Impacts of cluster on network topology structure and epidemic spreading

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

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 $v_{1}$

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

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

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

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