Design of a Trust Model and Finding Key-Nodes in Rumor Spreading

7/27/2019 Design of a Trust Model and Finding Key-Nodes in Rumor Spreading

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Design of a TrustModel and Finding Key-Nodes in Rumor Spreading Basedon Monte-Carlo Method* Lina Wang*t, Yuntao Yue*,Chi G u o * ~ , Xiaoying Zhang*Computer School ofWuhan University, Wuhan, P.R.China, 430079

E-mail: guochi[at]mail.whu.edu.cntThe Key Laboratory ofAerospace Information Security and Trust Computing ofMinistry ofEducation, Wuhan University, Wuhan, P.R.China, 430079E-mail: lnwang[at]whu.edu.cn

AbstractThis paper combines the results of research onsocial psychology, and has designed a trust model for

rumor spreading. It is considered that wheninformation exchanges between people, the trust ofinformation is related to the interpersonal closeness. Inaddition, this paper uses Monte-Carlo method to findthe key source nodes in rumor spreading by comparingthe total number of spread nodes and spreading time.We find that the key nodes that impact the rumorspreading are not necessarily those with higher degreeor betweeness. Our work will provide algorithmsupport for maintaining the credibility of networkinformation and ensuring the security of networkinformation content.Keywords: trust model, rumor spreading, key nodes,Monte-Carlo method1. Introduction

With the development of computer networks,communication between people is more and moreconvenient, and the spread of public opinion oncomputer networks has a bigger impact to the wholesociety [1]. In such a computer-based public opinionspread network (such as E-mail network, immediatemessage network, etc.), the function and influence ofeach node is different from each other [2]. Some nodesare key nodes in the spread of public opinion.Compared with ordinary nodes, they can spread publicopinion to more nodes in a short time. In order tomaintain the credibility of network information andensure the security of network information content, itis crucial to fmd out where these key nodes are.

Corresponding author978-1-4244-5113-5/09/$25.002009 IEEE

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The spread of public opinion based on computernetworks has the following characteristics: (1) Thespread network takes on the small world characteristics.Some studies showed that computer communicationnetworks have the small world characteristics, i.e, theyhave a shorter average path length and at the same timehave a higher clustering coefficient [3]. (2) The trustdegree between friends varies from each other. In thespread network of public opinion, the credibility ofpublic opinion content is closely related to its source.The closeness between friends is different, so the trustdegree between friends is also different, which impactsthe spread of public opinion (3) Everyone's personalityis different. The packets sending in computer networksare passive, but the spread of public opinion is active.Whether to spread or not depends on the personality ofeach individual. Some like to spread the news, whileothers are more cautious, who normally accept thenews passively [4]. Therefore, the spread of publicopinion is uncertain to some extent, and it can not besimply analyzed from the network topology. (4) Theinformation has the fresh feature. Public opinion,especially those spread on network tend to be fresh,that is, the attraction of public opinion to individuals isdecreasing as time passes. This phenomenon has animpact on the spread of public opinion.Therefore, the spread of public opinion has thefeature of randomness, uncertainty and dynamicity.The traditional researches on the spread of publicopinion usually analyzed the spread model of publicopinion and the t rend of spread speed [5][6]; there israrely a method about the excavation of network keynodes [7]. This paper presents a new method targetedat fmding out the key nodes.Because many characteristics of the spread ofpublic opinion are random, we adopt Monte Carlo


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2.2 The closeness between friends

in the real relationship network, and it can alsoexcavate the key nodes in the spread of public opinioneffectively and credibly. Each node in the networktopology represents an individual in the spreadnetwork.

The sociologists and psychologists have done alot of research on the closeness of interpersonalrelationships. lOS (Inclusion of Others in Self scale)Scale [9] is a good tool to measure the closenessrelationships . As shown in Figure 1, lOS scale isconstructed by seven pairs of double circle withincreasing overlapping level, forming a seven-pointequidistant scale. The bigger the overlap between twocircles is, the closer the relation between self and the

m CD (])

sampling method. The principle is that the probabilityof an event can be estimated by its frequency ofoccurrence in a large number of trials; when the samplesize is large enough, the frequency can be taken as itsprobability of occurrence [7]. We obtain theexpectation of the performance of public opinionspread through a large number of simulationexperiments samples.The spread of public opinion is related to thescope of social psychology to some extent. The relatedresearch has given us much inspiration, particularlywhen analyzing the closeness of interpersonalrelationships. In the following, we will present amathematical model portraying the humanrelationships.This paper is organized as follows: in the secondsection, we talk about the design of trust model inpublic opinion spread; then we discuss the spreadsimulation algorithm design based on Monte Carlomethod; after that, the data of simulation experimentsis analyzed; and finally, a brief summary is given interms of our works and we point out the future work. 2 3

