Matrix Model of Trust Management in P2P Networks

Transcription

1 Matrix Model of Trust Maagemet i P2P Networks Miroslav Novotý, Filip Zavoral Faculty of Mathematics ad Physics Charles Uiversity Prague, Czech Republic miroslav.ovoty@mff.cui.cz Abstract The trust maagemet i P2P etworks allows to establish the trust relatioships amog the peers ad decide whether the cosidered trasactio will be realized. It tries to esure safety i the extremely usafe eviromets of P2P etworks. I this paper we focus o the formal descriptio of requiremets of trusted P2P etworks ad we propose the model which is able to measure them. Our model should be adaptable to the previously published methods ad allows to compare them. The fudametals of our model are trust matrices which record the trust relatioships betwee peers. We defie similarity betwee trust matrices for purpose of compariso of differet maagemet systems. O the basis of this similarity we propose the criterio which is able to measure successfuless of trust maagemet. Trust maagemet; P2P Networks; Reputatio; Access cotrol; I. INTRODUCTION Trustworthiess i P2P etwork is a research area that is icreasigly receivig attetio of researchers. I cotrast to a cliet-server model, the umber of peers i the etwork is rapidly chagig ad peers ofte cooperate with other peers, which they ever cooperate with them before. The trust maagemet i P2P etworks allows to establish the trust relatioships amog the peers ad decide whether the cosidered trasactio will be realized. It tries to esure safety i the extremely usafe eviromets. The malicious peers i the P2P etwork have various possibilities to subvert the system due to etwork's ope ad aoymous ature. May papers attemptig to solve buildig trusted P2P etworks have bee published i recet years with varyig degrees of success. Reputatio based methods have prove to be the most effective [1]. Nevertheless, there is a umber of differet approaches which adapt reputatio maagemet to differet P2P eviromets or threat scearios [2]. Previous researchers use the simulatio to prove the potetial beefit of their proposal. The simulatios are orieted towards the ratio of usuccessful trasactio to successful oes. I the most cases, the commuicatio overhead or respose time are also take ito accout. But the idividual simulatio models are differet ad it is ot possible to use their output for a geeral compariso of reputatio maagemet. It is difficult to compare idividual methods because there are o fixed criteria which are able to measure the efficiecy of the reputatio maagemet. We have oly a vague otio of 'trusted P2P etwork'. The reputatio maagemet methods have bee built uder the premise that this otio is well uderstood. For the compariso of differet methods, we eed to determie the criteria of the trusted P2P etwork ad be able to measure them. I this paper we focus o the formal descriptio of requiremets of trusted P2P etworks ad we propose the model for measurig them. Our model should be adaptable to the previously published methods ad allows to compare them. We do ot moitor the ratio of usuccessful to successful trasactio, but we focus o the compariso of a trust value obtaied from reputatio maagemet with true peer's behavior. The paper is orgaized as follows. Sectio 2 briefly overviews the related work. I sectio 3 we preset our model ad itroduce a simple example i sectio 4. Sectio 5 aalyses the results which come from our model. Ad sectio 6 cocludes the paper ad discusses future work. II. RELATED WORK The problem of belief i the P2P etworks has bee itesively studied. The situatio whe oe peer must cooperate with a totally ukow parter imposes may challeges ad ot all have bee solved yet. Sharig experiece amog peers is a coveiet way to esure at least some trustworthiess amog the peers. This system is called reputatio-based trust maagemet. The opposite approach is policy-based trust maagemet which relies o mechaisms such as siged certificates ad trusted certificatio authorities. Ufortuately, such mechaisms is ot applicable i all P2P etwork hece the reputatio-based approach is widely used. Wheever we will talk about trust maagemet i this paper we will have reputatio-based trust maagemet i mid. May reputatio systems with various ideas ad targeted issues have bee published i recet years. Some of them use some kid of cetral etity, like the System TrustMe [3] which eed a bootstrap server or [4] usig a cetral reputatio computatio aget. O the other had, most reputatio systems calculate reputatio i a fully distributed maer [5], [6], [7] ad [8]. There are a wide area of approaches used i reputatio systems, some of them use algorithm similar to pagerak [5],

