Weakly interacting oscillators on dense random graphs

TL;DR

This paper proves a Law of Large Numbers for weakly interacting oscillators on dense graphs, showing convergence to a mean-field limit described by a non-linear Fokker-Planck equation, including general graph sequences like exchangeable random graphs.

Contribution

It extends mean-field theory to unlabeled graph limits without regularity assumptions, encompassing a broad class of graph sequences including exchangeable random graphs.

Findings

Empirical measures converge to the mean-field limit under graph convergence.

The limit is characterized by a non-linear Fokker-Planck equation weighted by the graphon.

Identifies graph sequences where convergence to the McKean-Vlasov equation occurs.

Abstract

We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e., two particles are interacting if and only if they are connected in the underlying graph. We establish a Law of Large Numbers for the empirical measure of the system that holds whenever the graph sequence is convergent in the sense of graph limits theory, i.e., to a graphon. The limit is shown to be the solution of a non-linear Fokker-Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on unlabeled graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the mean-field limit, i.e., to the solution of a classical McKean-Vlasov equation.