Controlling Contagion Processes in Activity Driven Networks

The vast majority of strategies aimed at controlling contagion processes on networks consider the connectivity pattern of the system either quenched or annealed. However, in the real world, many networks are highly dynamical and evolve, in time, concurrently with the contagion process. Here, we derive an analytical framework for the study of control strategies specifically devised for a class of time-varying networks, namely activity-driven networks. We develop a block variable mean-field approach that allows the derivation of the equations describing the coevolution of the contagion process and the network dynamic. We derive the critical immunization threshold and assess the effectiveness of three different control strategies. Finally, we validate the theoretical picture by simulating numerically the spreading process and control strategies in both synthetic networks and a large-scale, real-world, mobile telephone call data set.

Figure 1

Schematic representation of activity driven model and control strategies. (a),(b) Temporal network at two different time steps $T_1$ and $T_2$. (c) Integrated network over a certain period of time. The size and color of each node describes its activity, while the width and color of each link describes the weight. (d)–(f) Random, targeted, and egocentric control strategy respectively. Immunized nodes are plotted as red squares; probes as yellow triangles.

Figure 2

(a)–(c) Phase space of a SIS process under random, targeted, and egocentric control strategy, respectively. Considering $N={10}^4$, $m=3$, $\epsilon ={10}^{-3}$, and activity distributed as $F(a)\sim a^{-2.2}$, we plot $I_{\infty }$ as a function of $\beta /\mu$ and $p$. The solid (red) lines represent the critical value $p_c$. (d) Comparison of the stationary state of a SIS model with and without control strategy, $I_{\infty }^p/I_{\infty }^0$, as a function of $p$ when $\beta /\mu =0.81$. In green triangles, we consider the random strategy, in blue diamonds the targeted strategy, and in orange circles the egocentric strategy. Each plot is made averaging ${10}^2$ independent simulations started with 1% of random seeds.

Figure 3

Comparison of the stationary state of a SIS model with and without control strategies. We plot the ratio $I_{\infty }^p/I_{\infty }^0$ for the random (egocentric, targeted) strategy as a function of $p$ with green triangles (orange circles, blue diamonds). In each case, the value $I_{\infty }^p$ describes the final fraction of diseased nodes when one of the three strategies is applied. The value $I_{\infty }^0$ describes the final fraction of diseased nodes without interventions. Every simulation was initiated with 1% infected seed, executed ${10}^2$ times for $T={10}^4$ steps with $\beta /\mu =2.5$. Each step was integrated for $6\times {10}^2$ seconds, and periodic temporal boundary condition was applied.