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Epidemic spreading driven by biased random walks

Cunlai Pu, Siyuan Li, Jian Yang

Nanjing University of Science and Technology, Nanjing 210094, China

Abstract

Random walk is one of the basic mechanisms found in many network appli-cations. We study the epidemic spreading dynamics driven by biased ran-dom walks on complex networks. In our epidemic model, each time infectednodes constantly spread some infected packets by biased random walks totheir neighbor nodes causing the infection of the susceptible nodes that re-ceive the packets. An infected node get recovered from infection with a fixedprobability. Simulation and analytical results on model and real-world net-works show that the epidemic spreading becomes intense and wide with theincrease of delivery capacity of infected nodes, average node degree, homo-geneity of node degree distribution. Furthermore, there are correspondingoptimal parameters such that the infected nodes have instantaneously thelargest population, and the epidemic spreading process covers the largestpart of a network.

Keywords: epidemic spreading, biased random walks, complex networks

1. Introduction

Unexpected outbreak of many epidemics in biological systems[1, 2] andthe spread of viruses in technology systems[3, 4, 5] result in a lot of deathor great damage in related systems. The study of epidemiological mod-els has a long history, especially in the field of social science[6, 7]. TheSIR (susceptible-infected-removed) model and the SIS (susceptible-infected-susceptible) model are two representative models which capture the basicproperties of epidemic spreading through the transition among several dis-ease states[8, 9]. In SIR model, a susceptible individual will become infectedwith certain rate when it has contact with infected individuals. An infectedindividual will get immunity to the disease or die at some constant rate,

Preprint submitted to Elsevier August 5, 2018

and becomes a removed node which means it can not get infected again.Therefore, the spread of disease will terminate when all the infected individ-uals are removed from the disease. Differently in SIS model, there are justtwo states, susceptible and infected. A recovered individual can get infectedagain. If the fraction of infected individuals is large enough the disease willspread indefinitely, otherwise it will die out after sometime. Initially, the epi-demic models are considered under the homogeneous mixing hypothesis[10],in which it assumes that each time an arbitrary individual has an equal op-portunity to contact with everyone else in the population. Later, results fromnetwork science community demonstrate that most real-world networked sys-tems have heterogeneous topological structures[11, 12, 13], and this greatlypromote many mathematicians and physicists to explore epidemic modelson heterogeneous random networks[14, 15, 16, 17] by means of mean-fieldapproximation[18, 19, 20], generating functions formalism[21] and percola-tion theory[22]. It was found that for random networks with strongly het-erogeneous degree distribution, like many real-world networks, the epidemicthreshold is absent, which means epidemics always have a finite probabilityto survive indefinitely[11, 18].

Besides of topological properities, traffics in networks also have great im-pacts on the epidemic spreading. For instance, in the Internet computerviruses transmit from a node to another one with data packets. Withouttransmission of packets, viruses can not spread even if the two nodes arephysically connected. Another example is the air traffics speed the spread ofdisease among different spatial areas. The combination between epidemicspread and traffic dynamics were first considered in the metapopulationmodel[23] which characterizes the dynamics of systems composed of sub-populations. Then, Meloni et al[24] studied the impact of traffic dynamicson the spread of virus in the Internet, in which the information packets aretransmitted with the shortest path protocol. Later, many mechanisms wereproposed to suppress the traffic-driven epidemic spreading, for instance con-trolling the traffic flow[25], the routing strategy[26, 27], or the heterogeneouscuring rate[28], deleting some particular edges[29], etc.

Random walk is one of the basic mechanisms related to spreading processes[30,31, 32]. For example, a mobile phone virus may randomly dial some phonenumbers from the directory. Some computer viruses propagate randomly byemail or other online communication tools. Therefore, the role of randomwalks in the epidemic spreading should be explored. We propose an epidemicmodel driven by biased random walks. In our model, an infected node sends

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infected packets at a constant rate to its neighbor nodes through biasedrandom walks. A susceptible node gets infected after receive the infectedpackets, and will be removed from the set of infected nodes with a constantrate. We inverstigate the spreading dynamics, the optimal control parameterof our model and the influence of network topologies on our model.

