{
  "id": "1403.1274",
  "version": "v2",
  "published": "2014-03-05T21:15:27.000Z",
  "updated": "2014-03-07T05:27:33.000Z",
  "title": "Almost optimal sparsification of random geometric graphs",
  "authors": [
    "Nicolas Broutin",
    "Luc Devroye",
    "Gabor Lugosi"
  ],
  "categories": [
    "math.PR",
    "cs.DM",
    "cs.NI",
    "math.CO"
  ],
  "abstract": "A random geometric irrigation graph $\\Gamma_n(r_n,\\xi)$ has $n$ vertices identified by $n$ independent uniformly distributed points $X_1,\\ldots,X_n$ in the unit square $[0,1]^2$. Each point $X_i$ selects $\\xi_i$ neighbors at random, without replacement, among those points $X_j$ ($j\\neq i$) for which $\\|X_i-X_j\\| < r_n$, and the selected vertices are connected to $X_i$ by an edge. The number $\\xi_i$ of the neighbors is an integer-valued random variable, chosen independently with identical distribution for each $X_i$ such that $\\xi_i$ satisfies $1\\le \\xi_i \\le \\kappa$ for a constant $\\kappa>1$. We prove that when $r_n = \\gamma_n \\sqrt{\\log n/n}$ for $\\gamma_n \\to \\infty$ with $\\gamma_n =o(n^{1/6}/\\log^{5/6}n)$, then the random geometric irrigation graph experiences explosive percolation in the sense that when $\\mathbf E \\xi_i=1$, then the largest connected component has size $o(n)$ but if $\\mathbf E \\xi_i >1$, then the size of the largest connected component is with high probability $n-o(n)$. This offers a natural non-centralized sparsification of a random geometric graph that is mostly connected.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-07T05:27:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60C05"
    ],
    "keywords": [
      "random geometric graph",
      "optimal sparsification",
      "random geometric irrigation graph experiences",
      "largest connected component",
      "irrigation graph experiences explosive percolation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.1274B"
    }
  }
}