2 [9] others use fuzzy logic [10], [11], [12], game-theoretic model [13], [14] or ecological etwork [15]. This work does ot focus o a particular trust maagemet system or a particular aproach, but tries to create a model which allows to compare differet systems. Some attempts to compare them have bee made i previous work. Trust maagemet systems i the P2P etworks based o Distributed Hash Tables are aalyzed i [16]. The criteria used for the aalysis are related to a techical backgroud, like Possibility to weight recommedatios or Resposibility for the behavior of recommeded etities. The taxoomy which orgaized existig ideas ad threats i peer-to-peer trust maagemet is preseted i [17] ad the latest results i the area of trusted P2P etworks are criticized i [18]. But objective criteria to compare trust maagemet systems still missig. Curretly, we are ot able to compare differet trust maagemet systems accordig the criteria which express success i the trust maagemet tasks. The attempts to formalize trust ad reasoig about trust have bee made i a vector model [19]. This model use three parameters that cotribute towards defiig trust relatioships: experiece, kowledge ad recommedatio. It also defies a operatio o a trust vector which allows to determie the relative trustworthiess of two trustees. This model is exteded i [20] for esurig privacy. The other approach [21] tries to create a automated verificatio framework for security protocol based o a trust maagemet. It uses the same machiery applied for the formal verificatio of a security protocol. Aother formalism to model the trust gaiig process is based o the temporal logic [22] or the subjective logic [23]. III. MODEL DESCRIPTION A. Basic Defiitio For purpose of the reputatio maagemet we will cosider the P2P etwork as a commuity of odes which provide services for each other. The quality of provided services o various peers ca be diametrically differet ad also the quality of differet services o oe peer ca be differet. Our model must be able to describe differet aspects of peer behavior. First, we eed to have a cotext which the reputatio maagemet lives i. Defiitio 1: Let be the list of all services provided i the P2P etwork. We ca refer to a sigle service as i, where i desigates its positio i the list. Defiitio 2: Let A be the list of all aspects of service behavior which ca be moitored i the P2P etwork. We ca refer to a sigle aspect as a j, where j desigates its positio i the list A. Defiitio 3: The cotext C determies the services ad their aspects which are moitored i P2P etwork. It is a list of pairs ( i,a j), where i ad a j A. We ca refer to a sigle pair as c l, where l desigates its positio i the list C. The idea of services ad aspects gives us a flexible way to deal with the complexity of peer behavior. We are able to disassemble peer behavior to idividual properties ad trace trust for each of them separately. For example, the services ca be share computatio capacity, share files or provide some kid of iformatio. The aspects ca describe peer behavior as a service provider, as a service cosumer or as a feedback source. For example, the provider's aspects ca be: task is computed correctly, task is computed i time, shared files are ot ifected by viruses, ad provided iformatio are truth. The cosumer's aspects ca be: all required tasks are collected, private files are ot distributed to others or requests are distributed uiformly. If we talk about behavior of a feedback source, we ca use the aspect: provided feedbacks are true. The cotext coects services ad aspects. Obviously, a idividual aspect makes sese oly with some services. We illustrate this cocept by a example o the P2P etwork for distributed computig. The peers i this etwork are able to accept tasks from others, compute them ad retur the results to the submitters. The submitters collect the result, if they do ot, the result is discarded. The cotext would be: = ('share computatio capacity') A = ('task is computed correctly', 'task is computed i time', 'all required tasks are collected', 'provided feedbacks are true') C = ((s 1,a 1),(s 1,a 2),(s 1,a 3),(s 1,a 4)) For example, a iterpretatio of the property c 4 is 'provided feedback about share computatio capacity is true'. The ext step is to defie trust relatioship. The trust relatioship betwee a truster (a etity that trusts the other etity) ad a trustee (a etity that is trusted) is a basic cocept i trust maagemet. I our model, we keep trust relatioship for each property separately. This facilitates more detailed access cotrol ad represets the relatios amog trusts of idividual properties. We record the curret trust relatioship i a form of matrices, oe matrix for each property. Defiitio 4: The trust matrix i of dimesio x represets the trust relatioships for the property c i i the P2P etwork with peers. More precisely, the cell i(a,b) represets the trust relatioship of the peer a to the peer b i the property c i. The value of a trust relatioship is a real umber i the rage [-1,1]. The iterpretatio is as follows: If the value is 1, the peer is sure that the remote peer satisfies the property. If the value is -1, the peer is sure that the remote peer does ot satisfy the property. If the value is 0, the peer caot decide whether the remote peer satisfies the property or ot. It is also possible choose the rage [0,1] with eutral value 0.5. Both variats are equivalet, but we choose the rage

