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Journal of Statistical Physics

, Volume 140, Issue 2, pp 289–335 | Cite as

The Cut Metric, Random Graphs, and Branching Processes

  • Béla Bollobás
  • Svante Janson
  • Oliver RiordanEmail author
Article

Abstract

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).

Keywords

Random graph Phase transition Branching process 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Béla Bollobás
    • 1
    • 2
  • Svante Janson
    • 3
  • Oliver Riordan
    • 4
    Email author
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  2. 2.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  3. 3.Department of MathematicsUppsala UniversityUppsalaSweden
  4. 4.Mathematical InstituteUniversity of OxfordOxfordUK

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