{
  "id": "1703.05943",
  "version": "v1",
  "published": "2017-03-17T09:53:35.000Z",
  "updated": "2017-03-17T09:53:35.000Z",
  "title": "The dynamics of power laws: Fitness and aging in preferential attachment trees",
  "authors": [
    "Alessandro Garavaglia",
    "Remco van der Hofstad",
    "Gerhard Woeginger"
  ],
  "comment": "41 pages, 10 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-17T09:53:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "preferential attachment trees",
      "limiting degree distribution",
      "power laws",
      "power-law tail",
      "exponential tails"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}