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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Journal of Statistical Physics, Springer Verlag, 2016, 165 (4), pp.785-798. <10.1007/s10955-016-1652-3>
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Contributeur : Eric Luçon <>
Soumis le : mercredi 24 août 2016 - 09:56:06
Dernière modification le : mercredi 9 novembre 2016 - 20:13:51

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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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