A note on Fokker-Planck equations and graphons

TL;DR

This paper explores the relationship between graphon-based interaction networks and particle system behavior, providing simplified solution representations and approximation methods for complex heterogeneous interactions.

Contribution

It introduces a simplified solution form for graphon particle systems and demonstrates how step-kernels can approximate these systems with explicit convergence rates.

Findings

Different graphons can produce identical particle behaviors.

Independence of initial conditions from the graphon leads to mean-field behavior.

Step-kernels effectively approximate graphon particle systems.

Abstract

Fokker-Planck equations represent a suitable description of the finite-time behavior for a large class of particle systems as the size of the population tends to infinity. Recently, the theory of graph limits has been introduced in the classical mean-field framework to account for heterogeneous interactions among particles. In many instances, such network heterogeneity is preserved in the limit which turns from being a single Fokker-Planck equation (also known as McKean-Vlasov) to an infinite system of non-linear partial differential equations (PDE) coupled by means of a graphon. While appealing from an applied viewpoint, few rigorous results exist on the graphon particle system. This note addresses such limit system focusing on the relation between interaction network and initial conditions: if the system initial datum and the graphon degrees satisfy a suitable condition, a significantly simpler representation of the solution is available. Interesting conclusions can be drawn from this result: substantially different graphons can give rise to exactly the same particle behavior, e.g., if the initial condition is independent of the graphon, the particle behavior is indistinguishable from the well-known mean-field limit as long as the graphon has constant degree; step-kernels, a building block of graph limit theory, can be used to approximate the graphon particle system with a finite system of Fokker-Planck equations and an explicit rate of convergence.