3 [-1,1] because the iterpretatio is more atural. The egative values meas distrust ad coversely the positive values meas trust. The exact iterpretatio of other values ca deped o a idividual property. Geerally, the distace from zero ca be uderstood as a degree of certaity. Trust relatioships chage over time depedig o executed trasactios ad their outcomes. Hece, the trust matrices also chage. I our model, we wat to record trust relatioships i time t j, where j {0,1,2,...}. The time slot T j is a iterval [t j-1,t j] for j > 0. The otio j represets all trust matrices i time t j (that is at the ed of the time slot T j). The matrices 0 represet a iitial state. B. Modificatio of trust relatioships A sigle trasactio always ivolves two peers, source ad destiatio. The source is a peer, which cosumes a service (also cosumer), ad destiatio is a peer, which provides a service (also provider). It is a differet otatio tha truster ad trustee itroduced i sectio III.A. The otatio truster ad trustee is take from trust relatioship's poit of view ad the source ad destiatio from trasactio's poit of view. We study how a give trasactio iflueces a trust matrix i this sectio. Defiitio 5: Modificatio vector = (a,b,v 1,...,v m,w 1,..., w m) of dimesio 2m+2, m = C, describes the chages of trust relatioship betwee the truster a ad the trustee b. The values v i for i from 1 to m are real umbers betwee [-1,1] ad declare how much the trust relatioship will be chaged. The values w i are real o-egative umbers ad represets the weight of v i. Defiitio 6: The modificatio vectors a ' are compatible whe = (a,b,...) ad ' = (a,b,...). Hece, they describe trust relatioships betwee the same peers. Defiitio 7: The compositio operatio composes two compatible vectors ad '. The resultig vector is i the form: = a, b, v 1, v 2,.. v m, w 1, w 2,.., w m Where: v i = v i w i v ' i w ' i w i w ' i w i =w i w ' i This operatio is commutative, associative ad a result of composig two modificatio vectors is also a modificatio vector compatible with the operads. The fudametal of this operatio is a weighted average. The ifluece of a sigle trasactio o trust matrices ca be expressed by modificatio vectors. Cumulative effects of all trasactios i a specific time slot ca be expressed by compositio of modificatio vectors origiated from them. C. Access cotrol Oe of the crucial fuctios of trust maagemet is to establish rules decidig which trasactios are allowed. Those rules are based o trust relatioships ad ca be differet for each type of trasactio. I our model, the iput for access cotrol compoet is a trasactio request. Defiitio 8: The trasactio request q cotais four compoets: the specificatio of a trasactio type, the set S of source peers, the set D of possible destiatio peers ad a variable umber of parameters depeded o the trasactio type. A sigle trasactio request ca be composed ito S * D trasactios ad each trasactio ca produce several modificatio vectors. The objective of the access cotrol compoet is to pick up the allowed trasactios ad prepare the modificatio vectors for each trasactio. The trasactio costraits take ito accout the source rules ad the destiatio rules. The source rules limit the possible destiatios for a give source ad the destiatio rules limit the possible sources for a give destiatio. Defiitio 9: The source rule is a formula srule(s,d) that retur true if ad oly if the give trasactio ca be performed betwee peers s ad d. These rules are a part of trasactio specificatio. The destiatio rule drule(d,s) is defied aalogously. I a real etwork, the source rule srule(s,d) is evaluated o the source peer, hece i our model the source rule ca be calculated oly with the rows (s,_) i trust matrices. Likewise, the destiatio rule ca be calculated oly with the rows (d,_) i trust matrices. If oe source or destiatio rules for a give type of trasactio are defied we suppose that all trasactios of this type are allowed. Although, for each type of trasactio at least oe source rule should be defied otherwise the trust maagemet has o effect. Additioally, we allow more source rules to give the source more optios to choose a proper destiatio. The several source rules allows a peer to defie a softer restrictio i case the previous restrictio was too restricted ad smooth out all possible destiatios. We limit the destiatio rules to oe. The destiatio ca oly accept or refuse the request from a give source. We preset a simple algorithm for determiig the allowed trasactio. The source rules are distiguished by a differet priority, the rule with the highest priority is desigated as srule 0 (s,d). Let C r is a umber of defied source rules. For each s i S is created the set D i cotais the peers which are grated i the trasactio with s i. The procedure for determiig D i is as follows: D i = j=0 while D i = j C r do D i ={d D i ; srule j s i, d drule d, s i } j doe The trasactio is performed betwee peer s i ad every peers b, such that b D i.

