An Invariance Principle-Based Concentration Result for Large-Scale Stochastic Pairwise Interaction Network Systems

We study stochastic pairwise interaction network systems whereby a finite population of agents, identified with the nodes of a (directed) graph, update their states in response to both individual mutations and pairwise interactions with their neighbors. The considered class of systems includes the main epidemic models —such as the SIS, SIR, and SIRS models—, certain social dynamics models —such as the voter and antivoter models—, as well as evolutionary dynamics on graphs. Since these stochastic systems fall into the class of finite-state Markov chains, they always admit stationary distributions. We analyze the asymptotic behavior of the stationary distributions of stochastic pairwise interaction network systems in the limit as the population size grows large, while the interaction network maintains certain mixing properties. Our approach relies on the use of Lyapunov-type functions to obtain concentration results on these stationary distributions. Notably, our results are not limited to fully mixed population models, as they do apply to a much broader spectrum of interaction network structures, including, e.g., Erd¨ os-R´ enyi random graphs.