Interacting particle systems on sparse $W$-random graphs

TL;DR

This paper studies the large population limit of interacting particle systems on sparse random graphs, showing convergence to a graphon-based stochastic differential equation that captures complex interactions.

Contribution

It introduces a framework for analyzing particle systems on sparse graphs converging to graphons, including unbounded cases, using $L^p$ theory and Fubini extension.

Findings

Convergence of particle systems to graphon SDEs for sparse graphs.

Framework accommodates nonlinear interactions.

Handles unbounded graphons with $L^p$ theory.

Abstract

We consider a general interacting particle system with interactions on a random graph, and study the large population limit of this system. When the sequence of underlying graphs converges to a graphon, we show convergence of the interacting particle system to a so called graphon stochastic differential equation. This is a system of uncountable many SDEs of McKean-Vlasov type driven by a continuum of Brownian motions. We make sense of this equation in a way that retains joint measurability and essentially pairwise independence of the driving Brownian motions of the system by using the framework of Fubini extension. The convergence results are general enough to cover nonlinear interactions, as well as various examples of sparse graphs. A crucial idea is to work with unbounded graphons and use the $L^p$ theory of sparse graph convergence.