{
  "id": "2306.08113",
  "version": "v1",
  "published": "2023-06-13T20:05:29.000Z",
  "updated": "2023-06-13T20:05:29.000Z",
  "title": "Connectivity threshold for superpositions of Bernoulli random graphs",
  "authors": [
    "Daumilas Ardickas",
    "Mindaugas Bloznelis"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $G_1,\\dots, G_m$ be independent Bernoulli random subgraphs of the complete graph ${\\cal K}_n$ having variable sizes $x_1,\\dots, x_m\\in [n]$ and densities $q_1,\\dots, q_m\\in [0,1]$. Letting $n,m\\to+\\infty$, we study the connectivity threshold for the union $\\cup_{i=1}^mG_i$ defined on the vertex set of ${\\cal K}_n$. Assuming that the empirical distribution $P_{n,m}$ of the pairs $(x_1,q_1),\\dots, (x_m,q_m)$ converges to a probability distribution $P$ we show that the threshold is defined by the mixed moments $\\kappa_n=\\iint x(1-(1-q)^{|x-1|})P_{n,m}(dx,dq)$. For $\\ln n-\\frac{m}{n}\\kappa_n\\to-\\infty$ we have $P\\{\\cup_{i=1}^mG_i$ is connected$\\}\\to 1$ and for $\\ln n-\\frac{m}{n}\\kappa_n\\to+\\infty$ we have $P\\{\\cup_{i=1}^mG_i$ is connected$\\}\\to 0$. Interestingly, this dichotomy only holds if the mixed moment $\\iint x(1-(1-q)^{|x-1|})\\ln(1+x)P(dx,dq)<\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-13T20:05:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C82",
      "05C40"
    ],
    "keywords": [
      "bernoulli random graphs",
      "connectivity threshold",
      "superpositions",
      "independent bernoulli random subgraphs",
      "mixed moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}