TPS DR: A Universal Dimension Reducing Algorithm for

Appl. Math. Inf. Sci. 7, No. 1L, 161-167 (2013) 161

Applied Mathematics & Information SciencesAn International Journal

c⃝ 2013 NSPNatural Sciences Publishing Cor.

TPS DR: A Universal Dimension Reducing Algorithmfor Optimal Trust Path Selection in Complex SensorNetworkGuangquan Xu1,2, Xiuming Tian1, Xiaochun Cao1, Xiaohong Li1 and Zhiyong Feng1

1School of Computer Science and Technology, Tianjin University, Tianjin 300072, China2Shanghai Key Laboratory of Integrate Administration Technologies for Information Security, Shanghai Jiao Tong University, Shanghai200240, China

Received: 9 Sep. 2012, Revised: 2 Dec. 2012, Accepted: 16 Dec. 2012Published online: 1 Feb. 2013

Abstract: Energy efficiency, one of the key factors for sensor network, is a challenging work. It is agreed that sink routing is the maininfluencing factor for energy efficiency and selecting an optimal routing path becomes critical. It is hard for source participant (sinknodes, namely trustor) to select a trustworthy target participant (data collecting sensor nodes, namely trustee), especially when the scaleof sensor network becomes increasingly larger. An optimal trust path selection problem (TPS problem) is a Multi-Constrained OptimalPath (MCOP) problem, which is proved to be a NP-Complete problem. In this paper, we first introduce a concept of complex trustnetwork to model the relationships among network nodes in a complex sensor network. In this model, we represent the trust networkbased on high dimensional vector and matrix. Then, we propose an algorithm TPS DR (Trust Path Selection based on DimensionalityReduction) to simplify the trust network through reducing the dimensionality, i.e. cluster some similar nodes into a supernode. Ourexample demonstrates the usage and advantage of our model and algorithm.

Keywords: Dimensional Trust Vector, Complex Trust Network, Trust Path Selection, Multi-Constrained Optimal Path (MCOP)Selection Problem

1. Introduction

With the scale of sensor network becoming larger andlarger, the trust relationships between nodes areaccordingly becoming more complex and difficult tounderstand. As a result, the sink nodes can hardly find outthe optimal routing path. However there can be over tensof thousands of trust paths between sink node (so-calledtrustor) and its target node (data collecting sensor node,trustee) in large-scale networks [1].Then the problem ishow to find out an optimal trust path to get the trustworthytarget. Considering that trust is a complex conceptdepending on lots of complex aspects it may concerns,such optimal Trust Path Selection problem (TPS problem)is typically a Multi-Constrained Optimal Path (MCOP)selection problem, which is NP-Complete[2].

Some arts made some efforts in routing scheme ofcommunication network[3] After identifying that TPSproblem is a MCOP selection problem and thusNP-Complete, Korkmaz and Krunz[2] introduced a

corresponding solution, a heuristic algorithm (H MCOP),which was an attempt to minimize both the nonlinear costfunction (for the feasibility part) and the primary costfunction (for the optimality part). However, such analgorithm requires the state information to be accurate,not necessarily true in the real world. Li et al.[4]advanced a composite service selection algorithm basedon trust evaluation of Bayesian inference, and a MonteCarlo method (QoS constrained) based trust-orientedcomposite service selection algorithm was proposed. Theoptimal strategies in their later work[5] were advanced toimprove the efficiency of the algorithm. Liu et al.[5]considered the above mentioned aspects thoroughly andproposed an efficient approximation algorithmMONTE K. This algorithm was based on their newcomplex social network structure and a new concept ofQuality of Trust (QoT), which illustrated the ability toguarantee a certain level of trustworthiness in trustevaluation. They considered three QoT attributes, which

∗ Corresponding author e-mail: [email protected]⃝ 2013 NSP

Natural Sciences Publishing Cor.

