Physics Letters A 368 (2007) 458–463

www.elsevier.com/locate/pla

Influence of network structure on rumor propagation

Jie Zhou a, Zonghua Liu a,b,∗, Baowen Li b,c,a

a Institute of Theoretical Physics and Department of Physics, East China Normal University, Shanghai 200062, People’s Republic of Chinab Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, 117542-46 Singapore, Republic of Singapore

c NUS Graduate School for Integrative Sciences and Engineering, Singapore 117597, Republic of Singapore

Received 11 January 2007; accepted 11 January 2007

Available online 24 April 2007

Communicated by A.R. Bishop

Abstract

Rumor propagation in complex networks is studied analytically and numerically by using the SIR model. Analytically, a mean-field theory isworked out by considering the influence of network topological structure and the unequal footings of neighbors of an infected node in propagatingthe rumor. It is found that the final infected density of population with degree k is ρ(k) = 1− exp(−αk), where α is a parameter related to networkstructure. The number of the total final infected nodes depends on the network topological structure and will decrease when the structure changesfrom random to scale-free network. Numerical simulations confirm the theoretical predictions.© 2007 Elsevier B.V. All rights reserved.

PACS: 87.23.Ge; 89.75.Hc; 05.10.-a

1. Introduction

Complex networks have attracted an increasing attention inthe past few years [1–3]. It is found that most of the realis-tic networks such as the Internet, WWW, and biological net-works, etc., can be classified as either small world network orscale-free network. In order to understand the mechanism ofnetwork formation, Barabasi and Albert has proposed a model(BA model) to describe the preferential attachment [4]. Lateron, many modified models have been proposed to stress otheraspects of network growing [5–10]. Based on these models, thedynamical processes on networks have been studied. It is foundthat the network structure, such as the degree distribution, theclustering coefficient, and the assortativity, etc., can greatly af-fect its dynamics [1–3].

An interesting dynamical process on complex network is theepidemic spreading, which can be mainly characterized by twomodels. The first model is the two-state susceptible-infected-

* Corresponding author at: Institute of Theoretical Physics and Department ofPhysics, East China Normal University, Shanghai 200062, People’s Republic ofChina.

E-mail address: zonghualiu72@yahoo.com (Z. Liu).

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.01.094

susceptible (SIS) model [11–15]. This model describes sucha phenomenon that a susceptible node can become infectedand an infected node can recover and return to the suscepti-ble state, such as the computer virus and gonorrhea etc. Thesecond model is the three-state susceptible-infected-refractory(SIR) model which is well-known in mathematic epidemiol-ogy [16–22]. This model describes the phenomenon that theinfected nodes will become immunized or dead, such as theparotitis, the measles, SARS, and the influenza etc. The SIRmodel is different from the SIS model in the fact that the in-fected nodes will not return to the susceptible status but becomerefractory status. At a given time, each node in the network is inone of these three states. The epidemic spreading on complexnetworks have also been discussed on other models [23,24],such as on the SI model [25,26] and on the SIRS model [27].

Another interesting issue of the dynamical process is the ru-mor propagation. The SIR model can be also used to describethe rumor propagation on complex networks [28–31]. Supposethere is a rumor or news in the network. The person (node)who has heard it and wishes to spread it is in the infected sta-tus, the one who has not heard it is in the susceptible status,and the one who has heard it but is no longer interested inspreading it is in the refractory status. In a detailed spreading

J. Zhou et al. / Physics Letters A 368 (2007) 458–463 459

process on network, the propagation can be implemented byrandomly choosing a node to hold the rumor at the beginningand assuming only the neighbors of the node with the rumorhave a chance/possibility to contact the rumor. As time goeson, the rumor can infect the susceptible nodes that are con-nected to the node with the rumor and make them infected. Thepropagation process is over when there is no infected node inthe network. This problem was originally addressed by Sud-bury [28] and recently investigated by Zanette [29] and Liuet al. [30]. Sudbury’s case is equivalent to rumor spreadingon a complete random network, i.e., a homogeneous network.Sudbury find that the rumor can only be propagated to 80% pop-ulation [28]. Zanette studies the rumor propagation on smallworld networks by a mean-field equation nS = −nSnI /N ,nI = nSnI /N − nI (nI + nR)/N , and nR = nI (nI + nR)/N ,where N is the total population, nS(t), nI (t), and nR(t) are thenumbers of susceptible, infective, and refractory at time t , re-spectively. Zanette find that the percentage of nodes who havechance to hear the rumor is less than 80% [29]. Liu et al. [30]study the case of a general network and find that the finalpercentage of population who heard the rumor decreases witha network structure parameter p.

