On the Growth of a Superlinear Preferential Attachment Scheme

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

1 Scopus citations

Abstract

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
EditorsStefano Olla, Peter Friz, Wolfgang König, Chiranjib Mukherjee
PublisherSpringer New York LLC
Pages243-265
Number of pages23
ISBN (Print)9783030153373
DOIs
StatePublished - Jan 1 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
Volume283
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceConference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016
CountryGermany
CityBerlin
Period8/15/168/19/16

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Sethuraman, S., & Venkataramani, S. C. (2019). On the Growth of a Superlinear Preferential Attachment Scheme. In S. Olla, P. Friz, W. König, & C. Mukherjee (Eds.), Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016 (pp. 243-265). (Springer Proceedings in Mathematics and Statistics; Vol. 283). Springer New York LLC. https://doi.org/10.1007/978-3-030-15338-0_9