Diversity of multilayer networks and its impact on collaborating epidemics

Interacting epidemics on diverse multilayer networks are increasingly important in modeling and analyzing the diffusion processes of real complex systems. A viral agent spreading on one layer of a multilayer network can interact with its counterparts by promoting (cooperative interaction), suppressing (competitive interaction), or inducing (collaborating interaction) its diffusion on other layers. Collaborating interaction displays different patterns: (i) random collaboration, where intralayer or interlayer induction has the same probability; (ii) concentrating collaboration, where consecutive intralayer induction is guaranteed with a probability of 1; and (iii) cascading collaboration, where consecutive intralayer induction is banned with a probability of 0. In this paper, we develop a top-bottom framework that uses only two distributions, the overlaid degree distribution and edge-type distribution, to model collaborating epidemics on multilayer networks. We then state the response of three collaborating patterns to structural diversity (evenness and difference of network layers). For viral agents with small transmissibility, we find that random collaboration is more effective in networks with higher diversity (high evenness and difference), while the concentrating pattern is more suitable in uneven networks. Interestingly, the cascading pattern requires a network with moderate difference and high evenness, and the moderately uneven coupling of multiple network layers can effectively increase robustness to resist cascading failure. With large transmissibility, however, we find that all collaborating patterns are more effective in high-diversity networks. Our work provides a systemic analysis of collaborating epidemics on multilayer networks. The results enhance our understanding of biotic and informative diffusion through multiple vectors.

Figure 1

Bottom-top framework and SEIR model of multilayer networks. (a) An example of multilayer networks. (b) The description of the multilayer network in (a) by overlaid degrees and edge types. (c) The illustration of the state change of a node in a multilayer network according to the SEIR model.

Figure 2

Illustration of multilayer network diversity. (a) The demonstration network with high difference and evenness. (b) The demonstration network with low difference. (c) The demonstration network with low evenness. (d) Relation between difference and evenness. The shade area is all of the valid combination of difference and evenness, and we also show the position of three demonstration networks in the shade area.

Figure 3

Validation of the mean-field solution. Simulation results and mean-field solutions are compared for the three collaborating patterns. The simulation and analysis is performed on multilayer networks with a Poisson overlaid degree distribution and the number of layers fixed at 2. For random and concentrating collaboration, ${\beta }_0=1.1$, difference is 0.8 and evenness is 0.7. For cascading collaboration, ${\beta }_0=0.9$, difference is 0.55 and evenness is 0.9. Lines are mean-field solutions, and points are simulation results.

Figure 4

Contour plots of survival probability as function of structural evenness and likeness. (a) and (b) Random collaboration; (c) and (d) concentrating collaboration; and (e) and (f) cascading collaboration. The experimental network has two layers. The dispersibility ${\beta }_0$ is kept at 1.1 for random and concentrating collaboration and at 0.9 for cascading collaboration. The small transmissibility $\alpha$ is 0.5 for concentrating collaboration and 0.7 for random and cascading collaboration. The large transmissibility $\alpha$ is 0.9 for all collaborating patterns. The numbers indicate the values of corresponding contour lines, and the arrows indicate the approximative increasing gradient of contour lines.