{
  "id": "1712.00875",
  "version": "v1",
  "published": "2017-12-04T02:02:23.000Z",
  "updated": "2017-12-04T02:02:23.000Z",
  "title": "Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds",
  "authors": [
    "Florentin Münch",
    "Radoslaw K. Wojciechowski"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation. This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm. Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-04T02:02:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21"
    ],
    "keywords": [
      "general graph laplacians",
      "heat equation",
      "ollivier ricci curvature",
      "ollivier curvature",
      "diameter bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}