{
  "id": "1210.3830",
  "version": "v2",
  "published": "2012-10-14T20:14:16.000Z",
  "updated": "2015-03-17T09:39:15.000Z",
  "title": "Spatial preferential attachment networks: Power laws and clustering coefficients",
  "authors": [
    "Emmanuel Jacob",
    "Peter Mörters"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, 632-662",
  "doi": "10.1214/14-AAP1006",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent $\\tau>2$. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value $\\tau=3$. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-14T20:14:16.000Z",
      "abstract": "We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favouring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent {\\tau} > 2. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value {\\tau} = 3. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.",
      "comment": "25 Pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-17T09:39:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60C05",
      "90B15"
    ],
    "keywords": [
      "spatial preferential attachment networks",
      "clustering coefficient",
      "power law",
      "probability mechanism favouring short distances",
      "empirical degree distributions converge"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.3830J"
    }
  }
}