers (i.e., service providers) nd buyers (i.e., consumers or rters) re represented vi fctor grph on which they re rrnged s two sets of vrible nd fctor nodes tht re connected vi some edges. The reputtion vlues of the service providers cn be computed by messge pssing between nodes in the grph. In ech itertion of the lgorithm, ll the vrible vertices (service providers) nd subsequently ll the fctor vertices (the rters) pss new messges to their neighbors until convergence. We show tht the proposed itertive scheme is relible (in filtering out mlicious/unrelible rtings) while being computtionlly efficient (i.e., liner in the number of vribles). Thus, it cn be used s n effective nd sclble reputtion system in mny pplictions such s online services. 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-ITRM in detil. Next, in Section III, we evlute BP-ITRM vi computer simultions nd compre BP-ITRM with the existing nd commonly used reputtion mngement schemes. Further, we nlyze BP-ITRM using mthemticl model for the users. Finlly, in Section IV, we conclude our pper. A. Relted Wor Severl wors in the literture hve focused on building reputtion-mngement mechnisms [3]. The most fmous

2 nd primitive reputtion system is the one tht is used in eby. Other well-nown web sites such s Amzon, Epinions, nd AllExperts use more dvnced reputtion mechnism thn eby. Their reputtion mechnisms mostly compute the weighted verge of the rtings received for product (or peer) to evlute the reputtion of product (or peer). Hence, these schemes re vulnerble to collbortive ttcs by mlicious peers. Use of the Byesin Approch is lso proposed in [4]. Finlly, [5] proposed to use the Cluster Filtering method to distinguish between the relible nd unrelible rters. As we will illustrte vi computer simultions, ll existing methods re vulnerble to sophisticted ttcs such s RepTrp [6] since none of these schemes re designed considering the noise nd the incomplete informtion in the system. Inspired by the erlier wor on itertive decoding of error-control codes in the presence of stopping sets, we developed n lgebric itertive lgorithm for reputtion mngement [7] nd dversry detection [8]. Here, we propose new scheme bsed on the ey observtion tht the reputtion mngement problem cn be formulted s solving for mrginl lielihood functions of mny vribles by using the BP lgorithm. As we will illustrte, comprison with the existing schemes suggests tht the proposed BP-ITRM hs superior performnce (i.e., ccurcy, sclbility nd robustness ginst ttcs). II. BELIEF PROPAGATION BASED ITERATIVE TRUST AND REPUTATION MANAGEMENT In the reputtion mngement problem, we wish to me sttisticl inference bout the reputtions of service providers bsed on pst observtions. Tht is, given the pst dt evidence, wht is the lielihood (probbility) of the reputtion vlues being good or bd? Here, the interprettion of probbility is Byesin one. We formulte this problem s finding the mrginl probbility distributions problem. Unfortuntely, computing such probbility distributions is computtionlly prohibitive for lrge-scle systems. However, the proposed lgorithm (referred to s BP-ITRM) shows tht this problem cn be solved, fortuntely, by pplying the Belief Propgtion (BP) lgorithm (in liner complexity). We ssume two different sets in the reputtion system: i) the set of service providers, S, nd ii) the set of service consumers (rters), U. We note tht these two sets re not necessrily disoint. Trnsctions occur between service providers nd consumers, nd consumers (rters) provide feedbcs in the form of rtings bout service providers fter ech trnsction. 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. determining the trustworthiness of the rters by nlyzing their feedbc bout SPs. Let G be the reputtion vlue of the SP ( S) nd T i be the rting tht the rter i (i U) reports bout the SP ( S), whenever trnsction is completed between the two peers. Moreover, let R i denote the trustworthiness of the peer i (i 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 the rter i. We ssume there re u rters nd s SPs in the system (i.e., U = u nd S = s). We let G = {G : S} nd R = {R i : i U} 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 the rter i. 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 SP. 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 rters. Then, clerly, ech mrginl probbility function p(g T,R) my be obtined s follows 1 : p(g T,R) = p(g T,R). (1) G\{G } Unfortuntely, the number of terms in (1) grows exponentilly with the number of vribles, ming it infesible for lrge-scle systems. However, we propose to fctorize (1) to locl functions f i using fctor grph nd utilize the BP 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 [9], 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. T m b n c T nc Fig. 1: Fctor grph between the SPs nd the rters. Therefore, 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 U) hs report bout the SP ( S), we plce n edge with vlue T i from the fctor node i to the vrible node representing the SP. Next, we ssume 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 1 The nottion G\{G } implies ll vribles in G except G.

