{
  "id": "1411.3417",
  "version": "v1",
  "published": "2014-11-13T01:48:20.000Z",
  "updated": "2014-11-13T01:48:20.000Z",
  "title": "Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph",
  "authors": [
    "Shankar Bhamidi",
    "Nicolas Broutin",
    "Sanchayan Sen",
    "Xuan Wang"
  ],
  "comment": "99 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter $t$ (usually related to edge density) and a (model dependent) critical time $t_c$ which specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erd\\H{o}s-R\\'enyi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like $n^{2/3}$) and (b) the structure of components (rescaled by $n^{-1/3}$) converge to random fractals related to the continuum random tree. Till date, (a) has been proven for a number of models using different techniques while (b) has been proven for only two models, the classical Erd\\H{o}s-R\\'enyi random graph and the rank-1 inhomogeneous random graph. The aim of this paper is to develop a general program for proving such results. The program requires three main ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent (ii) scaling exponents of susceptibility functions are the same as the Erd\\H{o}s-R\\'enyi random graph and (iii) macroscopic averaging of expected distances between random points in the same component in the barely subcritical regime. We show that these apply to a number of fundamental random graph models including the configuration model, inhomogeneous random graphs modulated via a finite kernel and bounded size rules. Thus these models all belong to the domain of attraction of the classical Erd\\H{o}s-R\\'enyi random graph. As a by product we also get results for component sizes at criticality for a general class of inhomogeneous random graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-13T01:48:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "05C80"
    ],
    "keywords": [
      "erdős-rényi random graph",
      "inhomogeneous random graph",
      "scaling limits",
      "criticality",
      "attraction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 99,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.3417B"
    }
  }
}