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(IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 3, No.2, 2012 A New Approach of Trust Relationship Measurement Based on Graph Theory Ping He Department of Information Liaoning police Academy Dalian 116036, China Abstract—The certainty trust relationship of network node by using the method of graph theory. Then it measures the behavior has been presented based on graph theory, and a trusted degree of each node, and it also presents the trusted measurement method of trusted-degree is proposed. Because of measurement of the connection and hyper connection for node the uncertainty of trust relationship, this paper has put forward behavior of network. the random trusted-order and firstly introduces the construction of trust relations space (TRS) based on trusted order. Based on The object of this paper is to establishes dynamic trust all those above, the paper describes new method and strategy evaluation model based on node behavior characters, Through which monitor and measure the node behavior on the expectancy the construction of the relationship between practical node behavior character for trusted compute of the node. According to behavior characters and on-the-spot model, it sets up a couple manifestation of node behavior and historical information, adjust and predict the trusted-order of node behavior. The paper finally of mapping models of trust relationship, and sketches the establishes dynamic trust evaluation model based node behavior skeleton of relationship mapping inversion. characters, and then it discusses the trusted measurement The paper is organized as follows. Section 2 introduces the method which measures the connection and hyperlink for node behavior of network in trust relationship space. basic concept of trust relationship based on graph theory, and some properties of the evaluate principle of the trusted Keywords- Trust relations; trust relationship graph; trusted-order; network. Section 3 studies measurement of trusted relationship. random trust relationship. Section 4 we present dynamic trust evaluation model based on random trusted relationship. Section 5, conclusion puts forward I. INTRODUCTION the discoveries of this research and future research direction. In recent years, with the wide application of trusted II. BASIC CONCEPTS AND METHODS computing in the network security area, the studies of trust relationship in the node behavior of network have been made A. Certainty trusted relational graphs an important point [1, 2, and 3]. However, conventional environment of trust relationship conduction is “man around Suppose that a trusted network TN can be expressed as the model of experience”, which is difficult to deal with the trust featured by procedures of brain thinking. Because of the the corresponding graph G {V , E} , where almighty ability of experience model while processing V {v1 , v2 , , vn } TN , empiricism information, people depend on experience model to is a node set of and a great extent. When the results from experience model are E {e1 , e2 , , em } different from the reality significantly, people will doubt it. is an edge set. Moreover, e k : vi v j This is the reason of the occurrence of the incompatible （ k 1,2, , m ； i, j 1,2, , n) ， it problems when traditional information of trust relationship vi vj conducting ways is applied to process node behavior of represents that there is a trust relationship between and , network system. v v namely i trusts j . Therefore, the trusted graph G is called The trust network model is the prerequisite of trusted a directed graph. For example, given a trusted network with computing, how to evaluate the trust in the network, there is five vertices, based on the analyzing of network behaviors, the not a unity and general method up to now. In fact, the trust trust relationship of vertices is described as follows: network is a kind of network with trust relationships, trust relationship networks can be abstracted as a kind of topological v1 v5 , v2 v1 , v2 v5 ， relationships in mathematics. Recently, the research around the model of trust networks is come from different angles [1-6]. v3 v 2 , v3 v 4 , v3 v 5 ， But their common point is seeking a formal representation method reasonably. v4 v1 ， v4 v3 , v4 v5 ， Studies in literature [3] have shown