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arXiv · 2604.20984

Graphon Limits of Graph Reaction--Diffusion Equations

Edith J. Zhang·James Scott·Qiang Du
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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.

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Edith J. Zhang, James Scott, Qiang Du. 2026-04-22. Graphon Limits of Graph Reaction--Diffusion Equations. https://arxiv.org/abs/2604.20984

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