The main difference between Zanette’s and Liu’s worksis that the later considers the influence of the heterogeneousstructure. For a heterogeneous network, there can be a smallsubset of nodes with relatively large numbers of links. In anepidemic spreading process in heterogeneous networks, it ismore likely for such a heavily linked node to be infected andthen to infect the nodes that are connected to it. As a re-sult, more nodes can become refractory through these heav-ily linked nodes. In Liu et al’s work, this feature is consid-ered by introducing a weight factor ε(t) > 0 to enhance thetransformation from infected to refractory. That is, in their ap-proach, they use the ε(t)nR(t) to replace the nR(t). Hence, themean-field equation becomes nS = −nI (t)[1 − (ε(t)nR(t) +nI (t))/N ], nI = nI (t)[1 − 2(ε(t)nR(t)+nI (t))/N ], and nR =nI (t)(ε(t)nR(t)+nI (t))/N . The results obtained from this ap-proach depend on the weight factor ε(t). However, it is difficultto show how ε(t) depends on the topological structure of net-work in detail. Therefore the underlying mechanism of hownetwork structure influences the rumor spreading is still an openquestion.

In this Letter, in order to understand how the network struc-ture, such as the degree distribution, etc., influences the finalinfected population, we study the rumor propagation on hetero-geneous networks. In our approach, instead of introducing thefactor ε(t) to represent the influence of heterogeneous structureto the rumor propagation, we directly incorporate the degreedistribution into the SIR model by dividing the population intosubgroups with the same k and discussing the interactions be-tween different groups, and distinguish the unequal footingsof the neighbors of an infected node. Another feature of thisapproach is that, instead of studying the total final infected pop-ulation of the whole network, we investigate the dependenceof the final infected population on degree k. We find that thepercentage of the final infected people in those nodes with de-gree k increases nonlinearly with k. The total final infected

nodes can be simply expressed as a function of the percent-age of the infected people with degree k. It increases with thenetwork structure parameter p. Notice that the limit 80% inSudbury’s case comes from the complete random spreading,which is equivalent to a homogeneous network. However, therandom case of the general network with p = 0 is a randomgrowing network and cannot be considered as a rigorous ho-mogeneous network [32]. For implementing Sudbury’s case bynetwork, we construct a homogeneous network with the samedegree at each node and study the rumor propagation on it. Ourresults show that the rumor can be propagated to more peo-ple in the random growing network than that in the scale-freenetwork, and the homogeneous network is the best structure totransmit information and it can reach the limit of 80%. Com-puter simulations confirm all the theoretical predictions.

The Letter is organized as follows. In Section 2, we studythe rumor propagation on heterogeneous networks and presenta mean-field theory by taking into account the network struc-ture. In Section 3, we construct a homogeneous network withthe same degree at each node to study Sudbury’s case. In Sec-tion 4, we perform computer simulations to confirm the predic-tions given in Sections 2 and 3. The conclusions are given inSection 5.

2. Rumor propagation on heterogeneous networks

The principle of SIR model for rumor propagation on com-plex networks is as follows. Consider a population of N lo-cated at a complex network with N nodes where each nodesrepresents an individual. The interaction between individualscan only occur through the links. At each time step, eachnode/individual adopts one of three possible states: susceptible,infected, and refractory. The rumor can propagate from everyinfected node to one of its neighbors. For an infected node, therumor can be transmitted only to one of its (randomly chosen)neighbors at each time step. If the chosen node is in the suscep-tible status, it will become the infected status. If, on the otherhand, the chosen node is in the status of infected or refractory,the original node will lose its interest in the rumor and becomerefractory. Suppose, initially, one node is infected, acting as the“seed”, and all the other nodes are susceptible. As time goeson, more and more nodes become infected. After a certain time,there are not enough susceptible nodes to be infected, then thenumber of infected nodes will decrease. The spreading processcomes to the end when the number of infected nodes becomeszero. During this process, the susceptible nodes monotonouslydecrease and the refractory nodes monotonously increase.

