Diffusion and topology: Large densely connected bipartite networks

Viral diffusion is an important area of research in networks, with applications from epidemiology to cyber and cuberphysical systems security. To study the infection dynamics of a single virus in a complete network, it suffices to study the dynamics of a single statistic - the fraction of infected nodes in the network. Here, we consider multiple viruses (two viruses for simplicity.) To go beyond the complete network, we consider a (special type of) bipartite network and show that, in this case and with two viruses, the state of the network is a four dimensional Markov process that collects the fractions of infected nodes in each island (bipartite classes) for each type of infection. The dynamics of this Markov process is described by a system of coupled stochastic integral equations. The stochastic dynamics leads, in the large scale limit (mean field), to four coupled nonlinear ordinary differential equations. To study the system qualitative behavior and determine the asymptotic distributions of infected nodes by each virus in each island, we resort to studying simpler configurations whose dynamics bound the dynamics of the original system and exhibit the same attractor.