Optimal Trust Network Analysis with Subjective Logic

Transcription

1 The Second International Conference on Emerging Secrity Information, Systems and Technologies Optimal Trst Network Analysis with Sbjective Logic Adn Jøsang UNIK Gradate Center, University of Oslo Norway Tohid Bhiyan Faclty of Information Technology, QUT, Brisbane Astralia Abstract Trst network analysis with sbjective logic (TNA-SL) simplifies complex trst graphs into series-parallel graphs by removing the most ncertain paths to obtain a canonical graph. This simplification cold in theory case loss of information and thereby lead to sb-optimal reslts. This paper describes a new method for trst network analysis which is considered optimal becase it does not reqire trst graph simplification, bt instead ses edge splitting to obtain a canonical graph. The new method is compared with TNA-SL, and or simlation shows that both methods prodce eqal reslts. This indicates that TNA-SL in fact also represents an optimal method for trst network analysis and that the trst graph simplification does not affect the reslt. Introdction Trst networks consist of transitive trst relationships between people, organisations and software agents connected throgh a medim for commnication and interaction. By formalising trst relationships, e.g. as reptation scores or as sbjective trst measres, trst between parties within a domain can be derived by analysing the trst graph consisting of all the paths linking the parties together. TNA- SL (Trst Network Analysis with Sbjective Logic) [7, 6] takes directed trst edges between pairs as inpt, and can be sed to derive a level of trst between arbitrary parties that are interconnected throgh the network. Even in case no explicit trst paths between two parties exists, sbjective logic allows a level of trst to be derived throgh the defalt vacos opinions. TNA-SL therefore has a general applicability and is sitable for many types of trst networks. This can also be combined with Bayesian reptation systems [5]. In case of a complex network with dependent paths, TNA-SL reqires simplification of the trst graph into a SPG (irected Series-Parallel Graph) before compting the derived trst. The simplification consists of gradally removing the most ncertain trst paths ntil the whole graph can be represented in a series-parallel form. As this process removes information it cold intitively be considered sb-optimal. In this paper we describe a new method for trst network analysis which avoids this problem by splitting dependent trst edges into independent parts, so that each part can be taken into accont dring the comptation. This approach is considered optimal becase it does not remove information. Simlations show that thew new method prodces the same reslts as TNA-SL. This implies that the information removed throgh trst graph simplification is irrelevant for the trst network analysis, and that TNA-SL therefore prodces optimal reslts. 2 Transitive Trst Paths Trst transitivity means, for example, that if Alice trsts Bob who trsts avid, then Alice will also trst avid. This assmes that Alice is actally aware that Bob trsts avid. This cold e.g. be achieved throgh a recommendation from Bob to Alice as illstrated in Fig., where the indexes on each arrow indicate the seqence in which the trst relationships/recommendation are formed. Figre. Transitive trst principle The trst scope is the specific type(s) of trst assmed in a given trst relationship. In other words, the trsted party is relied pon to have certain qalities, and the scope is what the trsting party assmes those qalities to be. Let s assme that Alice needs to have her car serviced, so she asks Bob for his advice abot where to find a good /8 $ IEEE OI.9/SECURWARE

2 car mechanic in town. Bob is ths trsted by Alice to know abot a good car mechanic and to tell his honest opinion abot that. Bob in trn trsts avid to be a good car mechanic. Bob s trst in avid is fnctional whereas Alice s trst in Bob is referral becase Bob only refers to a mechanic. Both trst relationships are direct and have the same scope, namely that of being a good mechanic. Alice s trst in avid is indirect becase it is derived, and fnctional becase avid will actally do the job. This sitation is illstrated in Fig., where the indexes indicate the order in which the trst relationships and recommendations are formed. The examples above assme some sort of absolte trst between the agents along the transitive trst path. In reality trst is never absolte, and researchers have proposed to express trst as discrete verbal statements, as probabilities or other continos measres. TNA-SL assmes that trst relationships are expressed as sbjective opinions. 