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Some large deviation results for sparse random graphs
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  • Published: March 1998

Some large deviation results for sparse random graphs

  • Neil O'Connell1 

Probability Theory and Related Fields volume 110, pages 277–285 (1998)Cite this article

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Summary.

We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. The proof yields an asymptotic formula for the probability that the random graph is connected.

We also present an LDP and related result for the number of isolated vertices. Here we make use of a simple but apparently unknown characterisation, which is obtained by embedding the random graph in a random directed graph. The results demonstrate that, at this scaling, the properties `connected' and `contains no isolated vertices' are not asymptotically equivalent. (At the threshold probability they are asymptotically equivalent.)

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Authors and Affiliations

  1. BRIMS, Hewlett-Packard Labs, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK, , , , , , GB

    Neil O'Connell

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  1. Neil O'Connell
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Received: 14 November 1996 / In revised form: 15 August 1997

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O'Connell, N. Some large deviation results for sparse random graphs. Probab Theory Relat Fields 110, 277–285 (1998). https://doi.org/10.1007/s004400050149

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  • Issue Date: March 1998

  • DOI: https://doi.org/10.1007/s004400050149

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  • Mathematics Subject Classification (1991): 60F10
  • 05C80
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