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

International audienceWe 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 Erdos-Renyi 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