On the Growth of a Superlinear Preferential Attachment Scheme

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations


We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree, while all others have finite degree. In this note, we establish a.s. law of large numbers type limits and fluctuation results, as n ↑ ∞, for the counts of the number of nodes with degree k ≥ 1 at time n ≥ 1. These limits rigorously verify and extend a physical picture of Krapivisky et al. (Phys Rev Lett 85:4629–4632, 2000 [16]) on how the condensation arises with respect to the degree distribution.

Original languageEnglish (US)
Title of host publicationProbability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016
EditorsPeter Friz, Wolfgang König, Chiranjib Mukherjee, Stefano Olla
PublisherSpringer New York LLC
Number of pages23
ISBN (Print)9783030153373
StatePublished - 2019
EventConference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016 - Berlin, Germany
Duration: Aug 15 2016Aug 19 2016

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


ConferenceConference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016


  • Degree distribution
  • Fluctuations
  • Growth
  • Preferential attachment
  • Random graphs
  • Superlinear

ASJC Scopus subject areas

  • General Mathematics


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