An invariance principle based concentration result for large-scale stochastic pairwise interaction network systems

TL;DR

This paper establishes a concentration result for the stationary distributions of large-scale stochastic network systems with pairwise interactions, applicable to various models including epidemic, social, and evolutionary dynamics, under broad network conditions.

Contribution

It introduces an invariance principle-based approach to analyze the asymptotic behavior of stationary distributions in large stochastic networks, extending beyond fully mixed populations.

Findings

Concentration of stationary distributions as population size grows

Applicability to diverse network structures including Erdős-Rényi graphs

Unified analysis for epidemic, social, and evolutionary models

Abstract

We study stochastic pairwise interaction network systems whereby a finite population of agents, identified with the nodes of a graph, update their states in response to both individual mutations and pairwise interactions with their neighbors. The considered class of systems include the main epidemic models -such as the SIS, SIR, and SIRS models-, certain social dynamics models -such as the voter and anti-voter 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 these stationary distributions 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\"oos-R\'enyi random graphs.