162 G. Xu, X. Tian, X. Cao, X. Li, Z. Feng: TPS DR: A Universal Dimension Reducing...

however are not sufficient or accurate in describing such acomplex concept as trust. Furthermore, they have notconsidered the correlation between the QoT attributes,and the utility function is too simple to embrace all QoTattributes. In the work[6], the path with the maximal trustvalue was selected as the most trustworthy social trustpath. Unfortunately, some important aspects between theadjacent trust nodes and the recommendation roles of aparticipant have a significant influence on trustpropagation[7,8]. But these factors have not beenconsidered in existing social TPS solutions.

After all, trust cannot be described or evaluated byconsidering only several aspects. Furthermore, scientistsfrom multi-discipline have different viewpoints about theconcept of trust and the influence factors of trust[9].Practically, the limited number of influence factorsconsidered in the existing algorithms[2,5,11–15] can notdeliver enough performance. The probable solution is toconsider as many influence factors as possible to get theapproximate solution.

In this paper, considering the complexity of trust andits propagation[10], we will first introduce a concept ofcomplex trust network which contains the complex trustinformation, trust relationships and other importantaspects of trust and reputation. We then propose a modelto represent the trust network based on high dimensionalvector and matrix, finally the TPS problem is modeled asMulti-Constrained Optimal Path (MCOP) selectionproblem, which is a NP-Complete problem[2]. Therefore,we propose an approximate algorithm TPS DR, TrustPath Selection based on Dimensionality Reduction, tosimplify the trust network through reducing thedimensionality of the trust vectors so as to cluster somesimilar nodes into a supernode. Our exampledemonstrates the usage and advantage of our model.

2. Complex Trust Network

As for selecting the optimal trust path for sink node, thecritical problem is to construct a feasible trust network,which contains the needed trust information whilecontains the least unrelated information. In existing trustnetwork, nearly most researchers have considered onlyseveral aspects about trust to simply the process of trustpath selection. However, as we have argued in theprevious section, considering the reasonability andaccuracy, it is inappropriate to consider only partialinfluence factors when we solve such a problem as TPS.

In this paper, we will not consider the specific trustattributes but try to build up a prototype which containsenough attributes as demanded in various circumstances.That is to say, we aim at construct such a trust network tomodel a complex sensor network. Each trust nodecontains enough attributes and they are interconnected bysome trust relationships. For example, in Fig. 1, sourceand target participants, named trustor and trustee, containtens of thousands of attributes, while the trust

relationships are represented by the arrows between thetrust nodes. It should be noted that here we just give outthe prototype or template of the trust network, withoutconsidering the specific attributes and relationships. Inone case, if we regard trustor as an employer who isseeking a potential employee in the large scale socialnetwork, trustor can evaluate the trustworthiness oftrustee along some trust path selected by some strategies.In another case, we can regard trustor as one of someother participants, such as a traveler who is seeking atrustworthy travel route in the complex travel network. Ina word, our trust network can represent a lot of realapplications, without the requirement of considering thespecific attributes for each node or the relationshipsamong nodes. In this way, we can construct a complextrust network to describe the complex and uncertainrelationships between nodes in the online social network.For example, in Fig. 2, trustor S1 tries to select a trustpath to get trustee T1. It is known that, for a trust networkwith n nodes, there are at most N paths:

N = 1+C1n−2 +2!×C2

n−2 +3!×C3n−2 + · · ·+

(n−3)!×Cn−3n−2 +(n−2)!×Cn−2

n−2 ≥ 2n−2 (1)

If n=100, then N ≥ 298, which is a quite large numberthat we can hardly examine all solution space quickly.Therefore, to solve such a solution explosion problem, wemust reduce the solution space to an acceptable size.

2.1. Trust Vector Representation

As we mentioned above, trust is a complex conceptinfluenced by dozens of aspects. General solutions tried tofind out the key influence factors, however it is notenough and also inconsistent with the facts in most cases.In this paper, we will not care about what on earth theinfluence factors are, and the only thing we want to do isdescribing each trust node with a high dimensional vectorA:

A = [X1,X2 . . . ,Xn]T (2)

Where [.]T denotes the transpose of the vector,X1, . . . ,Xn are the values of all complete trust attributesinvolved. In this way, we can construct a matrix N torepresent the complex trust network:

N =

T11 . . . T1n...

