Paper 4 - A New Approach of Trust Relationship Measurement Based on Graph Theory

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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



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                                                                   (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 )



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                                                               (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




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                                                           (IJACSA) International Journal of Advanced Computer Science and Applications,
                                                                                                                      Vol. 3, No.2, 2012

                                Pk I                                                                REFERENCES
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                         k  I T P k I
                                                                           Technical Literature Publishing House, 1989 (Chinese).
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  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.
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                                                                           pp.313-335.
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                        V. CONCLUSIONS                                     dynamic trust relationship in Open distributed systems. Journal of
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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




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Description: 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.