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The Cut Metric, Random Graphs, and Branching Processes

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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).

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Béla Bollobás

  2. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Béla Bollobás

  3. Department of Mathematics, Uppsala University, P.O. Box 480, 751 06, Uppsala, Sweden

    Svante Janson

  4. Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, UK

    Oliver Riordan

Authors
  1. Béla Bollobás
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  2. Svante Janson
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  3. Oliver Riordan
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Corresponding author

Correspondence to Oliver Riordan.

Additional information

Research of B. Bollobás was supported in part by NSF grants DMS-0906634, CNS-0721983 and CCF-0728928, and ARO grant W911NF-06-1-0076

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Bollobás, B., Janson, S. & Riordan, O. The Cut Metric, Random Graphs, and Branching Processes. J Stat Phys 140, 289–335 (2010). https://doi.org/10.1007/s10955-010-9982-z

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  • DOI: https://doi.org/10.1007/s10955-010-9982-z

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