4 The ext task is a preparatio of modificatio vectors. Each trasactio type has modificatio templates which desigate truster, trustee ad weights. The values v i are obtaied by aother compoet ad will be icluded ito a modificatio vector later. The modificatio template together with the values v i creates a modificatio vector. We clarify this sectio i two examples. We use the same cotext as i the sectio III.A. ad describe two possible trasactios types i the etwork. The first trasactio type is calculate a task ad secod retur a feedback. For each of them we defie access rules ad modificatio vectors. The source rules for the first trasactio ca be: srule 0 s, d = 1 s, d s,d 0 srule 1 s, d = 1 s, d 0 2 s, d 0 srule 2 s, d = 1 s, d 0.1 The destiatio rule ca be: drule d, s = 3 d, s 0 The srule 0 has the highest priority. This rule chooses the best peers available for this trasactio. If oe of this peer is available due to destiatio rules o the remote side, the peer tries the rule with the secod highest priority srule 1. If this rule does ot retur ay usable peers, the peer tries the last desperate rule, which makes possible that the result will be icorrect. The destiatio rule limits sources to the peers which probably collect results, hece the cosumed work will ot be useless. The modificatio template for this trasactio ca be: 1 = s, d,,,,,1,1,0,0 2 = d, s,,,,,0,0,1,0 The first template describes the modificatio of trust relatioship from s to d. We ca see that it modifies oly properties c 1 ad c 2, other properties have weight 0. Similarly, the secod template modifies oly property c 3 of trust relatioships from d to s. The primary modificatio vectors (vectors origiated from trasactios) should have weights i the rage [-1,1]. Provided that the values v i (the evaluatio) are also i the rage [-1,1], the trust relatioships remai i the iterval [-1,1]. I the secod trasactio the peer s requires a feedback o the peer r from the peer d. The peer r is icluded ito the optioal parameters i the trasactio request. The source rules for this trasactio ca be: srule 1 s, d = 4 s, d 0.5 srule 2 s, d = 4 s, d 0 The destiatio rule is: drule d, s = true The modificatio templates ca be: 1 = s, d,,,,,0,0,0,1 2 = s, r,,,,, 4 s,d, 4 s, d, 4 s, d,0 The first template modifies the trust relatioship towards to a feedback source. It evaluates the trasactio itself. The secod vector icorporates the gaied data ito the trust relatioships. The weights of those data are trust values i the property c 4. The proper choice of access rules ad modificatio templates is crucial for effectiveess of whole system. We discuss system efficiecy i sectio V. D. Iteral trasactios The modificatio vectors which we describe i the above sectios do ot take ito accout the values of the previous trust relatioships. Each modificatio vector belogs to the trasactio which passes through the access cotrol. But the trust relatioships chage over time eve if there are o trasactios at all. We itroduce the otio of a iteral trasactio which is proceeded i each time slot ad for each pair of peers. The iteral trasactios provide a historical cotiuum of trust relatioships. The geeral tedecy is to attach greater importace to recet experiece tha the evets i the past. The iteral trasactios allow us to model such behavior. The same way as regular trasactios, the iteral trasactios are described by the modificatio templates. But i this case, the template determies all items icludig the values v i. Geerally, oly oe modificatio template is eeded, but we ca defie more of them. We discuss oe model of iteral trasactios which provides a historical cotiuum of trust relatioships. Eve if there are o trasactios i this time slot, the trust relatioship will chage. No trasactio meas that we lose certaity about peer's behavior, hece the value of trust relatioship approaches to ull. We also use the iformatio from the past to alter curret trust relatioship. Let suppose that we are i the time slot T j, the matrices j-1 represet trust relatioships from the previous time slot T j-1. The modificatio template for a iteral operatio betwee peers s ad d has a format: 1 = s, d, 1 j 1 s, d r 1,.., m j 1 s, d r m, w 1,.., w m This modificatio vector uses trust values from oe previous time slot adjusted by multiplicatio coefficiets r 1, r 2,.., r m i the rage [0,1] which idicate how much of previous trust i the property c i we use i ext time slot. The values w i's represet the weights of past experieces. Optioally, we ca use iformatio from more time slots. This modificatio template use iformatio from k previous time slots: 2 = s, d, v 1,..v m, w 1,.., w m Where: k v i = R i l j l i s, d l=1

