The nonlinear heat equation on W-random graphs

TL;DR

This paper establishes the continuum limit for coupled differential equations on W-random graphs, showing convergence of solutions from discrete models to an integral equation, thus justifying the thermodynamic limit for such systems.

Contribution

It introduces a rigorous derivation of the continuum limit for systems on W-random graphs, extending previous work on deterministic networks.

Findings

Solutions of discrete IVPs converge to the continuum solution

The continuum limit is represented as an evolution integral equation

Results justify the thermodynamic limit for large graph families

Abstract

For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in [9] justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs.