Journal of Statistical Physics

, Volume 165, Issue 4, pp 785–798 | Cite as

A Note on Dynamical Models on Random Graphs and Fokker–Planck Equations

  • Sylvain Delattre
  • Giambattista Giacomin
  • Eric Luçon
Article
  • 153 Downloads

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 \(\lim _{n \rightarrow \infty }p_n n= \infty \). The purpose of this note is 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.

Keywords

Interacting diffusions on graphs Mean field Nonlinear diffusion Fokker–Planck PDE Kuramoto models 

Mathematics Subject Classification

82C20 60K35 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sylvain Delattre
    • 1
  • Giambattista Giacomin
    • 1
  • Eric Luçon
    • 2
  1. 1.Laboratoire de Probabilités et Modèles Aléatoires, Sorbonne Paris Cité, UMR 7599Université Paris DiderotParisFrance
  2. 2.Laboratoire MAP5, Sorbonne Paris Cité, UMR 8145Université Paris DescartesParisFrance

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