{
  "id": "2105.05709",
  "version": "v2",
  "published": "2021-05-12T14:51:53.000Z",
  "updated": "2022-02-09T17:04:55.000Z",
  "title": "Graph distances in scale-free percolation: the logarithmic case",
  "authors": [
    "Nannan Hao",
    "Markus Heydenreich"
  ],
  "comment": "20 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\\in\\mathbb{Z}^d$ are linked by an edge with probability depending on i.i.d.\\ vertex weights and the Euclidean distance $|x-y|$. Depending on the various parameters involved, we get a rich phase diagram. We study graph distances and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-09T17:04:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "05C80"
    ],
    "keywords": [
      "scale-free percolation",
      "logarithmic case",
      "euclidean distance",
      "spatial random graph model",
      "light tail regime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}