2. SIR model

We improve the traditional SIR model by incorporating the traffic dynam-ics driven by random walks. The SIR model is one of the traditional epidemicspread models in literature. In the SIR model, there are three types of nodesincluding susceptible nodes, infected nodes and removed nodes. A suscep-tible node is susceptible to epidemics. An infected node is already infectedby the epidemic. A recovered node is the one that is removed from the setof infected nodes. Assume the numbers of susceptible, infected and removednodes at time t are denoted as S(t), I(t) and R(t) respectively. There arethree basic elements in the SIR model as follows[8]:

(1) Assume the number of nodes in the network is fixed to N . Then N =S(t) + I(t) +R(t) for all t.

(2) At time t, an arbitrary infected node infects the susceptible nodes by aratio β. Then the increased number of infected nodes at t is β∗s(t)∗I(t).

(3) At time t, the number of infected nodes removed is proportional to thetotal number of infected nodes I(t), which is λ ∗ I(t).

According to the three elements, the dynamics of the SIR model can beexpressed as follows[8]:

dIdt

= βSI − λI,dSdt

= −βSI,dRdt

= λI.

(1)

When time t is large enough, all the infected nodes will eventually becomesremoved nodes, and the epidemic spreading stops.

The SIR model is based on the assumption that a node has equal prob-ability to contact with every other node in a network. However, in realsituations, individuals often have heterogeneous numbers of contacts[11]. Afew individuals have large number of contacts which will get more contactsaccording to the rich-get-richer mechanism, while most of the individuals

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Figure 1: Illustration of the transitions among the susceptible, infected and removed nodesin our model.

have a few contacts. Especially in the Internet, the epidemic can not spreadfrom a node to another node unless there is transport of infected packetsbetween the two nodes. Additionally, in city networks even the cities arephysically well connected, an epidemic can not spread among cities unlessindividuals who get infected by the disease move among the cities. Therefore,epidemics are often correlated with traffics for their spreading.

3. Epidemic model driven by biased random walks

We consider the traffic dynamics driven by biased random walks in theepidemic spreading process. In our model, each time an infected nodes willdelivery constantly C infected packets to its neighbor nodes through biasedrandom walks. If an infected or removed node receives the packets, it willdrop the packets. If a susceptible node receives the packet, it becomes aninfected node and starts delivering infected packets from next time step.An infected node has the probability λ to become a removed node. Thetransitions among susceptible, infected and removed nodes are shown in Fig.1.

3.1. Dynamics of our model

Assume an arbitrary infected node a which has 〈K〉 neighbor nodes. 〈K〉is the average node degree of the network. The degrees of a’s neighbor nodesare k1, k2, · · · , k〈K〉 respectively. According to the biased random walk mech-anism, for a neighbor node i the probability that node a sends an infectedpacket to node i is as follows:

Pai =kαi

∑〈K〉j=1 k

αj

. (2)

4

Where α is the control parameter of the biased random walk. When α = 0,all the neighbor nodes have equal opportunity to receive an infected packetdelivered from node a which means they have equal probability to get in-fected. When α > 0, nodes of larger degree have larger probability to receivethe packet. When α < 0, nodes of smaller degree have larger probability toreceive the packet. Since a send C infected packets each time, the probabilitythat node i will not receive any packets from node a is:

Pai = (1−kαi

∑〈K〉j=1 k

αj

)C . (3)

Assume Xi is an random variable that represents the event that node i isinfected. Then Xi = 0 means node i hasn’t get any infected packet fromnode a, and node i is not infected. Xi = 1 means node i has received atleast one of the infected packets from node a, and node i is infected. Thenwe have:

