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

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• 作者：Sylvain Delattre ; Giambattista Giacomin ; Eric Luçon
• 关键词：Interacting diffusions on graphs ; Mean field ; Nonlinear diffusion ; Fokker–Planck PDE ; Kuramoto models
• 刊名：Journal of Statistical Physics
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：165
• 期：4
• 页码：785-798
• 全文大小：521 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613
• 卷排序：165

文摘

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.