Large population limits of Markov processes on random networks

TL;DR

This paper investigates how Markov processes on large random networks converge to mean-field models, providing conditions for this convergence and analyzing specific network types and heterogeneous populations.

Contribution

It establishes general conditions for the convergence of Markov process dynamics on random networks to mean-field limits, including various network models and heterogeneity.

Findings

Convergence conditions for mean-field limits are proved.

Analysis of voter model on different random graph models.

Extension to heterogeneous populations.

Abstract

We consider time-continuous Markovian discrete-state dynamics on random networks of interacting agents and study the large population limit. The dynamics are projected onto low-dimensional collective variables given by the shares of each discrete state in the system, or in certain subsystems, and general conditions for the convergence of the collective variable dynamics to a mean-field ordinary differential equation are proved. We discuss the convergence to this mean-field limit for a continuous-time noisy version of the so-called ``voter model'' on Erd\H{o}s-R\'enyi random graphs, on the stochastic block model, and on random regular graphs. Moreover, a heterogeneous population of agents is studied.