Figure 2 Number distribution of friends belonging todifferent degree. After many experiments, the parameters ofthe f3 distribution is set to a = 3,b = 2 , the distributionmatches the results in [10] well.Assumption 1: In order to explore the effects ofcloseness on the spread of public opinion, we assign

4 5 6 7Figure 1 The relationship between lOS scale andinterpersonal closeness.I indicates the relationship is cold,and 7 indicates closed. The closeness is increasing from I to7. Social psychology research showed that thedistribution of each person's friends belonging to theseseven closeness degrees is non-uniform [IO].Thecloseness distribution can be taken as a f3( a,b)distribution similarly, as shown in Figure 2.

I"

a- 3, b- 2

I~ 2 -

2.1 A carrier network of public opinion

2. Trust model design

Newman et al. have done a wide range ofempirical research on a large number of actual networktopology characteristics [8]. They pointed out that therelationship network depended on computer networkhas an obvious small-world effect, such as in the email network, the average degree (k) = 3.38 , theaverage path lengthL = 5.22, the clustering coefficientC = 0.13 .This paper adopts the WS small worldnetwork as the spread carrier, whose network topologyfeature is similar to the email network. The WS smallworld network model is proposed by Watts and Strogtzin 1998. It is constructed by re-connecting edgesrandomly by the probability of p in a regular graph.The WS small-world network has short average pathlength as well as high clustering coefficient.Using the WS small-world network as a spreadcarrier can simulate the spread of public opinion better

The model includes four parts: (1) A public opinionspread carrier network with small-world characteristics;(2) The model of closeness between friends, whichdetermines that the same message brings out differentinfluence when spread by different friends; (3) Thepreferred model of initiative spread of public opinion,it reflects whether the individual would like to spreadpublic opinion actively; (4) The attenuation of publicopinion.

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(1)

2.4 Fresh and trust attenuation of publicopinion

average weight (w) of the network. Then the meanvalue P_ avg of the normalized Poisson distributionshould be

the weight w i} on each edge ei } in the networktopology. w i} indicates the closeness of node i andnode j, the relationship will be closer when the weightis greater. We assign wij as 1-7 positive integerreferring to the lOS scale, the distribution of w.. in theIJentire network meets {3(3,2)

Assumption 2: In the process of public opinionspread, assume that node i transmits a message to nodej, then the trust to the message ofnodej is:

Wijm i j = ~L...J wi jj

When the node keeps receiving the same messagemany times, the trust to the message accumulatescontinually, which is also normal in the spread ofpublic opinion.

P _ avg = P _ maxi 2x(w) x(w)

=--=--=rpLWi X(w)so P _max = 2rp

The spread threshold of each individual is8i = Pr(A)k = Pr(A)'(P_max/max(Pr(A)))

= 2rpPr(A)max(Pr(A))

(2)

(3)

2.3 Design of spread critical value distributionIn the spread of public opinion, the personalcharacter is also a major factor. Some people like tospread the news while some people do not. We use a

threshold 8 to reflect this distinction. For an individuali, if its accumulated receiving trust to amessage M ji = L mji > 8i , we consider it chooses to

jspread this message to its own friends; otherwise, itdoes not.Considering those who love and those do not liketo spread the news very much at two extremes areminority, in our study, we use Poissondistribution Pr(A) to describe individual thresholddistribution of the entire network. Since in therelationship network, even those who do not like tospread the message very much will communicates withpeople sometimes, we normalize the common Poissondistr ibution to a set of real number n, and defme theminimum of n is 0 and the maximum is P_max ,P max l .

Definition 1: Consider the most commonsituation, those whose spread threshold is the meanvalue of the Poisson distribution should be actnormally in spread of messages, they should receive acertain amount of information and then spread out. Wedescribe this situation with a percentage rp, that is, ifeveryone has X friends, after he receives the messagesof x = X *rp friends, he will spread the messages out.

