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Nonuniform random geometric graphs with location-dependent radii

Iyer, Srikanth K and Thacker, Debleena (2012) Nonuniform random geometric graphs with location-dependent radii. In: ANNALS OF APPLIED PROBABILITY, 22 (5). pp. 2048-2066.

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Official URL: http://dx.doi.org/10.1214/11-AAP823

Abstract

We propose a distribution-free approach to the study of random geometric graphs. The distribution of vertices follows a Poisson point process with intensity function n f(center dot), where n is an element of N, and f is a probability density function on R-d. A vertex located at x connects via directed edges to other vertices that are within a cut-off distance r(n)(x). We prove strong law results for (i) the critical cut-off function so that almost surely, the graph does not contain any node with out-degree zero for sufficiently large n and (ii) the maximum and minimum vertex degrees. We also provide a characterization of the cut-off function for which the number of nodes with out-degree zero converges in distribution to a Poisson random variable. We illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero. Finally, we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually.

Item Type: Journal Article
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Additional Information: Copyright for this article belongs to INST MATHEMATICAL STATISTICS, CLEVELAND, USA
Keywords: Random geometric graphs;location-dependent radii;Poisson point process;vertex degrees;connectivity
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 17 Dec 2012 05:50
Last Modified: 17 Dec 2012 05:50
URI: http://eprints.iisc.ernet.in/id/eprint/45556

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