Diameter in ultra-small scale-free random graphs

Francesco Caravenna, Alessandro Garavaglia, Remco van der Hofstad (Corresponding author)

Research output: Contribution to journalArticleAcademicpeer-review

3 Citations (Scopus)
60 Downloads (Pure)

Abstract

It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k -(τ - 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2 log log n | log ( τ - 2 ) | and 4 log log n | log ( τ - 2 ) | , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order log log n precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2 / log d fwd . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

Original languageEnglish
Pages (from-to)444-498
Number of pages55
JournalRandom Structures and Algorithms
Volume54
Issue number3
DOIs
Publication statusPublished - May 2019

Keywords

  • configuration model
  • diameter
  • preferential attachment model
  • random graphs
  • scale free
  • ultra-small

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