Weakly interacting oscillators on dense random graphs

Abstract

We consider a class of weakly interacting particle systems of meanfield 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. The limit is shown to be the solution to a non-linear Fokker-Planck equation weighted by the (possibly random) graph limit. No regularity assumptions are made on the graphon limit so that our analysis allows for very general graph sequences, such as exchangeable random graphs. For these, we also prove a propagation of chaos result. Finally, we fully characterize the graph sequences for which the associated empirical measure converges to the mean-field limit, i.e., to the solution of the classical McKean-Vlasov equation.

Graphon particle system: Uniform-in-time concentration bounds

“…The study of graphon particle systems and associated finite particle models with meanfield heterogeneous interactions emerged recently ( [2,4,5,15,16,26,28]). There is also a growing number of applications of graphons in game theory; see e.g.…”

Learning Graphon Mean Field Games and Approximate Nash Equilibria

“…There have since been efforts to control cooperative graphon mean field systems with diffusive linear dynamics using spectral methods (Gao & Caines, 2019a;b). On the other hand, Bayraktar et al (2020); Bet et al (2020) consider large non-clustered systems in a continuous-time diffusion-type setting without control, while Aurell et al (2021b) and Aurell et al (2021a) consider continuous-time linear-quadratic systems and continuous-time jump processes respectively. To the best of our knowledge, only Vasal et al (2021) have considered solving and formulating a graphon mean field game in discrete time, though requiring analytic computation of an infinite-dimensional value function defined over all mean fields and thus being inapplicable to arbitrary problems in a black-box, learning manner.…”

A note on Fokker-Planck equations and graphons

“…In the last years, the study of interacting particle systems with a non-trivial dense network structure has been repeatedly addressed in the mathematical community: see, e.g., [4,10,21,9] for interacting diffusions, [5,6,20] for applications in mean-field games and [11] in the context of dynamical systems. Depending on the setting, many results on interacting particle systems are nowadays available [1,3,8,7,17] whenever the underlying graph sequence is converging, in some sense depending case by case, to a suitable object. More precisely, if the graph limit is a graphon, then, as the size of the system tends to infinity, the finite-time population behavior is suitably described by an infinite system of coupled non-linear Fokker-Planck equations, the coupling between equations being made by the graphon limit itself (see equation (1.1) for a simple example).…”

“…We refer to Proposition 1.3 for a precise statement which includes more general systems, including the degenerate case (σ ≡ 0). System (1.1) or, equivalently, the family of non-linear processes (1.2), have been proposed as a limit description in the literature [17,21,5,1,3], yet very little is known on their mathematical properties. We thus aim at mathematically addressing (1.1), firstly by showing the strong link with the graphon theory (including unlabeled graphons, see Proposition 1.5), secondly by proving that a simpler representation for both (1.1) and (1.2) exists, whenever the initial conditions and the underlying graphon satisfy a suitable condition, see Theorem 2.1 and, in particular, Corollary 2.3 and Proposition 2.6.…”