P (Xi = 0) = (1−kαi

∑〈K〉j=1

kαj

)C ,

P (Xi = 1) = 1− (1−kαi

∑〈K〉j=1

kαj

)C .(4)

Then the expected value of Xi is:

E(Xi) = 1− (1−kαi

∑〈K〉j=1 k

αj

)C . (5)

Assume a random variable Y that represents the number of neighbor nodesinfected by node a. Then the average value of Y is:

E(Y ) =

〈K〉∑

i=1

E(Xi)

= 〈K〉 −

〈K〉∑

i=1

(1−kαi

∑〈K〉j=1 k

αj

)C . (6)

Where the sum is over all the 〈K〉 neighbor nodes of node a. However, in theepidemic spreading process the neighbor nodes of node a may not be onlysusceptible nodes. To effevtively estimate the number of neighbor nodes thatnode a infects, we need to know the number of susceptible nodes among allthe neighbor nodes of node a. To estimate the total number of new infected

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nodes at time t, we count the ratio µ of susceptible nodes among neighbornodes of infected nodes in the network, which is as follows:

µ =

∑I(t)j=1 nj

∑I(t)j=1 kj

≈〈n〉

〈K〉. (7)

Where nj is the number of susceptible nodes among the neighbor nodes ofan infected node j. 〈n〉 represents the average number of susceptible nodesamong all the neighbors of infected nodes at time t. According to Eq. 6 andEq. 7, the total new infected nodes at time t is:

Inew(t) ≈ E(Y ) ∗ µ ∗ I(t)

≈ (〈K〉 −

〈K〉∑

i=1

(1−kαi

∑〈K〉j=1 k

αj

)C) ∗〈n〉

〈K〉∗ I(t). (8)

Combining Eq. 1 with Eq. 8, we get the dynamics equations of our modelas follows:

dIdt

= (〈K〉 −∑〈K〉

i=1(1−kαi

∑〈K〉j=1

kαj

)C) ∗ 〈n〉〈K〉

∗ I(t)− λI,

dSdt

= −(〈K〉 −∑〈K〉

i=1(1−kαi

∑〈K〉j=1

kαj

)C) ∗ 〈n〉〈K〉

∗ I(t),

dRdt

= λI.

(9)

We study the behaviours of S(t), I(t) and R(t) with increase of time t on alarge-scale scale-free network. In Fig. 2, S(t) decreases abruptly, and thensaturates with t. I(t) increases with t, then decreases with t, and finallysaturates. There is a peak of I(t) that corresponds to the instantaneousmaximum population of infected nodes. R(t) increases abruptly, and thensaturates with t, which is opposite to S(t). When t is large enough, theepidemic spreading process stops, and R(t) is number of all the nodes thathave ever been infected and removed finally from the disease. The trendsof the curves for biased random walks of α = −1 are similar with that ofsimple random walks of α = 0. Also, the simulation results and the analyticalresults obtained from Eq. 9 are consistent, as shown in Fig. 2.

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R(t

)

t

(f)

Figure 2: S(t), I(t), and R(t) vs. t for α = 0 and α = −1. A randomly selected node isset to be infected initially. Delivery capacity of infected nodes is C = 5. λ = 0.1. Thenetwork is generated by the static model[33]. The parameters are N = 10000, 〈k〉 = 5,and γ = 2.5.

0 20 40 60 80 100 120 140100020003000400050006000700080009000

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S(t

)

t

C=2 C=5 C=10 C=20 C=infinite

(a) (b) (c)

I(t)

t

R(t

)

t

Figure 3: S(t), I(t), and R(t) vs. t for various delivery capacities C. A randomly selectednode is set to be infected initially. α = 0. λ = 0.1. The network is generated by the staticmodel. The parameters of the network are N = 10000, 〈k〉 = 5, and γ = 2.5.