The previous works focus on the evolution of the total num-ber of infected nodes nR(t) [28–30]. It is difficult to incorporateexplicitly the network structure such as the degree distributioninto the mean-field equation. Considering the fact that the in-fected nodes may have different degrees, which is also true forthe susceptible and refractory nodes, we take an alternative ap-proach, namely, we study the evolution of those nodes with thesame degree k. Suppose the number of nodes with the samedegree k is Nk , we have Nk = P(k)N where P(k) denotes thedegree distribution. Let nk,S(t), nk,I (t), nk,R(t) be the numbers

460 J. Zhou et al. / Physics Letters A 368 (2007) 458–463

Fig. 1. Schematic illustration of the process of rumor transmitting: Supposethere is a link between node A and node B . At time t , node A gets the rumorand transmits it to node B . And then at time t + 1, node B will transmit therumor to one of its neighbors which includes node A. The key point is that thepossibility for node B to become the refractory is unity when B chooses A asits target and less than unity when B chooses one of the other neighbors of B

as its target.

of the susceptible, infected, and refractory nodes with degree k

at time t , respectively. We have the conservation equation

(1)Nk = nk,S(t) + nk,I (t) + nk,R(t).

Here in order to study the influence of network structure tothe infection process, we focus on the evolution of the sub-group where the degree distribution P(k) is very important inobtaining the expressions of nk,S , nk,I , and nk,R . For exam-ple, consider the evolution of nk,S(t). It may be infected notonly by the nk,I (t) but also by other nk′,I (t) with k′ �= k. Theinfluence from nk′,I (t) is directly related with the degree dis-tribution P(k) and its detailed form will be given in Eq. (2).On the other hand, the previous works on rumor propagationconsider all neighbors of an infected node on the equal footing,namely, any one of them has the same probability to becomeinfected or refractory when it is chosen as the receiver of infor-mation. However, this does not reflect the realistic situation ofrumor spreading where the neighbors may have different prob-abilities to become infected or refractory when they are chosenby the infected node. This can be explained with the help ofFig. 1. Consider two neighboring nodes A and B that are con-nected by a link in the network. Suppose node A is infected andtransmits the rumor to node B at time t . Then at time t +1, nodeB will choose one of its neighbors as the target to transmit therumor, which include node A. As A is the “father” of B , A willnot be at the same footing with the other neighbors of B . WhenA is chosen from the neighbors of B , B will become the refrac-tory with the probability of unity according to the propagationrules. When one of the other neighbors of B , except A, is cho-sen, B becomes the refractory or remain as infected, dependingon the status of the chosen node. That is, the probability forB to become refractory is less than unity. If the degree of B

is k, the probability for choosing A is 1/k and the probabilityof choosing the others is 1 − 1/k. Combining this analysis andthe homogeneous mixing hypothesis, we obtain the evolutionequations of nk,S , nk,I , nk,R ,

nk,S(t + 1) = nk,S(t) −∑k′

nk′,I (t)

(1 − 1

k′

)P(k′|k)

nk,S(t)

Nk

,

nk,R(t + 1)

= nk,R(t) + nk,I (t)

(2)×[

1

k+

(1 − 1

k

)∑k′

P(k′|k)nk′,I (t) + nk′,R(t)

Nk′

],

where P(k′|k) is the conditional probability for a link whichrepresents the possibility for a node with degree k to connecta node with degree k′ and reflects the influence of networkstructure, and the parts nk,S(t)/Nk and (nk′,I (t)+nk′,R(t))/Nk′come from the homogeneous mixing hypothesis. nk,I (t +1) canbe obtained from the conservation Eq. (1).

Eq. (2) is a discrete iterative map and can be converted intoa continuous equation as follows

nk,S = −∑k′

nk′,I (t)

(1 − 1

k′

)P(k′|k)

nk,S(t)

Nk

,

(3)

nk,R = nk,I (t)

[1

k+

(1 − 1

k

)∑k′

P(k′|k)nk′,I (t) + nk′,R(t)

Nk′

].