5 The multiplicatio vector R i of dimesio k represets the weights of past time slots i the property c i. To illustrate our cocept we use the followig example. We are i the time slot T j ad wat to compute the value v i of the iteral modificatio vector. We use the iformatio from four past time slots (k = 4) ad multiplicatio vector R i = (0.65, 0.17, 0.11, 0.01). The trust relatioships for property c i i the four previous time slots were (0.4, 0.2, 0.3, 0.5). The figure 1 illustrates the situatio. The computed value v i which will be used i the time slot T j is There are two reasoable rules for the multiplicatio vector R: R j > R j+1 for all j = 1 to k-1. This ca be iterpreted as: the ewer iformatio has greater weight tha older oe. R j < 1. This ca be iterpreted as: If there are ay other trasactios, the trust relatioships approach to ull. Figure 1. Example usage of iteral modificatio vectors We demostrated oe possibility to achieve a historical cotiuum i our trust model. But the desig of a iteral trasactio allows us to use a arbitrary model. E. Evaluatio of trasactio The ifluece of each trasactio o the trust relatioship is described by the modificatio vectors defied i sectio III.B. The defiitio of the trasactio cotais the templates of this vector. The templates desigate which peer will be a truster ad a trustee ad the weights (values w i). But the values v i deped o the behavior of the peers which participate o the particular trasactio executio ad must be calculated for each executio separately. I this sectio we focus o the modelig of peer behavior ad calculatio of the values v i. I the trust matrix model we distiguish peers accordig to its ature. There are two basic types of peers: completely hoest peers ad completely dishoest peers. The completely hoest peers always cooperate as a remote peer expects. The completely dishoest peers behave upredictable i all trasactios. The trust maagemet usually deals with completely dishoest peers easily. But there ca be a lot of peer types betwee those two extreme cases. For example, the peer behaves correctly ad icreases its reputatio. Whe it achieves a high reputatio, it starts to cause damage. This behavior is usually called treaso [17]. Aother commo problem is spreadig a false feedback [24]. Geerally, the malicious peers ca cooperate with each other ad implemet a very complicated algorithm to circumvet trust maagemet. The experieces idicate that such type of peers cause most of the problems [25]. The trust matrix model assumes that each peer type has its ow model which describes peer behavior as a participat i a trasactio. The peer model is a otrivial programmig compoet which is able to describe the complexity of the peer behavior. Each peer is associated with oe peer model. The objective of the peer model is to calculate the values v i for the trasactios which are iitiated by the peer which belogs to this peer model. The values v i ca express the evaluatio of the trasactio from oe participat's poit of view or the recommedatio o a remote peer. The peer models are resposible for all values which are used for alterig trust relatioships. They separate peer behavior from trust maagemet model ad allow to simulate differet peer behavior. The iterface betwee peer model ad trust maagemet represets the modificatio templates ad modificatio vectors. F. Trust Model At this momet, we have all prerequisites for costructio the matrix model. As we metioed above, the time axis is divided ito the time slots. We have a set of users' requests assiged to the time slots. Those users' requests do ot correspod exactly with the trasactios i the P2P etwork. For example, the user request dowload specific data ca lead ito several trasactios: fid a peer which maitai data, get refereces to a give peer ad trasfer data from a chose peer. Each of them ca be ispected by the access cotrol compoet ad each of them iflueces the trust matrices. Additioally, sometimes is ecessary to schedule two trasactios ito differet time slots because they are ot idepedet. For example, the result of trasactio get refereces to a give peer affects the decisio i the access compoet for the trasactio trasfer data from a chose peer. Hece, the oe user request ca occupy several time slots. I the matrix model we itroduce the compoet called trasactio preparatio which trasforms users' requests ito trasactio requests. The method of this trasformatio depeds o both the P2P layer ad maagemet system. The matrix model works i iteratios. Each iteratio step takes all trasactio requests belogig to the curret time slot ad trust matrices from the previous time slots ad produces ew trust matrices. Cosequetly, the trasactios which belog to oe time slot are executed cocurretly. It meas that their legitimacy is judged accordig to the same trust matrices ad their effects are projected ito the same trust matrices. We describe the iteratio step i more detail. The user requests are trasformed ito trasactio requests i the trasactio preparatio compoets. These requests are passed ito the access cotrol compoet which choose grated trasactios ad costruct modificatio templates. These templates are completed accordig the peer models (sectio III.E.). The iteral trasactios are also executed i each step. The modificatio vectors from iteral trasactios are merged with the vectors obtaied from the access cotrol compoet.

