## The Annals of Applied Probability

### Nonuniform random geometric graphs with location-dependent radii

#### 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 $nf(\cdot)$, 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.

#### Article information

Source
Ann. Appl. Probab., Volume 22, Number 5 (2012), 2048-2066.

Dates
First available in Project Euclid: 12 October 2012

https://projecteuclid.org/euclid.aoap/1350067993

Digital Object Identifier
doi:10.1214/11-AAP823

Mathematical Reviews number (MathSciNet)
MR3025688

Zentralblatt MATH identifier
1261.05093

#### Citation

Iyer, Srikanth K.; Thacker, Debleena. Nonuniform random geometric graphs with location-dependent radii. Ann. Appl. Probab. 22 (2012), no. 5, 2048--2066. doi:10.1214/11-AAP823. https://projecteuclid.org/euclid.aoap/1350067993

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