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Full Connectivity: Corners, Edges and Faces

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Abstract

We develop a cluster expansion for the probability of full connectivity of high density random networks in confined geometries. In contrast to percolation phenomena at lower densities, boundary effects, which have previously been largely neglected, are not only relevant but dominant. We derive general analytical formulas that show a persistence of universality in a different form to percolation theory, and provide numerical confirmation. We also demonstrate the simplicity of our approach in three simple but instructive examples and discuss the practical benefits of its application to different models.

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Notes

  1. If the convexity assumption is relaxed it is a semi-metric function, i.e. the triangle inequality may not be satisfied. For example, r ij is effectively infinite if there is an obstacle blocking the path of a wireless signal, but adding a node k that avoids the obstacle may permit indirect connection between i and j, thus in this case r ij >r ik +r kj . We defer the treatment of obstacles to a future paper.

  2. Other link models can also be considered. These may include single-input multiple-output (SIMO), multiple-input single-output (MISO), and multiple-input multiple-output (MIMO) (see also [39]).

  3. Typically η=2 corresponds to propagation in free space but for practical reasons it is often modeled as η>2 for cluttered environments.

  4. This was taken from a standard Rayleigh fading link model (also see [39]).

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Acknowledgements

The authors would like to thank the directors of the Toshiba Telecommunications Research Laboratory for their support.

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Authors and Affiliations

  1. Toshiba Telecommunications Research Laboratory, 32 Queen Square, Bristol, BS1 4ND, UK

    Justin Coon

  2. School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Carl P. Dettmann

  3. Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187, Dresden, Germany

    Orestis Georgiou

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  1. Justin Coon
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  2. Carl P. Dettmann
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  3. Orestis Georgiou
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Correspondence to Orestis Georgiou.

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Coon, J., Dettmann, C.P. & Georgiou, O. Full Connectivity: Corners, Edges and Faces. J Stat Phys 147, 758–778 (2012). https://doi.org/10.1007/s10955-012-0493-y

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