3 given the trustworthiness vlue nd the existing rtings of the ssocited rter. We now introduce the messges between the fctor nd the vrible nodes to compute the mrginl distributions using BP. To tht end, we choose n rbitrry fctor grph s in Fig. 1 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. Further, let Ξ = N \{} nd = N \{}. The BP lgorithm itertively exchnges the probbilistic messges between the fctor nd the vrible nodes in Fig. 1, 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 (ν) = {G (ν) : S} be the collection of vribles representing the vlues of the vrible nodes t the itertion ν 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, (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 (G(ν) ) = f (G,T,R (ν 1) ) µ (ν 1) x (G(ν 1) x ), G (ν 1) \{G (ν 1) } x (3) where G is the set of vrible nodes which re rguments of the locl function f t the fctor node. Further, R (ν 1) (the trustworthiness of rter 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 i N x {,1} T i x µ (ν 1) i (x). (4) The bove eqution cn be interpreted s one minus the verge inconsistency of rter clculted using the messges it received from its neighbors. Using (3), it cn be shown tht (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) )T + 1 R(ν 1) ] (1 T ) G (ν) R(ν 1) ] )(1 T ) (1 G (ν) ). (5) This resembles the belief/pleusbility concept of the Dempster- Shfer Theory [1]. Given T = 1, R (ν 1) cn be considered s the belief of the th rter tht the G (ν) vlue is one (t the ν th itertion). In other words, in the eyes of rter, the G (ν) vlue is equl to one with probbility R (ν 1). Thus, (1 R (ν 1) ) corresponds to the uncertinty in the belief of rter. In order to remove this uncertinty nd express p(g (ν) T,R (ν 1) ) s the probbilities tht G (ν) is zero nd Neighbors of SP re the set of rters who rted the SP while neighbors of rter re the SPs whom it rted. one, we distribute the uncertinty uniformly between two outcomes (one nd zero). Hence, in the eyes of the th rter, G (ν) vlue 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 =. 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 of the ν th itertion, the vrible nodes 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. Hence, the messge from vrible node to the fctor node t the ν th itertion is given by µ (ν) (G(ν) ) = 1 i (h) i Ξ h {,1} i Ξ i (G(ν) ) (6) 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. 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-ITRM is used s initil vlues in (5). 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 (ν), we first compute µ (ν) (G (ν) then we set G (ν) = 1 i= iµ(ν) ) using (6) but replcing Ξ with N, nd (i). III. EVALUATION OF BP-ITRM Here, we wish to evlute the proposed BP-ITRM using mthemticl model for the users. We consider slotted time throughout this discussion. For ech time-slot (or epoch), the itertive reputtion lgorithm is executed using the input prmeters R nd T to output the reputtion vlues fter convergence. We ssumed tht the rting vlues re either or 1. We let Ĝ denote the ctul vlue of the reputtion of the SP ( S), where Ĝ {,1} (1 represents good service qulity). Further, the qulity of ech service provider remins unchnged during time-slots. Rtings generted by the non-mlicious rters re uniformly distributed mong the SPs (i.e., their rtings/edges in the grph representtion re distributed uniformly mong SPs). We lso ssume the rting r h (provided by non-mlicious rter) is rndom vrible with Bernoulli distribution, where Pr(r h = Ĝ ) = p c nd Pr(r h Ĝ) = (1 p c). To fcilitte future references, frequently used nottions re listed in Tble I. U M : The set of mlicious rters U R: The set of non-mlicious rters r h : Report (rting) given by non-mlicious rter r m: Report (rting) given by mlicious rter d: Totl number of newly generted rtings, per time-slot, per non-mlicious rter b: Totl number of newly generted rtings, per time-slot, per mlicious rter TABLE I: Nottions nd definitions. A. Mlicious User Model We consider two mor ttcs tht re common for ny reputtion mngement mechnisms: Bd mouthing: Mlicious rters collude nd ttc the service providers with the highest reputtion by giving low rtings

