{
  "id": "2311.09317",
  "version": "v1",
  "published": "2023-11-15T19:23:08.000Z",
  "updated": "2023-11-15T19:23:08.000Z",
  "title": "Connectivity threshold for superpositions of Bernoulli random graphs. II",
  "authors": [
    "Mindaugas Bloznelis",
    "Dominykas Marma",
    "Rimantas Vaicekauskas"
  ],
  "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 $X_1,\\dots, X_m\\in \\{0,1,2,\\dots\\}$ and densities $Q_1,\\dots, Q_m\\in [0,1]$. Letting $n,m\\to+\\infty$ we establish the connectivity threshold for the union $\\cup_{i=1}^mG_i$ defined on the vertex set of ${\\cal K}_n$. Assuming that $(X_1,Q_1), (X_2,Q_2),\\dots, (X_m,Q_m)$ are independent identically distributed bivariate random variables and $\\ln n -\\frac{m}{n}E\\bigl(X_1(1-(1-Q_1)^{|X_1-1|}\\bigr)\\to c$ we show that $P\\{\\cup_{i=1}^mG_i$ is connected$\\}\\to e^{-e^c}$.The result extends to the case of non-identically distributed random variables $(X_1,Q_1),\\dots, (X_m,Q_m)$ as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-15T19:23:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C82",
      "05C40"
    ],
    "keywords": [
      "bernoulli random graphs",
      "connectivity threshold",
      "independent bernoulli random subgraphs",
      "superpositions",
      "identically distributed bivariate random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}