## Annals of Applied Probability

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

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

Srikanth K. Iyer and Debleena Thacker

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/11-AAP823

**Mathematical Reviews number (MathSciNet)**

MR3025688

**Zentralblatt MATH identifier**

1261.05093

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G70: Extreme value theory; extremal processes

Secondary: 05C05: Trees 90C27: Combinatorial optimization

**Keywords**

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

#### 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