3 Parallel Trst Combination It is common to collect advice from several sorces in order to be better informed when making decisions. This can be modelled as parallel trst combination illstrated in Fig.2, where again the indexes indicate the order in which the trst relationships and recommendations are formed. method, which is described in Sec.6, is based on sbjective logic which is sitable for analysing sch sitations. 4 Strctred Notation Transitive trst networks can involve many principals, and in the examples below, capital letters A,B,C and will be sed to denote principals instead of names sch as Alice and Bob. We will se basic constrcts of directed graphs to represent transitive trst networks, and add some notation elements which allow s to express trst networks in a strctred way. A single trst relationship can be expressed as a directed edge between two nodes that represent the trst sorce and the trst target of that edge. For example the edge [A,B] means that A trsts B. The symbol : will be sed to denote the transitive connection of two consective trst edges to form a transitive trst path. The trst relationships of Fig. is expressed as: ([A,]) = ([A,B]:[B,]). () Let s now trn to the combination of parallel trst paths, as illstrated in Fig.2. We will se the symbol to denote the graph connector for this prpose. The symbol visally resembles a simple graph of two parallel paths between a pair of agents, so that it is natral to se it for this prpose. In short notation, Alice s combination of the two parallel trst paths from her to avid in Fig.2 is then expressed as: ([A,]) = (([A,B]:[B,]) ([A,C]:[C,])) (2) 5 Trst Graph Simplification Figre 2. Parallel combination of trst paths Let s assme again that Alice needs to get her car serviced, and that she asks Bob to recommend a good car mechanic. When Bob recommends avid, Alice wold like to get a second opinion, so she asks Claire whether she has heard abot avid. Intitively, if both Bob and Claire recommend avid as a good car mechanic, Alice s trst in avid will be stronger than if she had only asked Bob. Parallel combination of positive trst ths has the effect of strengthening the derived trst. In the case where Alice receives conflicting recommended trst, e.g. trst and distrst at the same time, she needs some method for combining these conflicting recommendations in order to derive her trst in avid. Or Trst networks can have dependent paths. Sbjective logic reqires trst graphs to be expressed in a canonical form that has no dependent paths. An example of graph simplification is illstrated in Fig.3. Figre 3. Network simplification by removing weakest path The expression for the dependent graph on the left-hand side of Fig.3 wold be: ([A,]) = (([A,B]:[B,]) ([A,C]:[C,]) ([A,B]:[B,C]:[C,])) (3) 8

3 The problem with Eq.(3) is that the edges [A,B] and [C, ] appear twice. TNA-SL reqires graphs to be expressed in a form where an edge only appears once. This will be called a canonical expression, which is defined as follows: efinition (Canonical Expression) An expression of a trst graph in strctred notation where every edge only appears once is called canonical. A method for canonicalisation based on network simplification was described in [6, 7]. Simplification consists of removing the weakest, i.e. the least certain paths, ntil the network becomes a directed series-parallel graph which can be expressed on a canonical form. Assming that the path ([A, B]:[B,C]:[C, ]) is the weakest path in the graph on the left-hand side of Fig.2, network simplification of the dependent graph wold be to remove the edge [B,C] from the graph, as illstrated on the righthand side of Fig.3. Since the simplified graph is eqal to that of Fig.2, the formal expression is the same as Eq.(2). An alternative canonicalisation method called edge splitting will be described in Sec.7. The next section describes sbjective logic operators for analysing trst networks. 6 Trst Comptation with Sbjective Logic Sbjective logic is sitable for analysing trst networks becase trst relationships can be expressed as sbjective opinions with degrees of ncertainty. TNA-SL reqires trst relationships to be expressed as beliefs, and trst networks to be expressed as a SPG (irected Series-Parallel Graph) in the form of canonical expressions. 