6 Next, all compatible vectors are composed accordig the compositio operatio. The vectors which origiated from the compositio operatio desigates a ew trust matrix i this way: the value v i of modificatio vector (a,b,v 1,..,v m,w 1,..,w m) correspods to value Φ j i (a,b). Because the iteral trasactio is executed for each pair of peers i the etwork, we have a modificatio vector for each pair of peers ad all cells i a ew trust matrix are defied. The complete schema of proposed matrix model is illustrated i the figure 2. service s 1. If we wat to moitor aother service, we must defie a additioal property. It is also possible to defie a separate service for referece passig ad moitorig differet aspects of this service. We defie two types of trasactios: dowload iformatio about a stock ad dowload iformatio about a peer. We defie the access cotrol rules ad the modificatio templates for each type. Dowload iformatio about a stock : The possible aswers ca be a umerical expressio of the stock price or ukow. We suppose that the request is always directed to the peer which claims that it has required data. Hece, the aswer ukow ca be cosidered as a violatio of property c 2. We use followig modificatio templates ad access cotrol rules: The modificatio template: 1 = s, d,,,,1,1,0 The access cotrol rules: srule 1 s,d = 1 j 1 s, d 0.5 srule 2 s,d = 1 j 1 s, d 0 2 j 1 s, d 0 drule s, d = 2 j 1 d, s 0 The detailed explaatio of the access rule is described i a sectio III.C. The modificatio template directly icorporates the trasactio outcome ito trust relatioship i the properties c 1 ad c 2. Dowload iformatio about a peer : The destiatio's opiio o aother peer r is icluded i the aswer. The source icludes this opiio ito the ow trust relatioships towards the peer r. The source also modifies the trust relatioship i the property c 3 to destiatio. The modificatio templates are: 1 s, d,,,,0,0,1 IV. EXAMPLE USAGE I this sectio we describe a complete example usage of our model. Let imagie the P2P etwork for sharig iformatio about stock prices. Some peers have a direct access to some stock exchages ad aouce this fact to the etwork. Others oly retrasmit the iformatio. There are also peers which spread false data or do ot retrasmit obtaied data. We wat to restrict those peers. The cotext describig this situatio ca be: = 'share iformatio about a stock'. A = 'provided iformatio are correct', 'all aouced data are shared', 'provided feedbacks are true'. C = (s 1,a 1),(s 1,a 2),(s 1,a 3). Figure 2. Trust model schema We otice that the process of obtaiig refereces is a part of service s 1. We expect that the refereces are oly about the 2 s, r,,, The access cotrol rules:, 3 j 1 s, d, 3 j 1 s, d, 3 j 1 s, d srule 1 s,d = 3 j 1 s, d 0.6 srule 2 s,d = 3 j 1 s, d 0 drule 1 s, d = true We eed to establish a iteral trasactio to complete our model. We use simple time history with oly oe past time slot. The modificatio template is: 1 = s, d, 1 j 1 s, d 0.8, 2 j 1 s, d, 3 j 1 s, d 0.9,1,1,1 We ca see that the trust i property c 1 decreases faster tha i property c 3 ad the trust i property c 2 does ot decrease i time. I this example, we choose the weight 1 or 0 for all properties except the processig of refereces. But the proper desig of weights is importat for correct fuctio of the whole system. It is possible to chage weight dyamically based o the actual etwork coditios.

7 We fiish this sectio by demostratio of the processig of oe time slot T j. Let suppose that there are two operatios i this time slot. The first, the peer a requests iformatio about some stock from the peer b. This trasactio is evaluated as (v 1,v 2,0). The value i the property c 3 is 0 because this trasactio does ot give the iformatio about peer's behavior i the property c 3. The secod, the peer a requests iformatio about the peer b from the peer p. The peer p returs referece (r 1,r 2,r 3). The evaluatio of this trasactio will be cosidered i the ext time slot, this is a reaso why we do ot reflect it i this calculatio. The trust relatioships betwee peers a ad b after compositio of all modificatio vectors are: j 1 a, b = 0.8 j 1 1 a, b v 1 j 1 3 a, p r j 1 3 a, p j 2 a, b = j 1 2 a, b v 2 j 1 3 a, p r j 1 3 a, p j 3 a, b = 0.9 j 1 3 a, b j 1 3 a, p r 3 1 j 1 3 a, p The relatioships betwee peers s ad d, differet from a ad b, are chaged oly by the iteral trasactios: 1 j s, d =0.8 1 j 1 s, d 2 j s, d = 2 j 1 s, d 3 j s, d =0.9 3 j 1 s, d This simple example demostrates the possibilities of our model. The model desig is uiversal eough to describe may strategies used i the trust maagemet ad be implemeted i may differet etwork scearios. V. OUTPUT INTERPRETATION The output of our model is a set of trust matrices j, for each property oe trust matrix. The row i j a describe trust relatioships from the peer a to all other peers i the etwork i the time t j ad i the property c i. We defie several operatios over trust matrices which should help us to decide whether the trust maagemet is effective. A. Trust matrix similarity We wat to achieve a state whe the trust relatioships match the true peer behavior. At first, we defie a operatio which allows us to compare two trust matrices. We use this operatio for comparig trust matrices from two differet trust maagemet systems with the same set of trasactio requests ad the same evaluatio compoet. We use a cosie similarity to measure differece betwee two vectors: cos a, b = a b a b Defiitio 10: Let j i ad j i be trust matrices from two