Weakly interacting oscillators on dense random graphs

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.