How to determine if a random graph with a fixed degree sequence has a giant component

  • Felix Joos
  • Guillem Perarnau
  • Dieter Rautenbach
  • Bruce Reed
Article

Abstract

For a fixed degree sequence \({\mathcal {D}}=(d_1,\ldots ,d_n)\), let \(G({\mathcal {D}})\) be a uniformly chosen (simple) graph on \(\{1,\ldots ,n\}\) where the vertex i has degree \(d_i\). In this paper we determine whether \(G({\mathcal {D}})\) has a giant component with high probability, essentially imposing no conditions on \({\mathcal {D}}\). We simply insist that the sum of the degrees in \({\mathcal {D}}\) which are not 2 is at least \(\lambda (n)\) for some function \(\lambda \) going to infinity with n. This is a relatively minor technical condition, and when \({\mathcal {D}}\) does not satisfy it, both the probability that \(G({\mathcal {D}})\) has a giant component and the probability that \(G({\mathcal {D}})\) has no giant component are bounded away from 1.

Mathematics Subject Classification

05C80 05C82 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Felix Joos
    • 1
  • Guillem Perarnau
    • 1
  • Dieter Rautenbach
    • 2
  • Bruce Reed
    • 3
    • 4
    • 5
  1. 1.School of MathematicsUniversity of BirminghamBirminghamUK
  2. 2.Institute of Optimization and Operations ResearchUlm UniversityUlmGermany
  3. 3.CNRSNiceFrance
  4. 4.Kawarabayashi Large Graph Project, NIITokyoJapan
  5. 5.IMPARio de JaneiroBrazil

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