This set of equations govern the evolution of the populationwith degree k. It is a mean-field description with consider-ation of the network structure and the unequal footings be-tween the father node and the other neighbors. For the uncorre-lated network, the conditional probability satisfies P(k′|k) =k′P(k′)/〈k〉 [33,34]. Here we focus on the final density ofpopulation that has the chance to hear the rumor. Let T bethe time when the process of rumor spreading is over, i.e.,∑

k nk,I (T ) = 0. In obtaining the solution of Eq. (3) at timet = T , we have followed the approach in Refs. [29,30] by intro-ducing a set of auxiliary variables sk ≡ ∫ T

0 nk,I (t) dt . Assumethat the initial infected seed (at t = 0) has degree k0, we havethe initial conditions: nk,S = Nk , nk,I = 0, and nk,R = 0 fork �= k0 and nk,S = Nk − 1, nk,I = 1, and nk,R = 0 for k = k0.We thus obtain the solutions of Eq. (3)

nk,S(T ) = Nk exp

[ −k

〈k〉N∑k′

sk′(

1 − 1

k′

)],

nk,R(T ) = Nk

{1 − exp

[ −k

〈k〉N∑k′

sk′(

1 − 1

k′

)]},

(4)nk,I (T ) = Nk − nk,S(T ) − nk,R(T ) = 0.

By nk,I (T ) = 0 for different k, we may get a set of transcenden-tal equations on sk which can be accurately solved numerically.Hence, the final density of the refractory nodes with degree k is

(5)ρ(k) ≡ nk,R(T )

Nk

= 1 − e−αk,

where α = 1〈k〉N

∑k′ sk′(1 − 1

k′ ) depends on the network struc-ture. Obviously, ρ(k) will monotonously increase with k andapproaches to unity for large k. The total infected nodes duringthe spreading process is

(6)NR(T ) ≡∑

k

nk,R(T ) = N −∑

k

Nke−αk.

J. Zhou et al. / Physics Letters A 368 (2007) 458–463 461

The density of the total infected nodes is

(7)ρR ≡ NR(T )

N= 1 −

∑k

P (k)e−αk =∑

P(k)ρ(k).

which depends on the degree distribution P(k). Formulae (5)and (7) are the main results of this Letter.

3. Rumor propagation on homogeneous networks

There are two extreme cases for heterogeneous networks:random growing network with the Poisson distribution andscale-free network with power-law distribution. A general net-work may be in between these two extreme cases. Liu et al.present a growing network model to produce the structure ofa general network [9]. However, it is pointed out that even therandom growing network is not completely homogeneous [32].In order to implement the situation addressed by Sudbury [28],we construct a homogeneous network as follows: take N nodesand assign everyone the same degree k. Then we randomly adda link between two arbitrary chosen nodes, provided the num-ber of links at both of them is smaller than k. This processcontinues until every node has k neighbors. Therefore, all thenodes have the same degree in the resulted network. Accordingto our knowledge, it is the most “homogeneous” network withthe small-world property. Now we discuss a rumor propagationin this network.

With the same analysis as in the case of the heterogeneousnetworks, we can obtain the evolution equations of nk,S , nk,I ,and nk,R for the situation of homogeneous networks. As theconstructed homogeneous network has the same degree at eachnode, it is not necessary to distinguish the number of nodeswith different k. Let nS(t), nI (t), and nR(t) be the numbers ofthe susceptible, infected, and refractory nodes of the network attime t , respectively. We have the following evolution equations

nR(t + 1) = nR(t) + nI (t)

[1

k+

(1 − 1

k

)nR(t) + nI (t)

N

],

nS(t + 1) = nS(t) − nI (t)

(1 − 1

k

)(1 − nR(t) + nI (t)

N

),

(8)nI (t + 1) = N − nR(t + 1) − nS(t + 1),

where the part (nR(t) + nI (t))/N comes from the homoge-neous mixing hypothesis. Converting Eq. (8) into a continuousone, we have

nR = nI (t)

[1

k+

(1 − 1

k

)nR(t) + nI (t)

N

],

nS = −nI (t)

(1 − 1

k

)(1 − nR(t) + nI (t)

N

),

(9)nI = nI (t)

{1 − 2

[1

k+

(1 − 1

k

)nR(t) + nI (t)

N

]}.

By introducing the auxiliary variables s(t) ≡ ∫ t

0 nI (t′) dt ′ and

using the initial condition nI (0) = 1, nS(0) = N − 1, andnR(0) = 0 we obtain the solutions of Eq. (9)

nS(t) = (N − 1) exp

[− s(t)

(1 − 1

)],

N k

nR(t) = s(t) − (N − 1)

{1 − exp

[− s(t)

N

(1 − 1

k

)]},

(10)nI (t) = 1 − s(t) + 2(N − 1) exp

[− s(t)

N

(1 − 1

k

)].

