{
  "id": "2310.02785",
  "version": "v1",
  "published": "2023-10-04T13:03:46.000Z",
  "updated": "2023-10-04T13:03:46.000Z",
  "title": "Multivariate Regular Variation of Preferential Attachment Models",
  "authors": [
    "Anja Janßen",
    "Max Ziegenbalg"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We use the framework of multivariate regular variation to analyze the extremal behavior of preferential attachment models. To this end, we follow a directed linear preferential attachment model with a fixed number of outgoing edges per node for a random, heavy-tailed number of steps in time and treat the incoming edge count of all existing nodes as a random vector of random length. By combining martingale properties, moment bounds and a Breiman type theorem we show that the resulting quantity is multivariate regularly varying, both as a vector of fixed length formed by the edge counts of a finite number of oldest nodes, and also as a vector of random length viewed in sequence space. A P\\'{o}lya urn representation allows us to explicitly describe the extremal dependence between the degrees with the help of Dirichlet distributions. As a by-product of our analysis we establish new results for almost sure convergence of the edge counts in sequence space as the number of nodes goes to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-04T13:03:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "05C80",
      "60F25"
    ],
    "keywords": [
      "multivariate regular variation",
      "edge count",
      "random length",
      "sequence space",
      "directed linear preferential attachment model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}