{
  "id": "1311.3897",
  "version": "v2",
  "published": "2013-11-15T16:05:00.000Z",
  "updated": "2015-03-23T11:55:29.000Z",
  "title": "Connectivity of soft random geometric graphs",
  "authors": [
    "Mathew D. Penrose"
  ],
  "comment": "45 Pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n \\to \\infty$ the probability of full connectivity is governed by that of having no isolated vertices, itself governed by a Poisson approximation for the number of isolated vertices, uniformly over all choices of $p,r$. We determine the asymptotic probability of connectivity for all $(p_n,r_n)$ subject to $r_n = O(n^{-\\epsilon}),$ some $\\epsilon >0$. We generalize the first result to higher dimensions, and to a larger class of connection probability functions in $d=2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-15T16:05:00.000Z",
      "comment": "40 Pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-23T11:55:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60D05",
      "05C40",
      "60K35"
    ],
    "keywords": [
      "soft random geometric graphs",
      "connectivity",
      "uniform random points",
      "connection probability functions",
      "isolated vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.3897P"
    }
  }
}