In the final state with time t = T , we have nI (T ) = 0 and henceobtain the value of s(T ). Substituting s(T ) into the expressionsof nS(t) and nR(t) in Eq. (10) we can get the final number ofinfected nodes. The density of the total infected nodes is,

(11)ρR ≡ nR(T )

N≈ 3 exp

[− s(T )

N

(1 − 1

k

)]− 1,

which depends explicitly on the degree k. For the case oflarge k, Eq. (10) becomes Eq. (4) of ε = 1 in Ref. [30] whichgives ρR ≈ 0.8 for the homogeneous situation.

4. Numerical simulations

In this section, we perform numerical simulations to testifythe above theoretical predictions. We first construct a generalgrowing network by the approach given in Ref. [9] and thensimulate the rumor propagating on this network. The networkstructure is constructed as the following: We first take m nodesas the initial nodes and then add one node with m links at eachtime step. The m links from the added node go to m existingnodes with probability Πi ∼ (1 − p)ki + p, where ki is the de-gree of node i at that time and 0 � p � 1 is a parameter. The re-sulted network has an average degree 〈k〉 = 2m for large N . Ob-viously, (1 − p)ki in Πi represents the preferential attachmentand p in Πi the random attachment. That is, p is the probabil-ity that a new link is randomly connected to the existing node i

and (1 − p) is the probability that the new link is preferentiallyattached to node i. For p = 0, the model generates a strictlyscale-free network, while for p = 1, it generates a completelyrandom growing network. For 0 < p < 1, the resulting connec-tivity distribution is shown to be [9] P(k) ∼ [k +p/(1−p)]−γ ,where the scaling exponent γ is γ = 3 + p/[m(1 − p)]. Wesee that the power-law scaling for scale-free networks is re-covered for p = 0 and the distribution becomes exponentialP(k) ∼ e−k/m as p → 1.

Now we carry out the numerical simulations on the con-structed networks. Note that, because of the random compo-nents involved in the construction of the network and the nodefor the seed of rumor is randomly chosen, the final infecteddensity ρ(k) and ρR should be interpreted in a statistical way.Hence, we take an average over a number of realizations in ournumerical simulations.

First, we consider the scale-free network and let N = 1000,m = 5, and p = 0. We randomly choose a seed at t = 0 fromwhich the infection starts. At each time step, every infectednode contacts one of its neighbors. If this node is susceptible, itwill be infected; otherwise, the original infected node itself willlose interest in the rumor and become refractory. The processcontinues until time T at which there is not any infected node.We count the refractory nodes with degree k, nk,R(T ), and thetotal nodes with degree k, Nk . Then we calculate the final den-sity of infected nodes with degree k, ρ(k) = nk,R(T )/Nk . We

462 J. Zhou et al. / Physics Letters A 368 (2007) 458–463

Fig. 2. ρ(k) versus k. The results are obtained by 1000 realizations withN = 1000 and 〈k〉 = 10. “Circles” denote the numerical simulations and“squares” the theoretical predictions from Eq. (5), and (a) represents the case ofscale-free network with p = 0 and (b) the case of random network with p = 1.

find that ρ(k) increase monotonously with k when k < 35 andstay at ρ(k) = 1 or nearby when k � 35. The result is shownby “circles” in Fig. 2(a). Now we calculate the theoretical valueof ρ(k) through Eq. (5). The exponent α is obtained through itsexpression by calculating all the sk at t = T and then substitut-ing α into Eq. (5) to get ρ(k). “Squares” in Fig. 2(a) show howthe theoretical ρ(k) changes with k. Comparing the “circles”with the “squares” in Fig. 2(a) one can see that the theoreti-cal results agree very well with the numerical experiments. Wehave found the similar results for the random growing networkwith p = 1. Fig. 2(b) shows the results. The lines with “circles”denote the numerical simulations and the lines with “squares”the theoretical predictions. From Fig. 2(a) and (b) it is easy tosee that the theoretical results are always less than unity whilethe numerical results may reach unity for k > 30. The reason isthat the theoretical formula (5) is partially dependence on thehomogeneous mixing hypothesis which is based on an averageof infinite realizations. However, in our finite number of real-izations, those nodes with the heaviest links are infected everytime and leads to ρ(k) = 1.