6. Sbjective Logic Fndamentals Sbjective logic [3] is probabilistic logic that se opinions as inpt and otpt variables. Opinions explicitly express ncertainty abot probability vales, and can express degrees of ignorance abot a sbject matter sch as trst. An opinion is denoted by ωx A which expresses A s belief in the trth of proposition x. Alternatively, an opinion can focs on an entity X expressed as ωx A which can be interpreted as Party A believes that party X is honest and reliable regarding a specific scope,which can be interpreted as A s trst in X within the given scope. Binomial opinions are expressed as ω =(b, d,, a) where b, d, and represent belief, disbelief and ncertainty respectively, nder the constraint that b,d, [,] and b+d + =. The parameter a [,] is called the base rate, and is sed for compting an opinion s probability expectation vale that can be determined as E(ωx A )=b + a. More precisely, a determines how ncertainty shall contribte to the probability expectation vale E(ω A x ). In the absence of any specific evidence abot a given party, the base rate determines the defalt trst. The opinion space can be mapped into the interior of an eqal-sided triangle as illstrated in Fig.4 where the three parameters b, d and determine the position of the opinion point in the triangle, with ω x =(.7,.,.2,.5) as an example. Example opinion: ω x = (.7,.,.2,.5) Uncertainty isbelief Belief ax E( x) Probability axis ω x Projector Figre 4. Opinion triangle with example The base rate a x is indicated by a point on the probability axis, and the projector starting from the opinion point is parallel to the line that joins the ncertainty vertex and the base rate point on the probability axis. The point at which the projector meets the probability axis determines the expectation vale of the opinion, i.e. it coincides with the point corresponding to expectation vale E(ω A x ). The probability density over binary event spaces can be expressed as Beta PFs (probability density fnctions) denoted by Beta(α,β) [2]. Let r and s express the nmber of past observations of x and x respectively, and let a express the apriorior base rate, then α and β can be determined as: α = r + 2a, β = s + 2( a). (4) The following bijective mapping between the opinion parameters and the Beta PF parameters can be determined analytically [3]. b = r/(r + s + 2) d = s/(r + s + 2) = 2/(r + s + 2) a = base rate of x r = 2b/ s = 2d/ = b + d + a = base rate of x This means for example that a totally ignorant opinion with = and a =.5 is eqivalent to the niform PF Beta(,). It also means that a dogmatic opinion with = is eqivalent to a spike PF with infinitesimal width and (5) 8

4 infinite height expressed by Beta(bη, dη), where η. ogmatic opinions can ths be interpreted as being based on an infinite amont of evidence. After r observations of x and s observations of x with base rate a =.5, the a posteriori distribtion is the Beta PF with α = r + and β = s +. For example the Beta PF after observing x 7 times and observing x once is illstrated in Fig.5, which also is eqivalent to the opinion illstrated in Fig.4 Probability density Beta( p 8,2 ) Probability p Figre 5. A posteriori Beta(8,2) after 7 observations of x and observation of x A PF of this type expresses the ncertain probability that a process will prodce positive otcome dring ftre observations. The probability expectation vale of Fig.5 is E(p)= Operators for Trst Network Analysis Sbjective logic defines a nmber of operators[3, 8]. Some operators represent generalisations of binary logic and probability calcls operators, whereas others are niqe to belief theory becase they depend on belief ownership. This presentation focses on the transitivity (also called disconting) and the fsion (also called consenss) operators that are sed for trst graph analysis. The transitivity operator can be sed to derive trst from a trst path consisting of a chain of trst edges, and the fsion operator can be sed to combine trst from parallel trst paths. These operators are described below. Transitivity is sed to compte trst along a chain of trst edges. Assme two agents A and B where A trsts B, denoted by ωb A, for the prpose of jdging the trstworthiness of C. In addition B has trst in C, denoted by ωc B. Agent A can then derive