{
  "id": "2212.05551",
  "version": "v1",
  "published": "2022-12-11T17:31:18.000Z",
  "updated": "2022-12-11T17:31:18.000Z",
  "title": "Universality of the local limit of preferential attachment models",
  "authors": [
    "Alessandro Garavaglia",
    "Rajat Subhra Hazra",
    "Remco van der Hofstad",
    "Rounak Ray"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014). The degree distribution of this limiting random graph, which we call the random P\\'{o}lya point tree, has a surprising size-biasing phenomenon. Many of the existing preferential attachment models can be viewed as special cases of our preferential attachment model with i.i.d. out-degrees. Additionally, our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees. Our proof of local convergence consists of two main steps: a P\\'olya urn description of our graphs, and an explicit identification of the neighbourhoods in them. We provide a novel and explicit proof to establish a coupling between the preferential attachment model and the P\\'{o}lya urn graph. Our result proves a density convergence result, for fixed ages of vertices in the local limit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-11T17:31:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local limit",
      "universality",
      "preferential attachment fitness parameter",
      "study preferential attachment models",
      "existing preferential attachment models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}