Social contagions on time-varying community networks

Time-varying community structures exist widely in real-world networks. However, previous studies on the dynamics of spreading seldom took this characteristic into account, especially those on social contagions. To study the effects of time-varying community structures on social contagions, we propose a non-Markovian social contagion model on time-varying community networks based on the activity-driven network model. A mean-field theory is developed to analyze the proposed model. Through theoretical analyses and numerical simulations, two hierarchical features of the behavior adoption processes are found. That is, when community strength is relatively large, the behavior can easily spread in one of the communities, while in the other community the spreading only occurs at higher behavioral information transmission rates. Meanwhile, in spatial-temporal evolution processes, hierarchical orders are observed for the behavior adoption. Moreover, under different information transmission rates, three distinctive patterns are demonstrated in the change of the whole network's final adoption proportion along with the growing community strength. Within a suitable range of transmission rate, an optimal community strength can be found that can maximize the final adoption proportion. Finally, compared with the average activity potential, the promoting or inhibiting of social contagions is much more influenced by the number of edges generated by active nodes.

Figure 1

An illustration of the social contagion model on an activity-driven community network. The network is divided into two equal-sized communities $A$ and $B$, each with four nodes. The circle around a node indicates that the node is active. (a) At $t=0$, nodes 1 and 3 are randomly selected as seeds on community $A$, and the remaining nodes are susceptible. (b) At $t=1$, the instantaneous structure $G_1$ is generated, in which nodes 1 and 6 are activated with probability $a=0.25$, and $m=3$ edges are generated. Every edge connects to nodes in the same community with probability $u=0.6$ and with probability $1-u=0.4$ to the other community. Adopted nodes 1 and 3 transmit the behavioral information to their susceptible neighbors with $\lambda =0.8$. Note that node three transmits the information without activation. Node 6 receives two pieces of information, which reach the adoption threshold $\pi =2$, and thus it becomes adopted. Then, nodes 1 and 3 try to become recovered with $\gamma =0.1$. Finally, delete all edges generated at this time step. (c) At $t=2$, nodes 3 and 8 become active and form the instantaneous structure $G_2$. No information transmission succeeds this time. Then nodes 3 and 6 both become recovered successfully. (d) At $t=3$, nodes 4 and 5 are activated in the instantaneous structure $G_3$. The contagion process terminates since all adopted nodes become recovered.

Figure 2

The degree distributions of the instantaneous structure obtained from numerical simulations and theoretical predictions. The solid line (circles) and the dashed line (squares) represent the theoretical predictions (simulation results) of degree distribution with $m=5,\langle k\rangle =2$ and $m=10,\langle k\rangle =1.2$, respectively. Other parameters are set to be $N=10000, \mu =0.8$. The simulation results are averaged over 1000 realizations. All presented results have been normalized within the range $1\le k<20$.

Figure 3

The final adoption proportion (FAP) $R\left(\infty \right), R_A\left(\infty \right)$, and $R_B\left(\infty \right)$ vs information transmission probability $\lambda$ under different community strengths. In the figure, (a) $\mu =0.5$, (b) $\mu =0.9$, (c) $\mu =0.95$, and (d) $\mu =0.97$. The solid line (circles), dashed line (squares), and dotted line (diamonds) represent the theoretical predictions (simulation results) of $R\left(\infty \right), R_A\left(\infty \right)$, and $R_B\left(\infty \right)$, respectively. Simulation and theoretical suggested outbreak thresholds ${\lambda }_c$ are labeled by triangles and stars, which are recorded as soon as $R\left(\infty \right)$ is slightly larger than ${\rho }_0$, that is, $R\left(\infty \right)>0.04$, due to the difficulty of determining the outbreak threshold [38, 39]. The suggested thresholds are (a) ${\lambda }_c=0.54,0.60$; (b) ${\lambda }_c=0.40,0.44$; (c) ${\lambda }_c=0.38,0.42$; and (d) ${\lambda }_c=0.37,0.41$ for simulation and theoretical results, respectively. Other parameters are set to be $N=10000, {\rho }_0=0.03, a=0.2, m=5, \gamma =0.1$, and $\pi =3$, respectively.

Figure 4

Time evolutions of ${\rho }_A\left(t\right)$ and ${\rho }_B\left(t\right)$, in which (a) and (b) correspond to Figs. 3 and 3, respectively. The time series shown above are sampled under parameters $\lambda =0.6$ and time span $[0,300]$, and they are averaged over 200 realizations. The remaining parameters are set to be the same as those in Figs. 3 and 3.

Figure 5

The FAP $R\left(\infty \right), R_A\left(\infty \right)$, and $R_B\left(\infty \right)$ vs community strength $\mu$ under different information transmission rates (a) $\lambda =0.35$, (b) $\lambda =0.47$, (c) $\lambda =0.55$, and (d) $\lambda =0.7$. The solid line (circles), dashed line (squares), and dotted line (diamonds) represent the theoretical values (simulation values) of $R\left(\infty \right), R_A\left(\infty \right)$, and $R_B\left(\infty \right)$, respectively. Other parameters are set to be the same as in Fig. 3.

Figure 6

The FAP $R_A\left(\infty \right), R_B\left(\infty \right)$, and $R\left(\infty \right)$ vs community strength $\mu$ and information transmission rate $\lambda$. Color-coded values show simulation results in (a) $R_A\left(\infty \right)$, (c) $R_B\left(\infty \right)$, and (e) $R\left(\infty \right)$, and theoretical predictions in (b) $R_A\left(\infty \right)$, (d) $R_B\left(\infty \right)$, and (f) $R\left(\infty \right)$, respectively. Other parameters are set to be the same as in Fig. 3.

Figure 7

The behavioral information transmission rate $\lambda$ vs community strength $\mu$ for a given FAP (a) $R\left(\infty \right)=0.4$ and (b) $R\left(\infty \right)=0.7$. The dashed line (diamonds), solid line (circles), and dotted line (squares) denote the theoretical values (simulation values) of $m=2, a=0.5$; $m=5, a=0.2$; and $m=10, a=0.1$, respectively. Other parameters are set to be of $N=10000, {\rho }_0=0.03, \gamma =0.1$, and $\pi =3$, respectively.