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

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 Erdős-Rényi 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.

14 pages, 1 figure. A few corrections, streamlined some explanations and improved Corollary 1.2

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