{
  "id": "2002.10128",
  "version": "v1",
  "published": "2020-02-24T09:38:18.000Z",
  "updated": "2020-02-24T09:38:18.000Z",
  "title": "Poisson Approximation and Connectivity in a Scale-free Random Connection Model",
  "authors": [
    "Srikanth K. Iyer",
    "Sanjoy Kr. Jhawar"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study an inhomogeneous random connection model on the unit cube $S=\\left(-\\frac{1}{2},\\frac{1}{2}\\right]^{d},$ $d \\geq 2$. The vertex set of the graph is a homogeneous Poisson point process $\\mathcal{P}_s$ on $S$ of intensity $s>0$. Each vertex is endowed with an independent random weight distributed as $W$, where $P(W>w)=w^{-\\beta}1_{[1,\\infty)}(w)$, $\\beta>0$. Given the vertex set and the weights an edge exists between $x,y\\in \\mathcal{P}_s$ with probability $\\left(1 - \\exp\\left( - \\frac{\\eta W_xW_y}{\\left(d(x,y)/r\\right)^{\\alpha}} \\right)\\right),$ independent of everything else, where $\\eta, \\alpha > 0$, $r > 0$ is a scaling parameter and $d$ is the toroidal metric on $S$. We derive conditions on $\\alpha, \\beta$ such that under the scaling $r_s(\\xi)^d= \\frac{1}{c_0 s} \\left( \\log s +(k-1) \\log\\log s +\\xi+\\log\\left(\\frac{\\alpha\\beta}{k!d} \\right)\\right),$ $\\xi \\in \\mathbb{R}$, the number of vertices of degree $k$ converges in total variation distance to a Poisson random variable with mean $e^{-\\xi}$ as $s \\to \\infty$, where $c_0$ is an explicitly specified constant that depends on $\\alpha, \\beta, d$ and $\\eta$ but not on $k$. In particular, for $k=0$ we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large $s$. The Poisson approximation result is derived using the Stein's method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-24T09:38:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60G70",
      "60G55",
      "05C80"
    ],
    "keywords": [
      "scale-free random connection model",
      "connectivity",
      "vertex set",
      "poisson approximation result",
      "inhomogeneous random connection model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}