4 MAE ~ W=15% W=1% W=35% W=% W=4% W=3% W=5% Fig. : MAE performnce of BP-ITRM versus time when W of the existing rters re mlicious in RepTrp [6]. Avg. trustworthiness of mlicious rters.7.6 W=1% W=% W=3% Fig. 3: Averge trustworthiness of mlicious rters versus time for BP-ITRM when W of the existing rters re mlicious in RepTrp [6]. MAE BP ITRM ITRM Cluster Filtering Averging Scheme Byesin Approch Fig. 4: MAE performnce of vrious schemes when 3% of the existing rters become mlicious in RepTrp [6]. 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 SPs in the hope of improving their own chnces s providers. Bllot stuffing: Mlicious rters collude to increse the reputtion vlues of peers with low reputtions. In some pplictions, this could be mounted by group of selfish consumers ttempting to fvor their llies. We me the following ssumptions for modeling the dversry which is lso nown s the RepTrp ttc [6]. RepTrp is believed to be strong ttc ginst the reputtion mngement system. We ssumed tht the mlicious rters initite bd mouthing 3. Further, ll the mlicious rters collude nd ttc the sme subset Γ of SPs in ech time-slot (which represents the strongest ttc), by rting those SPs s r m =. In other words, we denote by Γ the set of size b in which every victim SP hs one edge from ech of the mlicious rters. The subset Γ is chosen to include those SPs who hve the highest reputtion vlues but received the lowest number of rtings from the non-mlicious rters (ssuming tht the ttcers hve this informtion). 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 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). B. Simultions We compred the performnce of BP-ITRM with three wellnown nd commonly used reputtion mngement schemes: 1) The Averging Scheme (which is widely used in eby or Amzon), ) Byesin Approch [4], nd 3) Cluster Filtering [5]. Further, we compred BP-ITRM with our previous method [7] (referred to s ITRM). We ssumed tht d (in Tble I) is rndom vrible with Yule-Simon distribution, which resembles the powerlw distribution used in modeling online systems, with the probbility mss function f d (d;ρ) = ρb(d,ρ+1), where B is the Bet function. Further, we set T = 5, b = 5, p c =.8, ρ = 1, U = 1 nd S = 1. We ssumed the dversry model in Section III-A. In the following, we mesured the 3 It is worth nothing tht even though we use the bd-mouthing ttc, similr counterprt results hold for bllot stuffing nd combintions of bd mouthing nd bllot stuffing. performnce of BP-ITRM, for ech time-slot, s the men bsolute error (MAE) G Ĝ, verged over ll the SPs tht re under ttc. The time-slots in ll the plots re shown fter subtrcting the offset-time T = 5. First, we evluted the MAE performnce of BP-ITRM U M U M+U R ), for different frctions of mlicious rters (W = t different time-slots in Fig.. We observed tht BP-ITRM provides significntly low errors for up to W = 4% mlicious rters. Next, in Fig. 3, we show the chnge in the verge trustworthiness (R i vlues) of mlicious rters with time. We conclude tht the trustworthiness vlues of the mlicious rters decrese over time, nd hence, the impct of the mlicious rtings vnishes over time. Further, Fig. 4 illustrtes the comprison of BP-ITRM with the other well-nown nd commonly used reputtion systems for bd mouthing when the frction of mlicious rters (W ) is 3%. It is cler tht BP-ITRM outperforms ll the other techniques significntly. In ll these simultions, the verge number of itertions for BP-ITRM is round 1 nd it decreses with time nd with decresing frction of mlicious rters. In most reputtion 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 effect of newcomer mlicious rters to the scheme, we introduced 1 more rters s newcomers. Hence, we hd U = rters nd S = 1 SPs in totl. We ssumed tht the rting vlues re either or 1, r h is rndom vrible with Bernoulli distribution s before, nd mlicious rters choose SPs from Γ nd rte them s r m = (this prticulr ttc scenrio does not represent the RepTrp ttc). In Fig. 5 we observed tht BP- ITRM still provides MAE BP ITRM ITRM Cluster Filtering Averging Scheme Byesin Approch significntly low Fig. 5: MAE performnce of vrious schemes when3% of the newcomer rters re mlicious. MAE nd preserves its superiority over the other schemes. From these simultion results, we conclude tht BP-ITRM significntly outperforms the Averging Scheme, Byesin

