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 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.
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Submitted on : Wednesday, August 24, 2016 - 9:56:06 AM
Last modification on : Thursday, April 11, 2019 - 4:02:09 PM

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Sylvain Delattre, Giambattista Giacomin, Eric Luçon. A note on dynamical models on random graphs and Fokker-Planck equations. Journal of Statistical Physics, Springer Verlag, 2016, 165 (4), pp.785-798. ⟨10.1007/s10955-016-1652-3⟩. ⟨hal-01355715⟩

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