her trst in C by disconting B s trst in C with A s trst in B, denoted by ωc A:B. The symbol is sed to designate this operator. The ncertainty favoring version of the transitivity operator [9] is expressed as: ω A:B C = ω A B ωb C bc A:B = ba B bb C dc A:B = ba B db C C A:B a A:B = da B + A B + ba B B C C = ab C. (6) The effect of transitivity disconting in a transitive chain is that ncertainty increases, not disbelief. Cmlative Fsion is eqivalent to Bayesian pdating in statistics. The cmlative fsion of two possibly conflicting opinions is an opinion that reflects both opinions in a fair and eqal way. Let ωc A and ωb C be A s and B s trst in C respectively. The opinion ωc A B is then called the fsed trst between ωc A and ωb C, denoting an imaginary agent (A,B) s trst in C, asifshe represented both A and B. The symbol issedto designate this operator. The cmlative fsion operator ωc A B = ωc A ωb C is expressed as: ω A B C b A B d A B A B ac A B = ω A C ωb C C =(bc AB C + bb C A C )/(A C + B C A C B C ) C =(dc AB C + db C A C )/(A C + B C A C B C ) C =(C AB C )/(A C + B C A C B C ) = ac A (7) where it is assmed that ac A = ab C. Limits can be compted [4] for C A = B C =. The effect of the cmlative fsion operator is to amplify belief and disbelief and redce ncertainty. 7 Trst Network Canonicalisation by Edge Splitting The existence of a dependent edge in a graph is recognised by mltiple instances of the same edge in the trst network expression. Edge splitting is a new approach to achieving independent trst edges. This is achieved by splitting a given dependent edge into as many different edges as there are different instances of the same edge in the trst network expression. Edge splitting is achieved by splitting one of the nodes in the dependent edge into different nodes so that each independent edge is connected to a different node. A general directed trst graph is based on directed trst edges between pairs of nodes. It is desirable not to pt any restrictions on the possible trst edges except that they shold not be cyclic. This means that the set of possible trst paths from a given sorce X to a given target Y can 82

5 contain dependent paths. The left-hand side of Fig.6 shows an example of a trst network with dependent paths. Figre 6. Edge splitting of trst network to prodce independent paths The non-canonical expression for the left-hand side trst network of Fig.6 is: [A,]= ([A,B] : [B,]) ([A,C] : [C,]) ([A,B] : [B,C] : [C,]) In this expression the edges [A,B] and [C,] appear twice. Edge splitting in this example consists of splitting the node B into B and B 2, and the node C into C and C 2. This prodces the right-hand side trst network in Fig.6 with canonical expression: [A,]= ([A,B ] : [B,]) ([A,C ] : [C,]) ([A,B 2 ] : [B 2,C 2 ] : [C 2,]) Edge splitting mst be translated into opinion splitting in order to apply sbjective logic. The principle for opinions splitting will be to separate the opinion on the dependent edge into two independent opinions that when cmlatively fsed prodce the original opinion. This can be called fission of opinions, and will depend on a fission factor φ that determines the proportion of evidence assigned to each independent opinion part. The mapping of an opinion ω =(b,d,,a) to Beta evidence parameters Beta(r, s, a) according to Eq.(5), and linear splitting into two parts Beta(r,s,a ) and Beta(r 2,s 2,a 2 ) as a fnction of the fission factor φ is: Beta(r,s,a ) : Beta(r 2,s 2,a 2 ) : r = φ2b s = φ2d a = a r 2 = ( φ)2b s 2 = ( φ)2d a 2 = a (8) (9) () () The reverse mapping of these evidence parameters into two separate opinions according to Eq.(5) prodces: b = d = ω : = b 2 = d 2 = ω 2 : 2 = a 2 = a a = a φb φ(b+d)+ φd φ(b+d)+ φ(b+d)+ ( φ)b ( φ)(b+d)+ ( φ)d ( φ)(b+d)+ ( φ)(b+d)+ (2) (3) It can be verified that ω ω 2 = ω, as expected. When deriving trst vales from the canonicalised trst network of Eq.(8) we are interested in knowing its certainty level as compared with a simplified network, as described in [6]. We are interested in the expression for the ncertainty of ω A corresponding to trst expression of Eq.(9). Since the edge splitting introdces parameters for splitting opinions, the ncertainty will be a fnction of these parameters. By sing Eq.(6) the expressions for the ncertainty in the trst paths of Eq.(9) can be derived as: A:B = db A + A B + b A B B A:C = dc A + C A + bc A C A:B 2:C 2 = b A B 2 d B 2 C 2 + d A B 2 + A B 2 +b A B 2 B 2 + b A B 2 b B 2 C 2 C 2 (4) By sing Eq.