In addition, assume that these people are the mostcommon network nodes, that is the degrees of thesenodes equal to the average degree of the network (k) ,and the weights of edges connecting these nodes are

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Definition 2: As the public opinion has the featureof fresh, the attractiveness to the individual isdecreasing over time. We introduce an attenuationcoefficient y to describe this trend. In each shot, thecumulative trust M?) of each node which has receivedthe public opinion will attenuate:

M?) =M iU- 1) 'y,O < Y < 1 (4)3. The simulation of rumor spreadingbased on Monte-Carlo method

Defin it ion 3: For each node in the network, wedefme a state S to indicate whether the node hasreceived the information. If S = 1, we consider thenode has received the information, while S = 0 standfor the node has not received the information.Defin ition 4: For each node in the network, wedefme a state R to indicate whether the node hasspread the information. IfR = 1, we consider the nodehas spread the information, while R = 0 stand for thenode has not spread the information.Defini tion 5: For each node in the network, wedefme a neighbor set I ' , that is, if there is an edge eij ,then join nodej into set r i .Assume that information received from the firstnode is sf , and it will certainly spread the information.Then the simulation algorithm of rumor spreading isdescribed as below.1. Set value for each edge of the network, andmake sure the distr ibution of the weight of all edgesmeets Poisson distribution.


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4.2.1 Identify the key nodes4.2 Experimental results

4.2.2 The impact of the critical value distributionparameter on the spread of public opinion

,

When the public opinion is spread from aparticular node i, we use the total number of nodesN received from the public opinion in the network andthe spread termination time t to describe the spreadperformance of the node. We need to fmd out the keynodes influencing the public opinion spread, whichsatisfy the bigger N and smaller t relatively.

Figure 3 Comparison of the N - t performance of allnodes in the network. The horizontal coordinate is time t ,thevertical coordinate is nodes number N , each node ismarked with its number order, and the parameter is set asA=5,ffJ = 0.3,y =0.9 .Figure 3 is the simulation results for the experiment.We can see the spread total number of nodes Nand

the spread time t vary from different network. Thenodes covered by the solid line in Fig.3 can beconsidered as the key nodes in the spread network,such as No.94,98,99.

2. Set spreading threshold for each node in thenetwork according to formula (3), and set Sand R toofor each node.3. Define two node sets: A,B4. Set time t = 0, and the total number of the

nodes which has received the rumor Num = I5. Set Sst = 1,Rst = 1, and join st into set A6. Get element i from set A7. Get elementj from set I',8. If R; = 0, then update the trust degree m; of

node i, according to formula (1).9. Set S; = 110. If mi > 0; and R; = 0, then join j into set B,

and set R ;= 1,Num = Num +1.11. If T, *0 , then tum to step 712. If A * 0 , then tum to step 613.Set A = B,B = 014. Attenuate the trust degree of the node whoseS = 0 according to formula (4)15. Set t=t+l .16. If the number of the nodes whose S = 0 is 0,

or A = 0 , then quit.17. Otherwise, tum to step 6In this way, after a simulation of the spreading,we would have the total number Num of nodes that

can get the rumor and the spread time t when therumor is spread form node st . We hope to find somekey nodes, which the public opinion can be spread tomore nodes in less time.We use Monte-Carlo method to simulate thespreading that begins from all the nodes respectivelyfor many times, and use the average of the totalnumber Num and the time t as the desired outcomes.

Figure 5 shows the impact of the changingparameter Aof Poisson distribution on the spread.We can see that as A decreases, the public opinion

can be spread to more nodes. As shown in Figure 4,this is because when A decreases, the peaks of Poissondistribution become higher, and less nodes have a highspread threshold. In this way, it is easier for the publicopinion to spread in the network.

4. Data analysis of the simulation result4.1 Parameter description

We use the topology generators, set the re-connectprobability p = 0.08 , and generate network topology ofWS small-world with 100 nodes. The topologycharacteristics are as below: the average degree(k ) =3.35 , the average path length L =5.13 , theclustering coefficient C = 0.14 , which is basicallysimilar to the real email network in [8].In our experiment, the parameters of f3distribution is fixed asa =3,b=2 . We discuss the keyfactor impacting the spread of public opinion and fmdout the key nodes by changing other parameters. 5 '0

o A - Ioo ).. _ 10

15 20Figure 4 Poisson distribution[ll]

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public opinion, so it increases difficulties for publicopinion to spread in the network.4.2.4 The impact of attenuation coefficient on thespread of public opinion

- - : - - - ' ' ' '- - -(a)

Figure 7 shows the impact of the change ofattenuation coefficient r on the spread of publicopinion.

" .- -: - - l- -: - . - :-- i' o - - '\---I:I(b)Figure 5 The impact of Poisson distribution on thespread of public opinion. Parameter settings:

rp = 0.3, r = 0.9 (a) A. = 3 , b) A. = 7 .4.2.3 The impact of control parameter rp on thespread of public opinion

' _ ' .'- - ' - - io- - ' ,-- - ,' - - ,',- - ..(a)

."".. . :0_ .,"l:.- .. . , "' ,! :, -~ ~ ~ =. ....._ ~.-(b)Figure 7 The impact of the change of attenuation

coefficient r on the spread of public opinion,Parameter settings: A. = 5, rp = 0.3 (a) r= 0,7 ,(b) r = 0.5 .