differet trust maagemets i the time t j, ad i the property c i. The otatio j i(k) expresses the k'th row i the matrix j i. The similarity vector i property c i is defied as follows: si m i i j, i j = cos i j 1, i j 1,.., cos i j, i j The l'th compoet of this vector represets the similarity of two trust maagemets from the peer l's poit of view. If the l'th compoet is 1, the effect of both trust maagemets for the peer l is idistiguishable. The value -1 implies that these trust maagemets give the peer l the opposite result. Geerally, the egatives values idicate dissimilarity, positive values idicate similarity ad the value 0 idicates idepedece. This vector ca be calculated for each property which is i both systems. Hece, we are able to partially compare two trust maagemets with overlappig property sets. We aggregate this vector ito a sigle value by comparig it with the vector of 1. Defiitio 11: The similarity value i the property c i betwee j i ad j i is defied as follows: i i j, i j =cos sim i i j, i j, 1 This value reflects the similarity of trust maagemets from all peers' poit of view with the same iterpretatio as oe compoet of the similarity vector. Now we are able to compare the output of two differet trust maagemet systems. But we eed to be able to decide which system is better. For purpose of this we defie trust matrices which represet the ideal trust maagemet system. I such system the trust relatioships are established accordig the true peer characteristic. Defiitio 12: The behavior matrix i of dimesio x represets the peer characteristic i the property c i. The cell i(a,b) represets behavior of peer a to peer b i the property c i. The peer behavior is expressed by a real umber i the rage [-1,1]. The iterpretatio is similar to trust matrix, the oly differece is that the trust matrix reflects the expected behavior resulted from the trust maagemet ad the behavior matrix reflects the true peer's ature. There three most importat values are give: 1: The peer a behaves always hoest to peer b. -1: The peer a behaves always dishoest to peer b. 0: There are o iteractios betwee peer a ad b. Because the peer behavior ca be variable i the time, we use otatio i j to express the behavior matrix i the time slot T j. We ca calculate the accuracy of trust maagemet for property c i i the time t j as the similarity value betwee the trust matrix i j ad the behavior matrix i j: i j = i i j, i j The perfect trust maagemet system has this value 1. These values ca be cosidered as oe criterio of successfuless of trust maagemet system.

8 The matrix similarity ca be also used i a sigle trust maagemet to compare matrices from differet time slots. This makes us possible to measure chages i the trust relatioships caused by a sequece of trasactios. B. Trust ad distrust I this sectio we focus o the trust matrix itself. We defie several operatios with a trust matrix which ivestigate trust relatioships from peer poit of view. We are iterested i the amout of trust which the etwork reposes i the peer ad which idividual peers repose i the etwork. I the followig text the p j deotes the j-th peer i the etwork. Defiitio 13: The trust i the property c l which the etwork reposes i the peer a is deoted as TP l(a) ad defied as follows: TP l a = j=1 max l p j, a,0 The TP meas 'trust from the etwork to the peer'. We take ito accout oly positive trust relatioships which represet expected hoest behavior. We ca defie the distrust i the similar way. Defiitio 14: The distrust i the property c l which the etwork reposes i the peer a is deoted as DP l(a) ad defied as follows: DP l a = j=1 mi l p j, a,0 Those values give us a idea of total trust or distrust of oe peer i the etwork. They ca be used for ratig of malicious techiques. If the malicious peers get a high TP value, the trust maagemet failed to stop them. O the cotrary, if the malicious peers get a high DP value, it implies that the trust maagemet succeeds i avoidig their malicious behavior. We otice that for all peers ad all properties it holds TP l(a) + DP l(a) 1. So far we defied trust ad distrust i idividual properties, ow we aggregate these values for the whole peer. We eed a weight vector W=(W 1,..,W C ), which determies the importace of idividual properties. This vector must satisfy the coditio: C W l =1 l=1 The value W l determies the weight of the property c l. Defiitio 15: The trust which the etwork reposes i the peer a is deoted as TP(a) ad defied as follows: C TP a = W l tp l a l=1 Defiitio 16: The distrust which the etwork reposes i the peer a is deoted as DP(a) ad defied as follows: C DP a = W l dp l a l=1 Those two values idicate how the peer persuaded the etwork eighbors of its trustworthiess. The higher TP value meas more trustful peer. We ca also defie those values i a reverse order. Defiitio 17: The trust i the property c l which the peer a reposes i the etwork is deoted as TN l(a) ad defied as follows: TN l a = j=1 max l a, p j,0 Defiitio 18: The distrust i the property c l which the peer a reposes i the etwork is deoted as DN l(a) ad defied as follows: DN l a = j=1 mi l a, p j,0 The TN meas 'trust from peer to etwork' ad DN is 'distrust from peer to etwork'. The values TN(a) a DN(a) are defied aalogously to defiitio 15 respective 16 usig the same weight vector W. These values idicate how the peer believes i the services provided by the etwork eighbors, how trustful is the etwork eviromet from sigle peer's poit of view. The higher TN value meas the