. . ....

Tm1 · · · Tmn

(3)

Here Ti j is the trust relationship vector between trustnodes i and j, if i ̸= j, Ti j stands for the trust degree thattrustor i transfer to trustee j, otherwise Ti j is theself-confidence of trust node i. Please note that thedifference between two concepts of trust vector of a node(A) in Equation (2) and trust relationship between nodes(Ti j) in Equation (3).


Appl. Math. Inf. Sci. 7, No. 1L, 161-167 (2013) / www.naturalspublishing.com/Journals.asp 163

Figure 1 The structure of our trust network.

2.2. The Similarity of Trust Nodes

In order to simplify the complex trust network, we needto cluster the similar trust nodes into a collective (orsupernode). The first question remains: how do weassemble the similar trust nodes into a supernode? Themethod we here use is through judging the similaritydegree of trust nodes. If the similarity is high, then suchnodes can be clustered into a collective, such is thesuper-node. Then how to compute the degree of similarityof each trust node? In this paper, we judge the similarityby computing the dot product of two nodes vectors, if thevalue is close to one, then the degree of similarity of suchtwo nodes is high; otherwise the degree is low. Pleasenote that all trust vectors should be normalized beforethey are computed by dot product. For example, there aretwo trust nodes represented by the corresponding vectorsA and B:

A = [X1,X2 . . . ,Xn]

B = [Y1,Y2 . . . ,Yn]

, then normalize them by:

A′ =[X1,X2 . . . ,Xn]

|A|=

[X1,X2 . . . ,Xn]√X2

1 +X22 + · · ·+X2

n

(4)

B′ =[Y1,Y2 . . . ,Yn]√

Y 21 +Y 2

2 + · · ·+Y 2n

(5)

So the dot product (similarity degree SAB) is computed asfollows:

SAB = A′ •B′ =X1Y1+X2Y2+···+XnYn√(√

X21 +X2

2 +···+X2n

)(√Y 2

1 +Y 22 +···+Y 2

n

) (6)

Definition 1 (similar node) If the dot product of twotrust vectors is close to 1, then the two trust nodes (A andB) are called similar nodes, one is the similar node of theother, that is: A ∼= B , here ∼= is symbol of similarity.

Of course, the threshold value is preferred by differentparticipants. If two trust nodes are similar, then they havealso similar or congenial values of trust. In the process of

Figure 2 An example of complex trust network

trust path selection, if there is no equivalent node toreplace some selected node, then similar node can bereplaced approximately.

Definition 2 (the matrix of similarity) For thecomplex trust network, we use the matrix of similarity SNto describe its similarities between all nodes, that is:

SN =

S11 . . . S1n...

. . ....

Sm1 · · · Smn

(7)

Here Si j is the similarity of trust nodes i and j (i ̸= j),whose value is decided by Equation (6). Obviously, Si j = 1if condition i = j is met, this is a matter of course. ThenEquation (7) is modified as:

SN =

1 . . . S1n...

. . ....

Sm1 · · · Smn

(8)

Furthermore, considering it is such an undirectedrelationship as similarity, we believe that:

Si j = S ji (9)

Therefore, if SN is a square matrix (m = n), it is asymmetric matrix, then we can only consider the left orright lower triangular equivalent form.

2.3. The Equivalence of Trust Node

Besides the relationship of similarity, there is anotherrelationship of equivalence. These two concepts are



related tightly, similar nodes can be equivalent if somepre-conditions are met, and on the contrary equivalentnodes are surely similar.

Definition 3 (equivalent node) If the degree ofsimilarity of two trust nodes (A and B) is high enough,then such two trust nodes can be regarded as equivalentnodes, one is the equivalent node of the other, that is:A ≡ B, here ≡ is symbol of equivalence.

