A Lyapunov Approach to Stochastic Interaction Dynamics Over Large-Scale Networks

We study stochastic interaction network models whereby a finite population of agents, identified with the nodes of a graph, update their states in response to pairwise interactions with their neighbors as well as spontaneous mutations. These include the main epidemic models, such as the Susceptible-Infected -Susceptible, the Susceptible-Infected-Recovered, and the Susceptible-Infected-Recovered-Susceptible models. We analyze the asymptotic behavior of such systems on Erdös-Rényi random graphs, in the limit as the population size grows large. Our approach is based on the use of (approximate) Lyapunov functions for Markov chains through which we can obtain stability results in terms of the corresponding invariant probabilities and on specific concentration results for Erdos-Renyi random graphs.