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I(t)

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R(t)

t

(c)

=-1 =0 =1

EpinionsEpinions =-1 =0 =1

=-1 =0 =1

S(t)

t

=-1 =0 =1

Epinions

(d)

I(t)

t

(e)

R(t)

t

(f)

Figure 4: S(t), I(t), and R(t) vs. t for various α. A randomly selected node is set to beinfected initially. C = 5. λ = 0.1. The network for (a), (b) and (c) is generated by thestatic model. The parameters of the network are N = 10000, 〈k〉 = 5, and γ = 2.5. TheEpinions network for (d), (e) and (f) has 75, 879 nodes and 508, 960 edges.

3.2. Factors of our model

Delivery capacity C of infected nodes is a critical factor in our epidemicspreading model. The larger the delivery capacity of an infected node, themore susceptible neighbor nodes an infected node will likely infects each time.When C → ∞, Inew(t) → 〈n〉 ∗ I(t). Then Eq. 9 is reduced as follows:

dIdt

= 〈n〉 ∗ I(t)− λI,dSdt

= −〈n〉 ∗ I(t),dRdt

= λI.

(10)

We show simulation results of S(t), I(t) and R(t) for different value of C inFig.3. Clearly, the larger C, the faster S(t), I(t) and R(t) convergence, andthe epidemic spreads. The larger C, the larger population of nodes that hasever been infected which is inferred from S(t) and R(t) when t is large enough.The larger C, the larger instantaneous population of infected nodes which isindicated from the peak of I(t). The parameter α is another key factor inour epidemic model which determines the probability that the neighbors ofan infected node get infected when C is a limited constant. In Fig. 4, we

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see that α is correlated with the instantaneous maximum number of infectednodes Ipeak, and the range of the spread that is reflected by the ultimatenumber of removed nodes Rend. α = −1 corresponds to a larger Ipeak anda larger Rend than α = 0 and α = 1. This indicates that random walksbiased on small-degree nodes favors the epidemic spreading which is holdfor the model network and the Epinions network, as shown in Fig.4. Thenwe investigate the optimal parameters αopt that lead to the maximum Ipeakand the maximum Rend on the model network and the Epinions network.In Fig.5, we see Ipeak and Rend as a function of α. Clearly, Ipeak and Rend

increase with α first, then decrease with α respectively. There are αopt thatcorrespond to maximum Ipeak and maximum Rend respectively. We presentthe results of maximum Ipeak, maximum Rend, and the corresponding αopt forsome real-world networks, as shown in Table 1.

Table 1: maximum Ipeak and maximum Rend with corresponding optimal parameters α

for real-world networks. C = 5. λ = 0.1.NAME NODES EDGES Ipeak αopt Rend αopt

Oregon-1 10790 22469 2903.33 -0.4 6010.37 -0.8Gnutella 62586 147892 37833.43 -0.8 58155.71 -1.4Epinions 75879 508837 35274.13 -0.8 61560.15 -1.2Wiki-Vote 7115 103689 4294.22 -1 6623.23 -1.4

Yeast 2361 7182 1314.21 -0.8 2214.63 -1.2email-Enron 36692 183831 12789.5 -0.8 24254.25 -1.4Facebook 4039 88234 2105.11 -0.8 3920.21 -0.4Geom 7343 11898 1778.5 -0.4 3202.5 -1

Political blogs 1222 19021 813.26 -1.2 1192.14 -2.2Power grid 4941 6594 1278.36 0.4 4304.27 -0.2

4. Impacts of networks structures on our model

We investigate the influence of topological properties of complex networksincluding average node degree and degree distribution, on the behaviors ofour epidemic spreading model. We focus on the spontaneous number ofinfected nodes Ipeak and the final population of nodes that have ever beeninfected Rend, as well as the related optimal parameters αopt. According Eq.

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Ipeak

Rend

I peak

(R

end)

Model network

I peak

(R

end)

Ipeak

Rend

(a)

Epinions

(b)

Figure 5: Ipeak and Rend vs. α. A randomly selected node is set to be infected initially.C = 5. λ = 0.1. The networks are the same as in Fig. 4. The results are the average of100 independent runs.

