Graphon Limits of Graph Reaction--Diffusion Equations

TL;DR

This paper establishes the convergence of solutions of graph-based reaction-diffusion equations to a limiting nonlocal graphon RD equation as the underlying graphs grow large and converge in cut norm.

Contribution

It introduces a new framework connecting graph reaction-diffusion equations with graphon limits and proves convergence results for both deterministic solutions and stochastic processes.

Findings

Solutions of graph RD equations converge in $L^p$ norm to the graphon RD equation.

Stochastic processes on graphs converge in probability to the graphon RD equation solution.

Provides a large numbers limit connecting stochastic processes and graphon equations.

Abstract

A graph reaction--diffusion (RD) equation is a system of differential equations that is defined on the nodes of a graph. Consider a sequence of growing graphs that converges in cut norm to a limiting graphon. We show that the solutions of the sequence of graph RD equations converge in $L^p$ norm, for $p \in [1,\infty]$, to the solution of a limiting nonlocal RD equation, which we call a graphon RD equation. Furthermore, we show a large numbers result for a stochastic particle process that consists of a random walk and a birth-death process on graphs. For a sequence of graphs that converge in cut norm to a limiting graphon, the sequence of stochastic processes converges in probability to the solution of the graphon RD equation.