higher opportuity to get required service. C. Globality There are other criteria of the trust matrix which are ot directly related with the trust relatioships betwee peers but describe characteristics of the whole trust maagemet. We ca ivestigate how the trust relatioships to the same trustee differ o various trusters. I the best sceario, all these trust relatioships are equal because the peer trustworthiess is absolute. Defiitio 19: The globality of the trust relatioship from the peer a to the peer b i the property c l expresses the differece betwee this trust relatioship ad all trust relatioships i the etwork where the trustee is the peer b. It is expressed by the formula: g l a, b =1 l p j, b l a, b j=1 2 1 If the value g l is 1, the trust relatioship is global i the property c l. It meas that all peers i the etwork have the same trust i the peer b as the peer a. I the matrix terms, all cells i the trust matrix i the colum b are equal. How the value g l

9 decreases, the differece betwee trusts grows. If the value g l is 0, the peer a has a opposite trust relatioship to the peer b tha other peers. This value ca give us a basic idea of the correctess of the trust relatioship. If more peers share the same or similar opiio about a remote peer, it is more likely that this opiio is correct. We ca also defie the globality value for all trust relatioships i the give property. Defiitio 20: The globality of the trust relatioships i the property c l is expressed by the formula: G l =1 1 2 k=1 i=1 j=i 1 l p i, p k l p j, p k 3 If the value G l is 1, the trust maagemet is global i the property c l. It meas that all trust relatioships with the same trustee have the same trust value. I the other words, all peers i the etwork agree o the trustworthiess of the sigle peer. I the matrix terms, the values i the matrix colums are equal. If we suppose that the idetical trustworthiess is based o the same iformatio, the globality gives us a idea about spreadig iformatio i the etwork. The high globality value meas that the iformatio about a sigle trasactio is projected oto the trust relatioships betwee may peers. The global trust maagemet meas that iformatio about all trasactio is distributed to all peers. I the distributed eviromet it is difficult to create global trust maagemet because spreadig iformatio about each trasactio to the all peers fast eough is impossible without excessive traffic. The cetralized trust maagemet is aturally global. All iformatio is stored i oe cetral poit which is resposible for calculatio ad distributio the trust relatioships. VI. CONCLUSION AND FUTURE WORK We have proposed the model of trust maagemet which is adaptable to the previously published methods ad allow us to compare them. The fudametals of our model are trust matrices which record the trust relatioships betwee peers i a specific time. Every chage i those matrices is expressed by a special vector. This vector covers all possible ways how the trust matrix ca be modified: the trasactios outcomes, the recommedatios obtaied from other peers, ad the spotaeous chages caused by time. The cocept of modificatio vectors allows to ivestigate the ifluece of oe trasactio o trust betwee peers. We also separate trust ito properties, each property relates to some aspect of peer behavior. We are able to trace each aspect separately ad examie peer behavior i more detail. The multidimesioal trust is able to better capture the complexity of peer behavior. The trust matrix model allows to defie access rules which work with idividual aspects ad improves the system robustess. The proposed criteria are defied separately for each property hece we are able to judge the effectiveess of the system i each property ad detect the weak poit more easily. Last but ot least we are able to study depedecies betwee properties. We defie the similarity betwee trust matrices. We ca compare trust matrices from differet trust maagemets or the trust matrix from oe trust maagemet ad the trust matrix which describe true pear behavior. The similarity value expresses the ratio of similarity betwee matrices. This value allows to compare the result of two trust maagemets i a term of success i buildig trust relatioships. We also defie the trust, distrust, ad the globality value which should help us better uderstad the processes i the reputatio maagemet. A lot of work remais to be doe. So far we focus oly o the trust maagemet itself. As a result of sectio III.F., there are two other compoets which we are ot able to model yet. The trasactio requests ad their assigmet to the time slots represet users' behavior, i.e. users' requests o services i the etwork. These requests desigate which trasactios will be executed ad ifluece the amout of iformatio available to the trust maagemet. The process of decidig which peer will be asked for recommedatio is also part of this compoet. Hece the schedule of trasactio requests has a great ifluece o the resultig trust matrices ad eed to be described i more detail. Additioally, the proposed matrix model eed to be validated with series of experimets which allow to asses that the proposal is valid ad computable i a reasoable time. I our future work we focus o the previously published reputatio techiques ad measure some characteristic defied i this work. We also cocetrate o the kow malicious techiques ad ivestigate the resistace of the various reputatio techiques to these threats. 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