{
  "id": "1509.07033",
  "version": "v1",
  "published": "2015-09-23T15:35:25.000Z",
  "updated": "2015-09-23T15:35:25.000Z",
  "title": "Random networks with preferential growth and vertex death",
  "authors": [
    "Maria Deijfen"
  ],
  "journal": "Journal of Applied Probability 47, 1150-1163 (2010)",
  "categories": [
    "math.PR"
  ],
  "abstract": "A dynamic model for a random network evolving in continuous time is defined where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function $b$ of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function $d$ of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution $\\{p_k\\}$ is derived and analyzed for a number of specific choices of $b$ and $d$. When $b(i)=i+\\alpha$ and $d(i)=\\beta$ -- that is, linear preferential attachment for the newborn and random deaths -- then $p_k\\sim k^{-(2+\\alpha)}$. When $b(i)=i+1$ and $d(i)=\\beta(i+1)$, with $\\beta<1$, then $p_k\\sim (1+\\beta)^{-k}$, that is, if also the death rate is proportional to the fitness, then the power law distribution is lost. Furthermore, when $b(i)=i+1$ and $d(i)=\\beta(i+1)^\\gamma$, with $\\beta,\\gamma<1$, then $\\log p_k\\sim -k^\\gamma$ -- a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-23T15:35:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random network",
      "vertex death",
      "preferential growth",
      "existing vertex",
      "asymptotic empirical fitness distribution"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150907033D"
    }
  }
}