{
  "id": "1607.05224",
  "version": "v1",
  "published": "2016-07-18T18:15:01.000Z",
  "updated": "2016-07-18T18:15:01.000Z",
  "title": "A note on dynamical models on random graphs and Fokker-Planck equations",
  "authors": [
    "Sylvain Delattre",
    "Giambattista Giacomin",
    "Eric Luçon"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "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 Erd\\H{o}s-R\\'enyi 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-18T18:15:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C20",
      "60K35"
    ],
    "keywords": [
      "random graphs",
      "fokker-planck equations",
      "dynamical models",
      "finite time horizon",
      "nonlinear diffusion process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}