{
  "id": "0909.1472",
  "version": "v3",
  "published": "2009-09-08T12:13:00.000Z",
  "updated": "2012-11-22T08:53:22.000Z",
  "title": "Novel scaling limits for critical inhomogeneous random graphs",
  "authors": [
    "Shankar Bhamidi",
    "Remco van der Hofstad",
    "Johan S. H. van Leeuwaarden"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/11-AOP680 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2012, Vol. 40, No. 6, 2299-2361",
  "doi": "10.1214/11-AOP680",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent \\tau. We investigate the case where $\\tau\\in(3,4)$, so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by $n^{-(\\tau-2)/(\\tau-1)}$, converge to hitting times of a \"thinned\" L\\'{e}vy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59]. Our results should be contrasted to the case \\tau>4, so that the third moment is finite. There, instead, the sizes of the components rescaled by $n^{-2/3}$ converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812-854] for the Erd\\H{o}s-R\\'{e}nyi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682-1703] and Turova [(2009) Preprint].",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-11-22T08:53:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical inhomogeneous random graphs",
      "novel scaling limits",
      "infinite third moment",
      "van der hofstad",
      "van leeuwaarden"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.1472B"
    }
  }
}