Some pessimistic trustors can sometimes believe thattwo nodes are equivalent only if the value of similarity ishigher than 0.9, while the optimistic ones may assume thethreshold value is 0.6. That is to say, the threshold value isvague and different between various trustdecision-makers (trustors). We define the equivalence oftwo trust nodes as such: if two nodes are equivalent, thenthe value of equivalence is one, otherwise zero, that is:

EAB =

{1, A ≡ B0, otherwise (10)

Definition 4 (the matrix of equivalence) we use thematrix of equivalence EN to describe its similaritiesbetween all nodes,

EN =

E11 . . . E1n...

. . ....

Em1 · · · Emn

(11)

Here Ei j is the equivalence of trust nodes i and j, whosevalue is decided by Equation (10). Obviously, Ei j = 1 (i =j), and Ei j = E ji. Then Equation (11) is modified as:

EN =

1 . . . E1n...

. . ....

Em1 · · · Emn

(12)

Similarly, if EN is a square matrix (m = n), it is asymmetric matrix, then we can only consider the left orright lower triangular equivalent form.

Generally, if the matrix of similarity is given, then wecan derive the matrix of equivalence by fuzzy inferringmethods, surely the results can be influenced greatly byselecting different membership functions.

3. An Example

For another example as Fig. 3 shows, there are two trustnodes A and B denoted by (for simplicity, we assume thereare only ten kinds of influence factors) in the trust networkshown as Fig. 3.

A = [0.6,0.4,0.7,0.7,0.2,0.8,0.9,0.6,0.5,0.7]

B = [0.3,0.4,0.6,0.5,0.9,0.4,0.7,0.8,0.2,0.8]

First we can get |A| |B| ≈ 2.02∗1.91 = 3.8582, the dotproduct is:

SAB = A′ •B′ =3.38

3.8582≈ 0.8761

Similarly, if there are four trust nodes in a trust networkand the other values of similarity are respectively given asFig. 3, then the matrix of similarity SN is:

SN =

SAA SAB SAC SADSBA SBB SBC SBDSCA SCB SCC SCDSDA SDB SDC SDD

=

1 0.8761 SAC SADSBA 1 SBC SBDSCA SCB 1 SCDSDA SDB SDC 1

=

1 0.8761 0.8873 0.78960.8761 1 0.2453 0.65430.8873 0.2453 1 0.92350.7896 0.6543 0.9235 1

If we set the threshold value of equivalence is 0.8,

namely, our fuzzy inferring rule is: if the value ofsimilarity is equal to or higher than 0.8, then such twonodes are equivalent. Thus, we can get the matrix ofequivalence EN as follows:

EN =

EAA EAB EAC EADEBA EBB EBC EBDECA ECB ECC ECDEDA EDB EDC EDD

=

1 1 1 01 1 0 01 0 1 10 0 1 1

Obviously, we can see that all nodes are equivalent,

and then such a trust network can be looked as a super-node.

Considering the fact that there is different personalpreference between various trustors in differentcircumstances, it depends on personal preference in aspecific case that whether such two nodes be clusteredinto a collective (super-node) or not.

4. TPS DR: Optimal Trust Path SelectionAlgorithm

We design our algorithm on condition that the followingproposition is true, which is about the judgment ofequivalent node.

Proposition 1 (judgment of equivalence) If twonodes are equivalent, then their previous nodes and nextnodes are equivalent respectively, and vice versa. Notethat similarity is directed while equivalence is undirected,and we adopt trust node-pair graph method to judge theequivalence just like using state-pair graph in timedcircuit.

Our algorithm studies from the related principle indigital logics, which is used to delete the extra states instates graph. The algorithm can be described as follows.Step 1 Define related variables.

Related variables include: vector representation oftrust relationship-Ti j, similarity degree/matrix (Si j/SN),



Figure 3 an example of trust network with four nodes

equivalence degree/matrix (Ei j/EN]), and threshold valueof similarity (ToS), Complex Trust Networkrepresentation (CTN).Step 2 Initialize and assign for equivalence matrix.

a) Set a threshold value of similarity, ToS, any twonodes with greater value than which are regarded asequivalence, otherwise is inequivalent.

b) If two nodes are equivalent, Ei j=1; otherwise Ei j=0.Step 3 Judgment for equivalent nodes.