6, we have:

E(Y ) = 〈K〉 −

〈K〉∑

i=1

(1−kαi

∑〈K〉j=1 k

αj

)C

= 〈K〉 −

〈K〉∑

i=1

(1− Ckαi

∑〈K〉j=1 k

αj

+C(C − 1)

2(

kαi

∑〈K〉j=1 k

αj

)2 − · · ·)

=

〈K〉∑

i=1

(Ckαi

∑〈K〉j=1 k

αj

−C(C − 1)

2(

kαi

∑〈K〉j=1 k

αj

)2 + · · ·). (11)

When 〈K〉 → ∞, we get:

E(Y ) ≈

〈K〉∑

i=1

Ckαi

∑〈K〉j=1 k

αj

≈ C (12)

Eq. 12 means that generally the number of nodes that an infected nodesinfects in one time step increases with average degree 〈K〉, and tends to C,which is further confirmed in Fig. 6. Also, for the whole epidemic spreadingprocess, the maximum Ipeak and the maximum Rend increase substantially,then saturate with 〈K〉 respectively, as shown in Fig. 7 and 8. Their corre-sponding optimal parameters αopt are generally negative and decrease with

10

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1

2

3

4

5

E(Y

)

<K>

=0, C=5

Figure 6: E(Y ) vs. 〈K〉 obtained from Eq. 6.

〈K〉. This indicates that when the networks become dense, random walksshould be more biased on small-degree nodes to make the epidemic spread-ing more intense and wide. These results are consistent both for randomnetworks (Fig. 7) and scale-free networks (Fig. 8). We also investigatethe impact of degree distribution on the epidemic spreading dynamics. InFig. 9, the maximum Ipeak and the maximum Rend increase abruptly, thensaturate with γ respectively, and this means when the degree distributionbecomes homogeneous, the spread of the epidemic becomes more fierce andwide in the network. The optimal parameters for Ipeak and Rend increase withγ. This indicates when the network becomes homogeneous, the extent thatrandom walks are biased on small-degree nodes to get a wide and fierce epi-demic spreading decreases. need less However, the fluctuations in the curvesare clear.

5. Conclusions and discussions

In summary, we investigate the epidemic spreading on complex networksincluding model networks and real-world networks. In our model, the epi-demic spreading goes with packets transmission driven by biased randomwalks. Analytical and simulation results demonstrate that Epidemic spread-ing becomes fierce and wide with increase of delivery capacity of infectednodes, average node degree, and the homogeneity of the network. The opti-mal parameters of the biased random walks in epidemic spreading are gener-ally negative values. This means the random walks are biased on small-degreenodes to make an intense and wide spread of the epidemic. However, the bi-

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(b)

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(c)

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maximum Rend

(d)

Figure 7: αopt and the corresponding Ipeak and Rend vs. 〈K〉 for random networks.The networks are generated by the ER (Erdos-Renyi) model[34], and the network size isN = 5000. A randomly selected node is set to be infected initially. C = 5. λ = 0.1. Theresults are the average of 104 independent runs.

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Figure 8: αopt and the corresponding Ipeak and Rend vs. 〈K〉 for scale-free networks.The networks are generated by the static model, the network size is N = 5000, and thepower-law parameter is γ = 2.5. A randomly selected node is set to be infected initially.C = 5. λ = 0.1. The results are the average of 104 independent runs.

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opt

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(d)

Figure 9: αopt and the corresponding Ipeak and Rend vs. power-law parameter γ. Thenetworks are generated by the static model, the network size is N = 5000, and the averagenode degree is 〈K〉 = 5. A randomly selected node is set to be infected initially. C = 10.λ = 0.1. The results are the average of 104 independent runs.

ased random walks are based on only degrees of the nearest neighbor nodesin our model. The effects of biased random walks with more topologicalinformation on the epidemic spreading still need to be explored.

