A note on dynamical models on random graphs and Fokker-Planck equations

TL;DR

This paper investigates how well large random graph-based diffusion models approximate mean field models, using Fokker-Planck equations, and discusses limitations for finite system sizes relevant to simulations and biological systems.

Contribution

It establishes the proximity of random graph models to mean field models via Fokker-Planck PDEs and highlights the limitations for finite system sizes.

Findings

Proximity holds on finite time horizons for large graphs.

Both systems are described by Fokker-Planck PDEs in the infinite limit.

Finite size effects limit applicability to real-world systems.

Abstract

We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erd\H{o}s-R\'enyi graphs with edge probability $p_n$, $n$ is the number of vertices, such that $\lim_{n \to \infty}p_n n= \infty$. The purpose of this note it twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE (or, equivalently, by a nonlinear diffusion process) in the $n=\infty$ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with $N$ large but finite, for example the values of $N$ that can be reached in simulations or that correspond to the typical number of interacting units in a biological system.