Epidemic spreading on interconnected networks

Many real networks are not isolated from each other but form networks of networks, often interrelated in nontrivial ways. Here, we analyze an epidemic spreading process taking place on top of two interconnected complex networks. We develop a heterogeneous mean-field approach that allows us to calculate the conditions for the emergence of an endemic state. Interestingly, a global endemic state may arise in the coupled system even though the epidemics is not able to propagate on each network separately and even when the number of coupling connections is small. Our analytic results are successfully confronted against large-scale numerical simulations.

Figure 1

Phase diagram showing the healthy phase and the endemic phase for the case of a random coupling (top), linear correlations (middle), and superlinear correlations between external and internal degrees (bottom). In all cases, networks A and B are identical, with an exponential degree distribution with $\langle k_{aa}\rangle =\langle k_{bb}\rangle =10$ and minimum degree 2. The area depicted in solid gray corresponds to the healthy phase and the transition to the endemic state is represented by the black curve. The dashed orange area corresponds to the endemic state that needs the contribution of both internal and external links and the white area to the case when the pure bipartite network alone is able to sustain the endemic state. The horizontal dashed line marks the point where external connections outnumber internal ones. Notice that when $\beta$ increases, nodes with high internal degrees also have high external degrees with high probability. Because of this, these nodes are excellent spreaders. As a consequence, the network needs smaller values of $\Lambda$ and/or interconnectivity to be in the endemic phase.

Figure 2

Time evolution of the prevalence $\rho \left(t\right)$ below and above the critical point for $\Lambda =0.8$ for two identical networks interconnected with linear correlations, i.e., $\beta =1$. Each network has an exponential degree distribution with $\langle k\rangle =10$ and minimum degree 2. The size of each network is $N={10}^6$ and results are averaged over 100 different realizations of the process. Right at the critical point, the prevalence decays as $\rho \left(t\right)\sim t^{-1}$.

Figure 3

Critical lines for $\beta =0$, 1, and 2 for the same networks as in Fig. 2. Symbols correspond to numerical simulations, black lines are the theoretical predictions according to Eq. (9), and blue dashed lines are obtained by rescaling the $\Lambda$ axis as $\Lambda \to \Lambda {(1-\langle k\rangle /\langle k^2\rangle )}^{-1}$. Error bars are of the order of symbol sizes.