Spreading of computer viruses on time-varying networks

Social networks are the prime channel for the spreading of computer viruses. Yet the study of their propagation neglects the temporal nature of social interactions and the heterogeneity of users' susceptibility. Here, we introduce a theoretical framework that captures both properties. We study two realistic types of viruses propagating on temporal networks featuring $Q$ categories of susceptibility and derive analytically the invasion threshold. We found that the temporal coupling of categories might increase the fragility of the system to cyber threats. Our results show that networks' dynamics and their interplay with users' features are crucial for the spreading of computer viruses.

Figure 1

Lifetime of the SIS process (a)–(c) and contour plot of $R_0({\lambda }_1,{\lambda }_2)$ (d)–(f). In (a), (b), (d), and (e) nodes are randomly assigned to two categories, in (c) and (f) instead in decreasing order of activity. We set (a), (d) $p=0.9$, and (b), (c), (e), (f) $p=0.4$. In (a)–(c) we fix $N=2\times {10}^5$, $m=4$, $\alpha =2.1$, ${\mu }_1={\mu }_2={10}^{-2}$, ${\lambda }_2=0.2$, $Y=0.3$, and $0.5\%$ of random initial seeds. We plot the median and $50\%$ confidence intervals in ${10}^2$ simulations per point. The solid lines come from Eq. (2), and the dashed lines are the analytical threshold in case of a single category.

Figure 2

In (a) we plot the analytical value of $R_0$ as function of $p$. The shaded area describes the region where ${min}_x{\beta }_x/{\mu }_x\le R_0\le {max}_x{\beta }_x/{\mu }_x$. The dashed vertical line describes the analytical value of $p$ above which $R_0>{max}_x{\beta }_x/{\mu }_x$. We set ${\mu }_1={10}^{-2}$ and ${\mu }_2=5\times {10}^{-3}$. In (b) we plot $p^{*}$ as a function of ${\mu }_1$ and ${\mu }_2$. In both plots, we set $m=4$, ${\lambda }_1=0.9$, ${\lambda }_2=0.2$, and randomly assign nodes to two categories.

Figure 3

Lifetime of the process (a)–(c), $R_0({\mu }_1,{\mu }_2)$ (d)–(f). In (a), (b), (d), and (e) nodes are randomly assigned to two categories, in (c) and (f) instead in decreasing order of activity. We set (a), (d) $p=0.9$, and (b), (c), (e), (f) $p=0.4$. In (a)–(c) we set $N=2\times {10}^5$, $m=4$, $\alpha =2.1$, ${\mu }_1={10}^{-2}$, ${\mu }_2=5\times {10}^{-3}$, ${\lambda }_2=0.2$, $Y=0.3$, and $0.5\%$ randomly selected seeds. We plot the median and $50\%$ confidence intervals in ${10}^2$ simulations per point. The solid lines come from Eq. (2). The dashed lines are the analytical threshold in case of a single category of recovery rate characterized by the average value of the recovery rates. In the contour plot we set ${\lambda }_1=0.485$ and ${\lambda }_2=0.2$.

Figure 4

Lifetime of the SIS process for (a)–(c) $\tau =2$, 3, and 10 for two categories to which nodes are assigned randomly. Simulations are done setting $N=2\times {10}^5$, $m=4$, $\alpha =2.1$, $Y=0.3$, $\mu ={10}^{-2}$, ${\lambda }_2=0.3$, $p=0.5$, and $0.5\%$ random initial seeds. We plot the median and $50\%$ confidence intervals in ${10}^2$ simulations per point.