Acknowledgments

This work was supported by the Natural Science Foundation of China(Grant No. 61304154), the Specialized Research Fund for the Doctoral Pro-gram of Higher Education of China (Grant No. 20133219120032), and thePostdoctoral Science Foundation of China (Grant No. 2013M541673).

References

[1] D. G. Green, T. Bossomaier, eds, Complex systems: from biology tocomputation, IOS press, 1993.

[2] H. W. Hethcote, SIAM Rev. 42 (2000) 599.

[3] J. Balthrop, S. Forrest, M. E. J. Newman, et al, arXiv preprintcs/0407048, 2004.

13

[4] P. Wang, M. C. Gonzalez, C. A. Hidalgo, et al, Science 324 (2009) 1071.

[5] A. Vespignani, Nature Physics 8 (2012) 32.

[6] F. Brauer, C. Castillo-Chavez, Mathematical models in population bi-ology and epidemiology, Springer, 2011.

[7] L. F. Berkman, I. Kawachi, eds, Social epidemiology, Oxford UniversityPress, 2000.

[8] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and itsApplications, Griffin, London, 1975.

[9] R. M. May, Infectious diseases of humans: dynamics and control, OxfordUniversity Press, 1995.

[10] R. M. Anderson, R. M. May, Infectious diseases of humans, Oxford:Oxford university press, 1991.

[11] M. Newman, Networks: an introduction, Oxford University Press, 2010.

[12] S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes, Rev. Mod. Phys. 80(2008) 1275.

[13] A. Barrat, M. Barthelemy, A. Vespignani, Dynamical processes on com-plex networks, Cambridge University Press, Cambridge, 2008.

[14] T. Zhou, Z. Q. Fu, B. H. Wang, Progress in Natural Science 16 (2006)452.

[15] R. Yang, B. H. Wang, J. Ren, et al, Phys. Lett. A 364 (2007) 189.

[16] G. Yan, Z. Q. Fu, J. Ren, W. X. Wang, Phys. Rev. E 75 (2007) 016108.

[17] S. W. Chou, K. Wang, Q. Liu, et al, Acta Phys. Sin 61 (2012) 150201.

[18] R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86 (2001) 3200.

[19] Z. Yang, T. Zhou, Phys. Rev. E 85 (2012) 056106.

[20] F. D. Sahneh, C. Scoglio, P. Van Mieghem, IEEE/ACM Transactionson Networking (TON) 21 (2013) 1609.

14

[21] M. E. J. Newman, Phys. Rev. E 66 (2002) 016128.

[22] R. Cohen, K. Erez, D. ben Avraham, et al, Phys. Rev. Lett. 85 (2000)4626.

[23] V. Colizza, A. Vespignani, Journal of Theoretical Biology 251 (2008)450.

[24] S. Meloni, A. Arenas, Y. Moreno, PNAS 106 (2009) 16897.

[25] P. Bajardi, C. Poletto, J. J. Ramasco, et al, PLoS ONE 6 (2011) e16591.

[26] H. X. Yang, W. X. Wang, Y. C. Lai, et al, Phys. Rev. E 84 (2011)045101(R).

[27] H. X. Yang, Z. X. Wu, J. Stat. Mech. 3 (2014) P03018.

[28] C. Shen, H. Chen, Z. Hou, Phys. Rev. E 86 (2012) 036114.

[29] H. X. Yang, Z. X. Wu, B. H. Wang, Phys. Rev. E 87 (2013) 064801.

[30] L. Lovasz, Combinatorics: Paul Erdos is eighty 2 (1993) 1.

[31] J. D. Noh, H. Rieger, Phys. Rev. Lett. 92 (2004) 118701.

[32] M. Bonaventura, V. Nicosia, V. Latora, Phys. Rev. E 89 (2014) 012803.

[33] K.-I. Goh, B. Kahng, D. Kim, Phys. Rev. Lett. 87 (2001) 278701.

[34] P. Erdos, A. Renyi, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17.

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