5 Approch nd Cluster Filtering in the presence of ttcers. We identify tht our non-probbilistic itertive scheme ITRM [7] is the closest in performnce to BP-ITRM. This emphsizes the robustness of using itertive lgorithms for reputtion mngement. Finlly, ssuming u rters nd s SPs, we obtined the computtionl complexity of BP-ITRM s mx(o ( cu ),O ( cs ) ) in the number of multiplictions, where c is smll number representing the verge number of rting edges per rter. In contrst, Cluster Filtering suffers qudrtic complexity versus number of rters (or SPs). C. The ǫ-optiml Scheme Using the dopted models for vrious peers, it is nturl to s if BP-ITRM mintin ny optimlity in ny sense. We declre reputtion scheme to be ǫ-optiml if the men bsolute error (MAE) ( G Ĝ ) is less thn or equl to ǫ for every SP. Thus, for fixed ǫ, we wish to obtin the conditions for n ǫ-optiml scheme. It cn be shown tht, equivlently, we require BP-ITRM to itertively reduce the impct of mlicious rters nd decrese the error in the reputtion vlues of the SPs below ǫ until it converges. Lemm 1: (Sufficient Condition): The error in the reputtion vlues of the SPs decreses with ech successive itertions (until convergence) if G () > G (1) is stisfied with high probbility for every SP ( S) with Gˆ = 1 4. Proof: Let G (ω) nd G (ω+1) be the reputtion vlue of n rbitrry SP with G ˆ = 1 clculted t the (ω) th nd (ω + 1) th itertions, respectively. G (ω+1) stisfied t the (ω +1) th itertion. > G (ω) p cr (w+1) +1 R (w+1) 1 U R N p cr (w+1) +1+R (w+1) U M N 1+ p cr (w) +1 R (w) 1 ˆR (w) U R N p cr (w) +1+R (w) (w) U M N 1+ ˆR if the following is ˆR (w+1) ˆR (w+1) >, (7) where R (w) (w) nd ˆR re the trustworthiness vlues of relible nd mlicious rter clculted 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 U M N nd R (w+1) R (w) for U R N. Thus, (7) would hold for the (w+1) th itertion. On the other hnd, if G (ω) < G (ω 1) (w) (w+1), we get ˆR < ˆR for U M N nd R (w+1) < R (w) for U R N. Hence, (7) is not stisfied t the (w + 1) th itertion. Therefore, if G (ω) > G (ω 1) holds for some itertion ω, 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 () > G (1) with high probbility for every SP with Ĝ = 1 to gurntee tht BP-ITRM itertively reduces the impct of mlicious rters until it stops. Once the sufficient condition is met, the probbility for ǫ-optimlity cn be obtined 5. Now, we illustrte performnce of our scheme in terms of ǫ for which BP- ITRM is n ǫ-optiml scheme bsed on our nlyticl results. As before, we ssumed tht d is rndom vrible with Yule-Simon distribution (with ρ = 1). 4 The opposite must hold for ny SP with ˆ G =. 5 We omitted the expression due to pge limit. The prmeters we 1 used re U M + U R = 1, S = 1 1, ρ = 1, T = 5, b = 5 nd p c = In Fig. 6, we illustrte the verge 1 6 ǫ (ǫ v) for which BP-ITRM is n ǫ- 1 8 optiml scheme with high probbility for % mlicious rters different frctions of Fig. 6: The verge ǫ vlues for which BPmlicious rters. We ITRM is n ǫ-optiml scheme with high probbility versus frction of mlicious rters. observed tht BP- ITRM provides significntly smll error vlues for up to 3% mlicious rters. ε v IV. CONCLUSION In this pper, we introduced the first ppliction of the Belief Propgtion lgorithm to solve for the inference problem rising in reputtion systems. We presented the Belief Propgtion bsed Itertive Trust nd Reputtion Mngement Scheme (BP-ITRM). BP-ITRM is grph-bsed reputtion mngement system in which service providers nd rters re rrnged s two sets of vrible nd fctor nodes nd the reputtion vlues of service providers re computed by messge pssing between these nodes in the grph until the convergence. The proposed BP-ITRM is robust mechnism to evlute the qulity of the service of the service providers from the rtings received from the rters. Moreover, it effectively evlutes the trustworthiness of the rters. We showed tht the complexity of the proposed scheme grows only linerly with the number of service providers (or rters) in the system. Further, we studied BP-ITRM in detiled nlysis nd showed the robustness using computer simultions. We lso compred BP- ITRM with some well-nown reputtion mngement schemes nd showed the superiority of our scheme both in terms of robustness ginst vrious ttcs nd efficiency. REFERENCES [1] J. Perl, Probbilistic Resoning in Intelligent Systems: Networs of Plusible Inference. 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