that a trust relationship can be expressed with a graph. In the paper we study a v5 v1 ， v5 v3 . quantitative expression of trust relationship in network system 19 | P a g e www.ijacsa.thesai.org (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 3, No.2, 2012 Its adjacency matrix is 0 0 0 0 1 v5 v1 v5 v2 v4 v5 1 0 0 0 1 A 0 1 0 1 1 , v3 v1 v5 v1 v3 v1 v5 v1 v3 v5 v1 v3 1 0 1 0 1 1 0 1 0 0 v4 v5 and the trusted relational graph G is shown in Figure 1. ● v1 v1 v3 v5 v1 v3 v4 ● v5 ● ● v2 v5 v 2 v 4 v5 v1 v3 v5 v 2 v 4 v5 ● v3 Figure 2. Trusted relational tree Figure 1. Trusted relational graph III. MEASUREMENT IN TRUSTED RELATIONSHIP B. Trusted relational trees The indirect trusted vector of each node in the network (when K=2) is as follows: TD(2) ( 2，4，7，6，4 ) , T Given a trusted network with five vertices (As shown in Figure 1), Let D(1) AI ( 1，2，3，3，2 ) , where T then it is certain that TD( K 1) can be measure the trusted I ( 1，1，1，1，1 ) , then D(1) is a trusted level vector T level (trusted degree) accurately than TD(K ) . In most cases, of each node. Such as the number of trusted vectors about the we must think about the limit of TD(K ) , when K . v node v 4 and 5 is 3 and 2 respectively. According to this a In order to ensure the limit convergence, and furthermore, the measuring value of each node should be trusted degree, conclusion can be drawn that the trusted level of the node v 4 therefore this paper regard the following limit as the measuring is taller than the node v5 . of each node in the trusted relational graph in networks: TD( K ) . But, the number of trusted vectors about the node v5 and lim k I T TD( K ) v 2 are both 2, how to distinguish the difference of v5 and v 2 ? It will find in fig. 2 that the measurement of the trusted degree about each node is the number of the path in the v On the analysis of Figure 1 it was found that v1 and 5 directed tree, which takes each node for the root. And then the relations are extended to the general case, the definition is as have a trusted relationship with v 2 , and the number of trusted follows: v vectors about the node v1 and 5 is 1 and 2 respectively. The Definition 3.1 In a network with n nodes, the trusted v v v Node v1 and 3 have trusted relationships with 5 , and the capability (trusted degree) of a node i can be determining by the number of the path, which connects with the K-th path v number of trusted vectors about the node v1 and 3 is 1 and vi and starts from the node . This number is called the K-th 3 respectively. td k (vi ) trusted capability, and denoted as . Vector It can be seen that the indirect trusted level of v 2 is 3, and v5 TD(k ) {td k (v1 ), td k (v2 ), , td k (vn )} the indirect trusted level of is 4. Thereby, it may be taken v5 is called the K-th trusted capability vector of the trusted for granted that the trusted level of is taller than the node relational graph G . v2 . Definition 3.2 In a network of n vertices, a limit Based on the graph theory, the trusted path determines the td k (vi ) trusted level in trusted networks. The above analysis can be td (vi ) lim representing by the tree in figure 2. k n v1 v2 v3 td i 1 k (v i ) 20 | P a g e www.ijacsa.thesai.org (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 3, No.2, 2012 v The random trusted relationship is expressed by the random is entitled relatively limit trusted degree of the node i , it is graph for trusted relationship, it has respective trusted relational called a trusted degree for short. For that reason, we have graph in the basis of different network activities and space-time states. Furthermore, the extent of trusted relationships presents TD(k ) T (td (v1 ), td (v2 ), , td (vn ))T lim , certain probabilistic characteristics with the change of network k I T TD( k ) Pij (0 Pij 1) activities; we can use to express the arisen which is called a trusted vector of each node in the trusted v v relational graph, where I (1,1, ,1) . T probability of i j . Thereby the trusted relationships in the network consisted of n nodes can be expressed by a Theorem 3.1 Let G be a trusted relational graph of n family of trusted relational graphs, the family of trusted vertices, its adjacency matrix is A, if G is bidirectional G(n, ( P )) ij relational graphs are noted as , it is called a 1 is an eigenvector corresponding to connected and n 4 , random trusted relational graph. When vi v j , take for the biggest values of A, then T exists certainly and granted, Pij 0 ; When i, j 1,2 , n ， i j , T 1 /( I T 1 ) , moreover 1 /( I 1 ) 1 . T 0 Pij 1 , apparently, P [ Pij ] constitutes a square It can seem from theorem3.1, the K-th trusted degree matrix of order n, G(n, ( Pij )) is called a probability matrix. TD(k ) of each node is computable in the trusted relational Suppose the connection of each node be random and graph G with n vertices, and it can be obtain by the independence, then a definition is as follows: following algorithm: Definition 4.1 A directed and weighted graph, which （1）when k 0 ， TD(0) I ； weighted is a probability matrix P, and it is called a network （ 2 ） when k 1 ， 2 ， … ， TD(k ) ATD(k 1) ； expression of G(n, ( Pij )) that is noted as N (n, P) . TD (k ) TD(k ) /( I T TD(k ) ； Definition 4.2 The weighted product of each edge in the directed path L is called a transfer probability in N (n, P) . （ 3 ） when given precision e 0 ， calculated until k m ，if it is satisfied： It is called the k-th order dispersive degree of the node v i that the sum of all of transfer probabilities with k connective TD (k ) TD (k ) e ， paths, which starting from the node v i . Noted as N k (vi ) , and then stopped calculating to choose T TD (m) . N (k ) ( N k (v1 ), N k (v2 ), , N k (vn ))T . The algorithm given in theorem3.1 can be put to use in network according to different trusted levels. Regardless of the N k (v i ) Definition 4.3 The limit lim is called a limit connected meaning of network note, it always measures the k I T N (v ) k i trusted degree in the trusted relational graph. Meanwhile, the transfer probability of a node v i . trusted vector T can be regaled as a weighted vector, which expresses the trusted degree of each node in the trusted Based on the probability theory, it is well known that the relational graph. transfer probability of the path Lij (as dependence), which is IV. RANDOM TRUSTED RELATIONSHIP from the node v i to v j in N (n, P) , is the present Thinking about the trusted relational graph of discussion in probability of Lij in G(n, ( Pij )) , that is to say it is a vi v j the previous section, if it has , then it exists the probability of the directed connection (trusted relational vi vj chain)between the node v i and v j in the random trusted trusted relationship of completely specified between and It is called certainty trusted relational graph which has the relational graph G(n, ( Pij )) . It is still used td k (vi ) to trusted relationship of completely specified. In fact, trusted express the number of paths, which starting from the node v i relationship is uncertainty in lots of trusted networks. For example, the trusted relationship among people in the Internet, and taking with k paths. We can prove as follows: because of the vitality of network activities, the trusted relationship of network is uncertainty, for this reason, the Theorem 4.1 Let N (n, P) be disconnected, n 4 , then uncertain research methods is used to analyze the trusted the limit transfer probability of each node exists certainly, and relationship in network. equals to the limit 21 | P a g e www.ijacsa.thesai.org (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 3, No.2, 2012 Pk I REFERENCES lim . [1] Zhuo lingxiao. Discrete Mathematics, Shanghai:Shanghai Scientific and k I T P k I Technical Literature Publishing House, 1989 (Chinese). [2] Lin chuang, Peng xuemei. Research on Trustworthy Networks. Deduction 4.1 There has N (k ) P I in N (n, P) . k Chinese Journal of Computers. 2005, vol. 28, pp.751-758 (Chinese). [3] Ammann P, Wijesekera D, Kaushik S. Scalable Graph-based Network Theorem 4.2 Let N (k ) be the k-th order dispersive Vulnerability Analysis, Proceedings of the 9th ACM Conference on Computer and Comm. Security. New York, USA: ACM Press, 2002, degree vector of each node in N (n, P) , and let TD(k ) be pp.217-224. the number vector of starting from each node and taking with [4] Swiler L P, Phillips C, Gaylor T. A Graph-based Network Vulnerability k paths in G(n, ( Pij )) , then E (TD(k )) N (k ) is Analysis System, Sandia National Laboratories, Albuquerque, USA, Technical Report: SAND97-3010/1, 1998. obtained. This theorem explains that the k-th order dispersive [5] Theodorakopoulos G, Baras j s. On trust models and trust evaluation degree of the node v i in N (n, P) is the mathematical metrics for ad-hoc networks. IEEE Journal on Selected Areas in Communications. 