On the Growth of a Superlinear Preferential Attachment Scheme

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

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

Fingerprint

Preferential Attachment
Vertex of a graph
Law of large numbers
Degree Distribution
Graph Model
Graph in graph theory
Condensation
Random Graphs
Count
Directly proportional
Fluctuations
Verify

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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

On the Growth of a Superlinear Preferential Attachment Scheme. / Sethuraman, Sunder; Venkataramani, Shankar C.

Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016. ed. / Stefano Olla; Peter Friz; Wolfgang König; Chiranjib Mukherjee. Springer New York LLC, 2019. p. 243-265 (Springer Proceedings in Mathematics and Statistics; Vol. 283).

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

Sethuraman, S & Venkataramani, SC 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. Springer Proceedings in Mathematics and Statistics, vol. 283, Springer New York LLC, pp. 243-265, Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016, Berlin, Germany, 8/15/16. https://doi.org/10.1007/978-3-030-15338-0_9
Sethuraman S, Venkataramani SC. On the Growth of a Superlinear Preferential Attachment Scheme. In Olla S, Friz P, König W, Mukherjee C, editors, Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016. Springer New York LLC. 2019. p. 243-265. (Springer Proceedings in Mathematics and Statistics). https://doi.org/10.1007/978-3-030-15338-0_9
Sethuraman, Sunder ; Venkataramani, Shankar C. / On the Growth of a Superlinear Preferential Attachment Scheme. Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016. editor / Stefano Olla ; Peter Friz ; Wolfgang König ; Chiranjib Mukherjee. Springer New York LLC, 2019. pp. 243-265 (Springer Proceedings in Mathematics and Statistics).
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