{
  "id": "2008.11557",
  "version": "v1",
  "published": "2020-08-26T13:38:59.000Z",
  "updated": "2020-08-26T13:38:59.000Z",
  "title": "Asymptotics for cliques in scale-free random graphs",
  "authors": [
    "Fraser Daly",
    "Alastair Haig",
    "Seva Shneer"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we establish asymptotics (as the size of the graph grows to infinity) for the expected number of cliques in the Chung--Lu inhomogeneous random graph model in which vertices are assigned independent weights which have tail probabilities $h^{1-\\alpha}l(h)$, where $\\alpha>2$ and $l$ is a slowly varying function. Each pair of vertices is connected by an edge with a probability proportional to the product of the weights of those vertices. We present a complete set of asymptotics for all clique sizes and for all non-integer $\\alpha > 2$. We also explain why the case of an integer $\\alpha$ is different, and present partial results for the asymptotics in that case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-26T13:38:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60F05"
    ],
    "keywords": [
      "scale-free random graphs",
      "asymptotics",
      "chung-lu inhomogeneous random graph model",
      "assigned independent weights",
      "graph grows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}