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Infinite random graphs and properties of metrics

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

We give a survey of recent developments in the theory of countably infinite random geometric graphs. Classical results of Erdős and Rényi establish that countably infinite random graphs are isomorphic with probability 1. Infinite random graphs have vertices identified with points in a metric space, and edges are added with a given probability dependent on the relative location of their endpoints. The probability that infinite random geometric graphs are isomorphic is considered. The metric spaces where such a unique isotype emerges are indeed fairly rare, and specifically arise in the context of finite-dimensional normed spaces equipped with the \(\ell_{\infty }\)-metric. We survey negative results for random geometric graphs in the cases of the Euclidean and hexagonal metric. Recent work which considers infinite random geometric graphs in the general setting of normed linear spaces is described. Open problems in the area are provided in the final section.

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Acknowledgements

The second author wishes to thank IMA, which she visited during the annual thematic program on Discrete Structures. IMA gave her the opportunity to present this work as part of the seminar series, and to discuss it with other visitors. Both authors acknowledge support from grants from NSERC.

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Authors and Affiliations

  1. Department of Mathematics, Ryerson University, Toronto, ON, Canada, M5B 2K3

    Anthony Bonato

  2. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada, B3H 3J5

    Jeannette Janssen

Authors
  1. Anthony Bonato
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  2. Jeannette Janssen
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Correspondence to Jeannette Janssen .

Editor information

Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Bonato, A., Janssen, J. (2016). Infinite random graphs and properties of metrics. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_11

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