We can see that the smaller the attenuationcoefficientr is, the fewer nodes the public opinion canspread to. This is also consistent with the definition ofattenuation coefficient, if the trust of individual to thepublic opinion declines faster, the public opinion ismore difficult to spread in the network.

Figure 6 shows the impact of the change of controlparameter rp on the spread of public opinion

-,(a)

-. ,.,

Table 1 The characteristics of the networktopologyofke d

4.2.5 The characteristics of the network topology ofkey nodesIn the previous study, as the experimentparameters change, the total spread nodes number Nand the spread time t of a single node also change.However, the key nodes remains fixed, which provesthat our method is universal.

no esNode degree Order of betweeness Order ofdegree betweeness39 8 7 528 3677 8 7 1412 798 12 1 2342 199 10 2 1633 3

' - -' - - ' - -111 .. . 7(b)Figure 6 The impact of the change of control

parameter rp on the spread o f public opinion.Parameter settings: A. = 5, r = 0.9 (a) rp = 0.2 , (b)rp = 0.4

As shown, with the increase of o , the publicopinion can be transmitted to fewer and fewer nodes.This is because when rp becomes greater, the nodeswill need to receive more times before spreading the

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As shown in table 1, we sort all nodes in thenetwork topology according to degree and betweeness,we fmd that the degree and betweeness of those keynodes are not the highest. This shows that the keynodes which influence the public opinion spread arenot entirely dependent on the static network topologycharacteristics, which means our approach has certainadvantages compared to the traditional method infmding out the key nodes based on network topology.5. ConclusionThis paper proposes a method to fmd the keynodes in the rumor spreading of computer networks,We fmd that the key nodes that impact the rumorspreading are not all have higher degree orbetweeness. Our approach combines the research ofsocial psychology, and designs a trust model forrumor spreading. We discuss the impact of eachcontrol parameters to the rumor spreading. Next, Wewill research the game condition of both side ofrumor spread in networks when the negative publicopinion is led into considered.6. AcknowledgmentThis work was funded by the major project ofChinese National Natural Science Foundation(90718006), the Chinese National Natural ScienceFoundation of China (60743003), and the Ph.D.Programs Foundation of Ministry of Education ofChina (20070486107).

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7. Reference[1]. Xide Qin, Dejiang Rao. "Public Opinion Course".Wuhan: Wuhan University Press, 1998.[2]. Junyan Su, Lingjiang Kong, Muren liu. "OpinionDynamic Model Based on Weighted Networks". Journal 0/Guangxi Normal University, 2006,24(2), 1-4.[3]. Watts, D.J, Strogatz, S.H. "Collective dynamics of'small-world' networks". Nature, 1998,393 (6684): 409-10[4]. QingfengWu, Lingjiang Kong, Muren Liu. "Influenceof person's character upon the evolution of the cellularautomatamodel for public opinion".Journal ofGuanxiNormal University.2004,22(4),5-9[5]. Zhaofeng Pan, Xiaofan Wang, Xiang Li. "SimulationInvestigation on Rumor Spreading on Scale-freeNetworkwith Tunable Clustering".Journal ofSystemSimulation,2006,18(8),2346--2348[6]. Changyu Liu, Xiaofeng Hu. "Public OpinionPropagation Model Based on Small World Networks".Journal ofSystem Simulation.2006,18(12),3608-3610[7]. E. Zio, C. Rocco, "Security Assessment in ComplexNetworks Exposed to Terrorist Hazard: A Methodologicalapproach based on Cellular Automata and Monte CarloSimulation" Complex Network and Infrastructure Protection(CNIP 2006),2006, Rome, Italy[8]. Newman M E 1."The structure and function of complexnetworks". SIAMReview,2003,45:167--256[9]. Aron A, Aron EN, Smollan D. "Inclusion of others inthe Self Scale and the structure of interpersonal closeness".Journal 0/Personality and Social Psychology, 1992,63:596612[10]. Li H Z. "Culture, gender and self-close-other(s)connectedness in Canadian and Chinese samples". EuropeanJournal ofSocial Psychology, 2002,32(1): 93--104[11]. Poisson distribution. Wikipedia:http://en.wikipedia.org/wiki/Poisson_distribution

Design of a Trust Model and Finding Key-Nodes in Rumor Spreading

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