Second, we investigate how ρR varies with the structureparameter p. In numerical experiments, we produce differentnetwork structures for different p and randomly choose dif-ferent seeds. Once the process of rumor spreading is over, wecount the number NR(T ) of the total final infected nodes forall the degree k and calculate the density ρR = NR(T )/N . Theresult is shown in Fig. 3 by the “circles”. For comparison, wealso calculate the ρR through Eq. (7) (see “squares” in Fig. 3).Obviously, in Fig. 3 “squares” are very close to “circles”. Fur-thermore, from Fig. 3 it is easy to see that ρR increases with thestructure parameter p. This can be explained as follows: For thenetworks with the same average degree, scale-free network hasmore nodes with larger degree than that in the random grow-ing network. Therefore, in scale-free network with p = 0 therumor can be easily transmitted to the hub nodes with the heav-iest degree and then to the other nodes. Once the hub nodesare in the infected or refractory status, the other infected nodes

Fig. 3. ρR versus p. The results are obtained by using 1000 realizationswith N = 104 and 〈k〉 = 10. “Circles” denote the numerical simulations and“squares” the theoretical predictions. The lines are drawn to guide the eye.

Fig. 4. ρR versus 〈k〉 for different types of networks. The three groups are (frombottom to up), scale free network (p = 0), random growing network (p = 1),and the network with the same degree at each node. The computation are doneby using 1000 realizations with N = 1000. The lines are drawn to guide theeye.

will be easy to become refractory as they have larger probabil-ity to be connected to the hub nodes than to the other nodes.When the infected nodes choose the hub nodes as their tar-gets of rumor spreading, themselves become refractory. Thisfeature disappears in the random growing network with p = 1,which results in a faster ending of rumor spreading in scale-freenetwork than in random growing network. Namely, the rumorin random growing network will survive longer and spread tolarger population than that in scale-free network which is justwhat we have observed in Fig. 3.

Third, we check how ρR changes with the average degree〈k〉. For a fixed parameter p, we construct different networkswith different m or 〈k〉 and then make numerical simulationsof rumor propagation on each network. The middle and lowerparts of Fig. 4 show the numerical results of ρR versus 〈k〉

J. Zhou et al. / Physics Letters A 368 (2007) 458–463 463

where “stars” represents the case of p = 0 and “pluses” thecase of p = 1. For comparison with the theoretical prediction,we also show the theoretical results from Eq. (7) in Fig. 4. The“triangles” represent the case of p = 0 and “asterisks” the caseof p = 1. Comparing the middle groups with the low group, wecan see that they are consistence and increase with 〈k〉, indicat-ing that larger 〈k〉 is closer to the case of homogeneous network.However, we find that they cannot approach the homogeneouslimit 0.8 even if we take a very large 〈k〉. Therefore, the case ofSudbury cannot be implemented in the general network.

Finally, we investigate the situation of the same degree ateach node. We construct a homogeneous network by using themethod given in Section 3 and make the numerical experimentsof rumor propagation in this network. We find that ρR increaseswith k and approaches the limit 0.8 when k is very large. The uppart of Fig. 4 shows the results where “circles” represents thenumerical simulation and “squares” the theoretical result fromEq. (11). This result confirms that the Sudbury’s case can be im-plemented by our homogeneous network. Comparing the threeparts of Fig. 4 one can see that the random growing networkhas stronger spreading ability than the scale-free network. Thehomogeneous network is the best structure for rumor spreading.

5. Conclusions

We have studied analytically and numerically the rumorpropagation on complex networks by the SIR model. Differ-ent from previous works, our theory takes into account the in-fluence of network topological structure and distinguishes theunequal footings between the father and the other neighboringnodes. We find that the degree distribution influences directlythe final density of infected nodes. The network with a smallerp has a larger ρR than the network with a larger p. Furthermore,we have studied the components of the final infected nodes andfound that ρ(k) increases with k by Eq. (5). Especially, we haveconstructed a homogeneous network with the same degree ateach node to explain the result of previous mean-field limit,ρR = 0.8. These results have revealed the mechanism of hownetwork structure influences ρR and have been confirmed bynumerical simulations.

Acknowledgements

This work was supported by the NNSF of China underGrant Nos. 10475027 and 10635040, by the PPS under GrantNo. 05PJ14036, by SPS under Grant No. 05SG27, and byNCET-05-0424, and in part by NUS Faculty Research Grantunder Grant No. R-144-000-165-101.

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