A scaling limit for the degree distribution in sublinear preferential attachment schemes

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Abstract

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to 'non-tree' evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C0-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003).

Original languageEnglish (US)
Pages (from-to)703-731
Number of pages29
JournalRandom Structures and Algorithms
Volume48
Issue number4
DOIs
StatePublished - Jul 1 2016

Keywords

  • Degree distribution
  • Dynamical system
  • Fluid limit
  • Infinite
  • Law of large numbers
  • ODE
  • Preferential attachment
  • Random graphs
  • Semigroup
  • Sublinear weights
  • Uniqueness

ASJC Scopus subject areas

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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