{
  "id": "0907.0897",
  "version": "v2",
  "published": "2009-07-05T22:44:28.000Z",
  "updated": "2009-08-18T11:31:39.000Z",
  "title": "Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1",
  "authors": [
    "Tatyana S. Turova"
  ],
  "comment": "Version 2: Added reference and correction",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the random graph on $n$ vertices $1, ..., n$. Each vertex $i$ is assigned a type $X_i$ with $X_1, ..., X_n$ being independent identically distributed as a nonnegative discrete random variable $X$. We assume that ${\\bf E} X^3<\\infty$. Given types of all vertices, an edge exists between vertices $i$ and $j$ independent of anything else and with probability $\\min \\{1, \\frac{X_iX_j}{n}(1+\\frac{a}{n^{1/3}}) \\}$. We study the critical phase, which is known to take place when ${\\bf E} X^2=1$. We prove that normalized by $n^{-2/3}$ the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion $B^a(s)$ with diffusion coefficient $\\sqrt{{\\bf E}X{\\bf E}X^3}$ and drift $a-\\frac{{\\bf E}X^3}{{\\bf E}X}s$. This shows that finiteness of ${\\bf E}X^3$ is the necessary condition for the diffusion limit. In particular, we conclude that the size of the largest connected component is of order $n^{2/3}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-08-18T11:31:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60G42"
    ],
    "keywords": [
      "critical inhomogeneous random graphs",
      "diffusion approximation",
      "asymptotic joint distributions",
      "discrete random",
      "largest connected component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0897T"
    }
  }
}