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December  2017, 22(10): 3749-3770. doi: 10.3934/dcdsb.2017187

Impacts of cluster on network topology structure and epidemic spreading

Shuping Li 1, and  Zhen Jin 2,,

1. 

Department of Mathematics, North University of China, Taiyuan Shan'xi 030051, China
2. 

Complex Systems Research Center, Shanxi University, Taiyuan Shan'xi 030006, China, and Department of Computer Science and Technology, North University of China, Taiyuan Shan'xi 030051, China

* Corresponding author: Zhen Jin

Received  October 2016 Revised  June 2017 Published  July 2017

Fund Project: The first author is supported by the National Natural Science Foundation of China under Grants (11331009,11571210,11571324), National Youth Natural Science Foundation (11301491), Shanxi Province Science Foundation for Youths (201601D021015).

Considering the infection heterogeneity of different types of edges (lines and edges in the triangle in a network), we formulate and analyze an novel SIS model with cluster based mean-field approach for a network. We mainly focus on how network clustering influences network structure and the disease spreading over the network. In networks with double poisson distributions, power law-poisson distribution, poisson-power law distributions and double power law distributions, we find that cluster is positive(the clustering coefficient is increasing on the expected number of triangles) when the average degree of lines is fixed and the moment of triangles is less than some threshold. Once the moment of triangles exceeds that threshold, cluster will become negative(the clustering coefficient is decreasing on the expected number of triangles). For the disease, clustering always increases the basic reproduction number of the disease in networks with whether positive cluster or negative cluster. It is different from existing results that cluster always promotes the disease spread in the homogeneous or heterogeneous network.

Keywords: Positive cluster, negative cluster, conditional probability, mean-field, basic reproduction number.
Mathematics Subject Classification: Primary:34D20;Secondary:92D30.
Citation: Shuping Li, Zhen Jin. Impacts of cluster on network topology structure and epidemic spreading. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3749-3770. doi: 10.3934/dcdsb.2017187
References:
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show all references

References:
[1]

K. Chung, Y. Baek, D. Kim, M. Ha and H. Jeong, Generalized epidemic process on modular networks Phys. Rev. E, 89 (2014), 052811, 7pp. doi: 10.1103/PhysRevE.89.052811.  Google Scholar

[2]

E. Coupechoux and M. Lelarge, How clustering affects epidemics in random networks, Advances in Applied Probability, 46 (2014), 985-1008.  doi: 10.1017/S0001867800007515.  Google Scholar

[3]

S. N. Dorogovtsev, Clustering of correlated networks Phys. Rev. E, 69 (2004), 027104, 5pp. doi: 10.1103/PhysRevE.69.027104.  Google Scholar

[4]

K. T. D. Eames, Modelling disease spread through random and regular contacts in clustered populations, Theor. Popul. Biol., 73 (2008), 104-111.  doi: 10.1016/j.tpb.2007.09.007.  Google Scholar

[5]

J. P. Gleeson, S. Melnik and A. Hackett, How clustering affects the bond percolation threshold in complex networks Phys. Rev. E, 81 (2010), 066114, 10pp. doi: 10.1103/PhysRevE.81.066114.  Google Scholar

[6]

T. House and M. J. Keeling, Insights from unifying modern appoximations to infections on networks, J. R. Soc. Interface, 8 (2011), 67-73.  doi: 10.1098/rsif.2010.0179.  Google Scholar

[7]

M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proc. R. Soc. London, Ser. B, 266 (1999), 859-867.  doi: 10.1098/rspb.1999.0716.  Google Scholar

[8]

L. A. Meyers, Contact network epidemilogy: Bond percolation applied to infectious disease prediction and control, Bull. Amer. Math. Soc., 44 (2007), 63-86.  doi: 10.1090/S0273-0979-06-01148-7.  Google Scholar

[9]

J. C. Miller, Spread of infectious disease through clustered populations, J. R. Soc. Interface, 6 (2009), 1121-1134.  doi: 10.1098/rsif.2008.0524.  Google Scholar

[10]

J. C. Miller, Percolation and epidemics in random clustered networks Phys. Rev. E, 80 (2009), 020901, 4pp. doi: 10.1103/PhysRevE.80.020901.  Google Scholar

[11]

C. Molina and L. Stone, Modelling the spread of diseases in clustered networks, J. Thero. Biol., 315 (2012), 110-118.  doi: 10.1016/j.jtbi.2012.08.036.  Google Scholar

[12]

M. E. J. Newman, Spread of epidemic disease on networks Phys. Rev. E, 66 (2002), 016128, 11pp. doi: 10.1103/PhysRevE.66.016128.  Google Scholar

[13]

M. E. J. Newman, The Structure and Function of Complex Networks, SIAM Rev., 45 (2013), 167-256.  doi: 10.1137/S003614450342480.  Google Scholar

[14]

