{
  "id": "2408.07475",
  "version": "v1",
  "published": "2024-08-14T11:37:43.000Z",
  "updated": "2024-08-14T11:37:43.000Z",
  "title": "A logical limit law for the sequential model of preferential attachment graphs",
  "authors": [
    "Alperen Özdemir"
  ],
  "comment": "42 pages",
  "categories": [
    "math.PR",
    "math.CO",
    "math.LO"
  ],
  "abstract": "For a sequence of random graphs, the limit law we refer to is the existence of a limiting probability of any graph property that can be expressed in terms of predicate logic. A zero-one limit law is shown by Shelah and Spencer for Erd\\\"{o}s-Renyi graphs given that the connection rate has an irrational exponent. We show a limit law for preferential attachment graphs which admit a P\\'{o}lya urn representation. The two extreme cases of the parametric model, the uniform attachment graph and the sequential Barab\\'{a}si-Albert model, are covered separately as they exhibit qualitative differences regarding the distribution of cycles of bounded length in the graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-14T11:37:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "preferential attachment graphs",
      "logical limit law",
      "sequential model",
      "uniform attachment graph",
      "zero-one limit law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}