{
  "id": "1406.1758",
  "version": "v1",
  "published": "2014-06-06T17:52:34.000Z",
  "updated": "2014-06-06T17:52:34.000Z",
  "title": "Scaling limits and influence of the seed graph in preferential attachment trees",
  "authors": [
    "Nicolas Curien",
    "Thomas Duquesne",
    "Igor Kortchemski",
    "Ioan Manolescu"
  ],
  "comment": "32 pages, 11 figures",
  "categories": [
    "math.PR",
    "cs.DM",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barab\\'asi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \\& R\\'acz concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barab\\'asi-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension $2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-06T17:52:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "preferential attachment trees",
      "seed graph",
      "scaling limits",
      "random compact metric space",
      "brownian continuum random tree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.1758C"
    }
  }
}