{
  "id": "1910.13227",
  "version": "v1",
  "published": "2019-10-29T12:37:30.000Z",
  "updated": "2019-10-29T12:37:30.000Z",
  "title": "Critical scaling limits of the random intersection graph",
  "authors": [
    "Lorenzo Federico"
  ],
  "comment": "30 pages, no figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We analyse the scaling limit of the sizes of the largest components of the Random Intersection Graph $G(n,m,p)$ close to the critical point $p=\\frac{1}{\\sqrt{nm}}$, when the numbers $n$ of individuals and $m$ of communities have different orders of magnitude. We find out that if $m \\gg n$, then the scaling limit is identical to the one of the \\ER Random Graph (ERRG), while if $n \\gg m$ the critical exponent is similar to that of Inhomogeneous Random Graphs with heavy-tailed degree distributions, yet the rescaled component sizes have the same limit in distribution as in the ERRG. This suggests the existence of a wide universality class of inhomogeneous random graph models such that in the critical window the largest components have sizes of order $n^{\\rho}$ for some $\\rho \\in (1/2,2/3]$, which depends on some parameter of the graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-29T12:37:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "90B15",
      "82B27"
    ],
    "keywords": [
      "random intersection graph",
      "critical scaling limits",
      "largest components",
      "wide universality class",
      "inhomogeneous random graph models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}