Here we exercise trust node-pair graph method.Considering the scale is often too large, we will divide theCTN into a fit number of sub-networks. The detailedflows are:

a) Construct part trust node-pair from the similaritymatrix, since the scale is large enough;

b) For each given node-pair till all node-pairs arecontained, considering the equivalence for all previousnodes and next nodes, if yes, then we can make sure theyare equivalent, otherwise turn to d);

c) Identify the equivalent node-pairs, based on whichwe modify the equivalence matrix EN and renew it in therelated context.

d) Checking whether there are new equivalent node-pair, if yes, turn to b);

e) If no, stop.Step 4 Reducing method for CTN.

For all equivalent nodes, all but one should be deleted.The previous and next nodes for the deleted nodes can beforwarded to their equivalent node still alive in CTN. Inthis way, we have finished simplifying the CTN, which isespecially meaningful for large scale CTN, hencecontributes much for our TPS problem.Step 5 Trust Path Selection process.

This process is out of our discussion scope in thisstage, we will consider it in more detail in the near future.

a) Use k nearest neighbor algorithm, k-NN.

b) Compute the absolute value of trust vector Ti j, calledutility function, which is different from that of others inthat we will not consider the detailed designation of thisfunction but just give a universal vector representation.

c) Search the path forwardly, till we have found thetarget node-trustee, or when we have arrived at the seventhsearch.

This process is about trust path selection, which is ourfuture goal after giving the simplified trust network. Ouralgorithm is based on the small world theory, that is, thereis a maximum step-number between the source and thetarget, the number is six. Therefore we will stop oursearch process at the seventh step.

5. Conclusions and future work

In this paper, we addressed the TPS problem for sinknode in complex sensor network, which is aMulti-Constrained Optimal Path (MCOP) selectionproblem, hence a NP-complete problem. Comparing withother work, our work will not consider the detailed trustcomputing method but regard the trust relationship ofeach sensor node as a trust vector, which makes ituniversal in trust metrics. To solve TPS problem, ourwork aims at reducing the scale of the complex sensornetwork, which is transferred to the treatment with itscomplex trust network. As for the properties of softwaresystems, trustworthiness is critical [16], so our near futuretasks is to reduce the scale of trust network to get thecomplete solution for TPS problem.

Our more future work includes adopting fuzzyreasoning to map the similarity matrix into equivalencematrix, applying various utility functions to verify theadvantage of our algorithm. Even more importantly, moreexamples should be taken to exemplify the usage of ouralgorithm and experiments should be made to comparethe other work.

6. Acknowledgements

The authors would like to thank the editors andanonymous referees for their suggestions and theremarkable improvements they brought to this paper. Thispaper has been supported by the National Natural ScienceFoundation of China (No. 61003080, 61070202), theOpening Project of Shanghai Key Laboratory of IntegrateAdministration Technologies for Information Security,Tianjin Research Program of Application Foundation andAdvanced Technology (No.10JCZDJC15700) and theprogram of the 985 project of Tianjin University.

References[1] J. Kunegis, A. Lommatzsch, and C. Bauckhang, The

Slashdot zoo: mining a social network with negative edges.In WWW-09 (2009), 741-750.



[2] T. Korkmaz, and M. Krunz, Multi-constrained optimal pathselection. In: INFOCOM-01 (2001), 834-843.

[3] Ahyoung Lee, Ilkyeun Ra, A Queuing Network ModelBased on Ad Hoc Routing Networks for MultimediaCommunications, APPLIED MATHEMATICS &INFORMATION SCIENCES, vol. 6, si: 1, JAN 2012,pp. 271-283.

[4] Lei Li, Yan Wang, and Ee-Peng Lim, Trust-OrientedComposite Service Selection with QoS Constraints, Journalof Universal Computer Science (2010), vol. 16, no. 13,1720-1744.

[5] Guanfeng Liu, Yan Wang and Mehmet Orgun, OptimalSocial Trust Path Selection in Complex Social Networks, theTwenty-Fourth AAAI Conference on Artificial Intelligence,Atlanta, Georgia, USA (2010), July 11-15, pp. 1391-1398.

