# American Institute of Mathematical Sciences

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

## Impacts of cluster on network topology structure and epidemic spreading

 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.

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
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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
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 $v_{1}$
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
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
) 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">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
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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