2006, vol.24, pp.318-328. expectation of the number of paths, which starting from the [6] Sun Y, Yu W. Han Z,Liu K J R. Information theoretic framework of node v i and taking with k paths in G(n, ( Pij )) . Since the trust modeling and evaluation for ad hoe networks. IEEE Journal on Selected Areas in Communications, 2006, vol.249, pp.305-319. measurement of trusted levels for a certain node can be [7] M. Gómez, J. Carbó, and C. B. Earle. Honesty and trust revisited: the expressed by the number of paths starting from the node. advantages of being neutral about other’s cognitive models. Hereby, we regard the limit transfer probability vector as the Autonomous Agents and Multi-Agent Systems, 2007, vol.15, pp.313-335. weighted vector T of a certain node in the random trusted relational graph. According to theorem 4.2, we can obtain the [8] Hofstede, G. J., Jonker, C. M., Meijer, S. and Verwaart, T. Modelling trade and trust across cultures, in Ketil St¿len et al., ed., `Trust same arithmetic as theorem 1, so for as changing the Management, 4th Inter-national Conference, iTrust 2006', Vol. 3986 of adjacency matrix A for the probability matrix P in Lecture Notes in Computer Science, Springer-Verlag Berlin N (n, P) . If and only if Pij 0 or Pij 1 , a random Hei-delberg, Pisa, Italy, 2006, pp. 120-135. [9] Y. Wang and M. P. Singh. Formal trust model for multiagentsystems. In trusted relational graph turns into a certainty trusted relational Proc. 20th International Joint Conference onArtificial Intelligence graph, so the latter is a special case of the former. (IJCAI), 2007, pp. 1551–1556. [10] Li Xiao-Yong, Gui Xiao-Lin. Research on adaptive prediction model of V. CONCLUSIONS dynamic trust relationship in Open distributed systems. Journal of Computational Information Systems, 2008, vol.4, In the development of the trusted computing, theoretical pp.1427-1434(Chinese). research lags behind practical. The trusted measurement is the AUTHORS PROFILE basic theory of the trusted computing, and is also a key technology in the process of development of the trusted He Ping is a professor of the Department of Information at Liaoning Police Academy, P.R. China. He is currently Deputy Chairman of the Centre of computing. In this paper, a certainty trusted network and a Information Development at Management Science Academy of China. In random trusted network were introduced respectively. Then a 1986 He advance system non-optimum analysis and founded research measurement method of the trusted degree was presented and institute. He has researched analysis of information system for more than 20 its arithmetic was described. These theories and methods will years. Since 1990 his work is optimization research on management help the development of the trusted computing. For future information system. He has published more than 100 papers and ten books, and is editor of several scientific journals. In 1992 awards Prize for the works, the methods will be optimized, which not only depict Outstanding Contribution Recipients of Special Government Allowances P. R. the fact but also can be used simply and practically. China 22 | P a g e www.ijacsa.thesai.org

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The certainty trust relationship of network node behavior has been presented based on graph theory, and a measurement method of trusted-degree is proposed. Because of the uncertainty of trust relationship, this paper has put forward the random trusted-order and firstly introduces the construction of trust relations space (TRS) based on trusted order. Based on all those above, the paper describes new method and strategy which monitor and measure the node behavior on the expectancy behavior character for trusted compute of the node. According to manifestation of node behavior and historical information, adjust and predict the trusted-order of node behavior. The paper finally establishes dynamic trust evaluation model based node behavior characters, and then it discusses the trusted measurement method which measures the connection and hyperlink for node behavior of network in trust relationship space.

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