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.

## 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 |
---|---|

Publication: | ANNALS OF APPLIED PROBABILITY |

Publisher: | INST MATHEMATICAL STATISTICS |

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.ac.in/id/eprint/45556 |

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