{
  "id": "2305.03607",
  "version": "v1",
  "published": "2023-05-05T15:10:34.000Z",
  "updated": "2023-05-05T15:10:34.000Z",
  "title": "Connectivity of inhomogeneous random graphs II",
  "authors": [
    "Jan Hladký",
    "Gopal Viswanathan"
  ],
  "comment": "20 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Each graphon $W:\\Omega^2\\rightarrow[0,1]$ yields an inhomogeneous random graph model $G(n,W)$. We show that $G(n,W)$ is asymptotically almost surely connected if and only if (i) $W$ is a connected graphon and (ii) the measure of elements of $\\Omega$ of $W$-degree less than $\\alpha$ is $o(\\alpha)$ as $\\alpha\\rightarrow 0$. These two conditions encapsulate the absence of several linear-sized components, and of isolated vertices, respectively. We study in bigger detail the limit probability of the property that $G(n,W)$ contains an isolated vertex, and, more generally, the limit distribution of the minimum degree of $G(n,W)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-05T15:10:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "connectivity",
      "inhomogeneous random graph model",
      "isolated vertex",
      "limit distribution",
      "limit probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}