M. E. J. Newman, Clustering and preferential attachment in growing networks Phys. Rev. E, 64 (2001), 025102, 13pp. doi: 10.1103/PhysRevE.64.025102.  Google Scholar

[15]

M. E. J. Newman, Random graphs with clustering Phys. Rev. E, 103 (2009), 058701, 5pp. doi: 10.1103/PhysRevLett.103.058701.  Google Scholar

[16]

M. E. J. Newman, Properties of highly clustered networks Phys. Rev. E, 68 (2003), 026121, 7pp. doi: 10.1103/PhysRevE.68.026121.  Google Scholar

[17]

M. E. J. Newman, Power laws, Pareto distributions and Zipf's law, Contemp. Phys., 46 (2005), 323-351.  doi: 10.1080/00107510500052444.  Google Scholar

[18]

R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks Phys. Rev. E, 63 (2001), 066117, 8pp. doi: 10.1103/PhysRevE.63.066117.  Google Scholar

[19]

R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), 3200-3203.  doi: 10.1103/PhysRevLett.86.3200.  Google Scholar

[20]

M. A. Serrano and M. Boguñá, Clustering in complex networks. I. General formalism Phys. Rev. E, 74 (2006), 056114, 9pp. doi: 10.1103/PhysRevE.74.056114.  Google Scholar

[21]

M. A. Serrano and M. Boguñá, Clustering in complex networks. Ⅱ. Percolation properties Phys. Rev. E, 74 (2006), 056115, 8pp. doi: 10.1103/PhysRevE.74.056115.  Google Scholar

[22]

B. Szendroi and G. Csányi, Polynomial epidemics and clustering in contact networks, Proc. R. Soc. London, Ser. B: Biological Sciences, 271 (2004), S364-S366.  doi: 10.1098/rsbl.2004.0188.  Google Scholar

[23]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[24]

E. M. Volz, J. C. Miller, A. Galvani and L. A. Meyers, Effects of heterogeneous and clustered contact patterns on infectious disease dynamics PLOS Comp. Biol. , 7 (2011), e1002042, 13pp. doi: 10.1371/journal.pcbi.1002042.  Google Scholar

[25]

D. J. Watts and S. H. Strogatz, Collective dynamics of "small-world" networks, Nature, 393 (1998), 440-442.  doi: 10.1038/30918.  Google Scholar

Figure 1.  Total degree distribution. $(a).$ $p_{l,r}$ is double poisson distributions, $(b).$ $p_{l,r}$ is power law-poisson distribution, $(c).$ $p_{l,r}$ is poisson-power law distribution, $(d).$ $p_{l,r}$ is double power law distributions
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Figure 3.  Transmission in the triangle. $v_{1}$ is not infected by the infected neighbor in the triangle, $v_{1}$ and $v_{2}$ are not infected by $v_{3}$, or $v_{1}$ is not infected by $v_{3}$, and, $v_{3}$ transmits the disease to $v_{2}$, while, $v_{2}$ doesn't transmit the disease to $v_{1}$. $v_{1}$ is directly infected by $v_{3}$, or, $v_{1}$ is not infected by $v_{3}$, and, after $v_{3}$ transmits the disease to $v_{2}$, $v_{2}$ transmits the disease to $v_{1}$
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Figure 2.  Illustration of clustered network. It is a regular clustered network where there is no common edges for any two triangles. Each node in this network is connected to two lines and one triangle, and the total number of nodes is a constant
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Figure 4.  Relations of the clustering coefficient $C$ and degree distributions. (a) and (b) describe variations of the clustering coefficient $C$ in the network with double poisson distributions and power law-poisson distribution respectively. (c) and (d) describe variations of the clustering coefficient $C$ in the network with double power distributions and poisson-power law distribution respectively, $\langle l\rangle=3.5125$ in (a), $\langle r\rangle=3.2415$ in (b) and (d), in (c), $\langle l\rangle$ is equal to 3.5125 and 10 in the network with double power law distributions and poisson-power law distribution respectively
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Figure 5.  Relations of the basic reproduction number $R_{0}$, degree distributions, and the clustering coefficient $C$. (a)-(d) shows that $R_{0}$ is increasing with the increasing of $\langle r\rangle$((a) and (c) in Fig. 5) or $\langle l\rangle$((b) and (d) in Fig. 5) in networks with different types of degree distributions. In addition, in (a) and (c), when the clustering coefficient $C$ is in a certain interval, the basic reproduction number $R_{0}$ is different for a given clustering coefficient $C$ because the distribution of triangles is different. This phenomenon is not outstanding when the number of triangles is power law distribution shown as (c) of Fig. 5. (b) and (d) of Fig. 5 show that $R_{0}$ is decreasing about $C$ when $\langle r\rangle$ is fixed in networks with four types of joint distributions
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Figure 6.  Symbol of $\frac{\partial R_{0}}{\partial C}$, when the average number of second neighbors is fixed and the joint degree distribution is double poisson distributions
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