{
  "id": "1005.1136",
  "version": "v5",
  "published": "2010-05-07T06:05:00.000Z",
  "updated": "2011-08-30T06:04:06.000Z",
  "title": "Random graphs with a given degree sequence",
  "authors": [
    "Sourav Chatterjee",
    "Persi Diaconis",
    "Allan Sly"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/10-AAP728 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2011, Vol. 21, No. 4, 1400-1435",
  "doi": "10.1214/10-AAP728",
  "categories": [
    "math.PR",
    "math.CO",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have graph limits in the sense of Lov\\'{a}sz and Szegedy with identifiable limits. This allows simple determination of other features such as the number of triangles. The argument proceeds by studying a natural exponential model having the degree sequence as a sufficient statistic. The maximum likelihood estimate (MLE) of the parameters is shown to be unique and consistent with high probability. Thus $n$ parameters can be consistently estimated based on a sample of size one. A fast, provably convergent, algorithm for the MLE is derived. These ingredients combine to prove the graph limit theorem. Along the way, a continuous version of the Erd\\H{o}s--Gallai characterization of degree sequences is derived.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2011-08-30T06:04:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "degree sequence",
      "random graphs",
      "natural exponential model",
      "maximum likelihood estimate",
      "graph limit theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.1136C"
    }
  }
}