Graph Limit for Interacting Particle Systems on Weighted Random Graphs

TL;DR

This paper establishes a framework for analyzing the large-population limit of interacting particle systems on weighted random graphs, demonstrating convergence to a deterministic graph-limit equation and extending results to switching graphs.

Contribution

Introduces a general framework for weighted random graphs extending graphons and proves convergence of particle systems to a deterministic graph-limit equation.

Findings

Finite particle systems converge to a deterministic graph-limit as population grows.

Weighted random graphs generalize traditional graphon models.

Switching weighted random graphs also converge to the same graph-limit.

Abstract

In this article, we study the large-population limit of interacting particle systems posed on weighted random graphs. In that aim, we introduce a general framework for the construction of weighted random graphs, generalizing the concept of graphons. We prove that as the number of particles tends to infinity, the finite-dimensional particle system converges in probability to the solution of a deterministic graph-limit equation, in which the graphon prescribing the interaction is given by the first moment of the weighted random graph law. We also study interacting particle systems posed on switching weighted random graphs, which are obtained by resetting the weighted random graph at regular time intervals. We show that these systems converge to the same graph-limit equation, in which the interaction is prescribed by a constant-in-time graphon.