Open Access
October 2012 Nonuniform random geometric graphs with location-dependent radii
Srikanth K. Iyer, Debleena Thacker
Ann. Appl. Probab. 22(5): 2048-2066 (October 2012). DOI: 10.1214/11-AAP823


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 $nf(\cdot)$, where $n\in \mathbb{N}$, and $f$ is a probability density function on $\mathbb{R}^{d}$. 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.


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Srikanth K. Iyer. Debleena Thacker. "Nonuniform random geometric graphs with location-dependent radii." Ann. Appl. Probab. 22 (5) 2048 - 2066, October 2012.


Published: October 2012
First available in Project Euclid: 12 October 2012

zbMATH: 1261.05093
MathSciNet: MR3025688
Digital Object Identifier: 10.1214/11-AAP823

Primary: 60D05 , 60G70
Secondary: 05C05 , 90C27

Keywords: connectivity , location-dependent radii , Poisson point process , Random geometric graphs , vertex degrees

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.22 • No. 5 • October 2012
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