[6] C. Hang, Y. Wang, and M. Singh, Operators for propagatingtrust and their evaluation in social networks, In: AAMAS-09(2009), 1025-1032.

[7] P. S. Adler, Market, hierarchy, and trust: The knowledgeeconomy and the future of capitalism, Organization Science(2001), 12(12): 215-234.

[8] R. Miller, D. Perlman, and S. Brehm, IntimateRelationships, McGraw-Hill College, 4th edition (2007).

[9] T3 Trust Theory. http://t3.istc.cnr.it/trustwiki/index.php/T3 Trust Theory (2011-5-12).

[10] J. Golbeck, and J. Hendler, Inferring trust relationships inweb-based social networks, ACM Transactions on InternetTechnology (2006), 6(4): 497-529.

[11] Annika Hinze and Qiu Quan, Trust- and Location-Based Recommendations for Tourism, On the Move toMeaningful Internet Systems: OTM 2009, Lecture Notesin Computer Science (2009), Volume 5870/2009, 414-422,DOI: 10.1007/978-3-642-05148-7 31.

[12] Wei Guo, Ren-Zuo Xu, Bin Liu, Research on SubjectiveTrust Routing Algorithm for Mobile Ad Hoc Networks,2010 6th International Conference on WirelessCommunications Networking and Mobile Computing(WiCOM), Chengdu, 23-25 Sept. (2010), pp. 1-6.

[13] Fernando Kuipers, Piet Van Mieghem, TurgayKorkmaz,Marwan Krunz, An Overview of Constraint-Based Path Selection Algorithms for QoS Routing, IEEECommunications Magazine, December (2002), pp. 50-55.

[14] Cuirong Wang, Xiaozong Yang, and Yuan Gao, A RoutingProtocol Based on Trust for MANETs, In: H. Zhuge andG.C. Fox (Eds.): GCC 2005, LNCS 3795 (2005), pp. 959-964.

[15] T. Yu, Y. Zhang, and K.J. Lin, Efficient Algorithms for WebServices Selection with End-to-End QoS Constraints, ACMTransactions on the Web, Vol. 1, No. 1, Article 6 (2007).

[16] Yanguo Michael Liu, Issa Traore, Properties forSecurity Measures of Software Products, APPLIEDMATHEMATICS & INFORMATION SCIENCES, vol. 1,no. 2, MAY 2007, pp. 129-156.

Guangquan Xu is aPh.D. and associate professorat the School of ComputerScience and Technology,Tianjin University, China. Hereceived Ph.D. degree fromTianjin University in March2008. He is a member of theCCF and ACM. His researchinterests include verified

software, trusted computing, trust and reputation.

Xiuming Tian isa master student in computerscience and technology at theSchool of Computer Scienceand Technology, TianjinUniversity, whose researchinterests include securityand privacy, social network.

Xiaochun Cao iscurrently a professorof Computer Science at theSchool of Computer Scienceand Technology, TianjinUniversity. He received theB.E. and M.E. degrees bothin computer science fromBeihang University (BUAA),Beijing, China, and the Ph.D.

degree in computer science from the University of CentralFlorida, Orlando, USA. His research interests includeinformation security, computer vision.

Xiaohong Li is aprofessor of the Schoolof Computer Scienceand Technology, TianjinUniversity, China. Shereceived Ph.D. degreefrom the School of ComputerScience and Technology,Tianjin University. Herresearch interests lie in

trusted software, and information security.



Zhiyong Feng is aprofessor and associate deanof the School of ComputerScience and Technology,Tianjin University, China. Hereceived Ph.D. degree fromthe School of MechanicalEngineering, TianjinUniversity. His researchinterests lie in computer

integrated manufacturing system (CIMS), knowledgeengineering, distributed AI, pervasive computing andinformation security.


TPS DR: A Universal Dimension Reducing Algorithm for

Documents

Transcript of TPS DR: A Universal Dimension Reducing Algorithm for