(7) and Eq.(4), the expression for the ncertainty in the trst network of Eq.(9) can be derived as: A = A:B A:C A:B 2 :C 2 A:B A:C +A:B A:B 2 :C 2 + A:C A:B 2 :C 2 2 A:B A:C A:B 2 :C 2 (5) By sing Eq.(2), Eq.(4) and Eq.(5), the ncertainty vale of the derived trst ω A according to the edge splitting principle can be compted. This vale depends on the edge opinions and on the two splitting parameters φb A and φc. As an example the opinion vales will be set to: ω A B = ω B = ω A C = ω C = ω B C =(.9,.,.,.5) (6) The compted trst vales for the two possible simplified graphs are: (ω A B ω B ) (ω A C ω C ) =(.895,.,.5,.5) (7) ω A B ω B C ω C =(.729,.,.27,.5) (8) 83

6 The ncertainty level A when combining these two graphs throgh edge splitting as a fnction of φb A and φc is shown in Fig.7 Uncertainty A of network simplification. This shows that network simplification with TNA-SL in fact does prodce optimal reslts. Or analysis was based on a fixed set of edge opinion vales. Becase of the large nmber of parameters involved, it is a relatively complex task to verify if or conclsion is valid for all possible trst edge opinion vales, so a complete stdy mst be the sbject of ftre work. The present stdy has given a strong indication that trst network simplification prodces optimal reslts even thogh edges are removed from the trst graph. References φ A B φ C.8 [] B. Christianson and W. S. Harbison. Why Isn t Trst Transitive? In Proceedings of the Secrity Protocols International Workshop. University of Cambridge, 996. [2] M.H. egroot and M.J. Schervish. Probability and Statistics (3rd Edition). Addison-Wesley, 2. Figre 7. Uncertainty A as a fnction of φ A B and φ C The conclsion which can be drawn from this is that the optimal vale for the splitting parameters are φb A = φc = becase that is when the ncertainty is at its lowest. In fact the ncertainty can be evalated to A =.5. This is eqivalent to the trst network simplification of Eq.(7) where the edge [B,C] is completely removed from the lefthand side graph of Fig.6. The least optimal vales for the splitting parameters is when φb A = φc =, reslting in A =.27. This is eqivalent to the absrd trst network simplification of Eq.(8) where the edges [A,C] and [B,], and thereby the most certain trst paths are completely removed from the left-hand side graph of Fig.6. Given the edge opinion vales sed in this example, ([A, B]:[B,C]:[C, ]) is the least certain path of the left-hand side graph of Fig.6. It trns ot that the optimal splitting parameters for analysing the right-hand side graph of Fig.6 prodces the same reslt as network simplification where this particlar least certain path is removed. 8 iscssion and Conclsion We have described edge splitting as a new principle for trst network analysis with sbjective logic. This method consists of splitting dependent trst edge opinions in order to avoid dependent paths, which can be considered optimal becase it does not case any loss of information. Or analysis and simlation have shown that edge splitting prodces the same reslt as the previosly described TNA-SL method [3] A. Jøsang. A Logic for Uncertain Probabilities. International Jornal of Uncertainty, Fzziness and Knowledge-Based Systems, 9(3):279 3, Jne 2. [4] A. Jøsang. Probabilistic Logic Under Uncertainty. In The Proceedings of Compting: The Astralian Theory Symposim (CATS27), CRPIT Volme 65, Ballarat, Astralia, Janary 27. [5] A. Jøsang, T. Bhiyan, Y. X, and C. Cox. Combining Trst and Reptation Management for Web-Based Services. In The Proceedings of the 5th International Conference on Trst, Privacy & Secrity in igital Bsiness (TrstBs28), Trin, September 28. [6] A. Jøsang, E. Gray, and M. Kinateder. Simplification and Analysis of Transitive Trst Networks. Web Intelligence and Agent Systems, 4(2):39 6, 26. [7] A. Jøsang, R. Hayward, and S. Pope. Trst Network Analysis with Sbjective Logic. In Proceedings of the 29 th Astralasian Compter Science Conference (ACSC26), CRPIT Volme 48, Hobart, Astralia, Janary 26. [8] A. Jøsang, S. Pope, and M. aniel. Conditional dedction nder ncertainty. In Proceedings of the 8th Eropean Conference on Symbolic and Qantitative Approaches to Reasoning with Uncertainty (ECSQARU 25), 25. [9] A. Jøsang, S. Pope, and S. Marsh. Exploring ifferent Types of Trst Propagation. In Proceedings of the 4th International Conference on Trst Management (itrst), Pisa, May