Nonuniform random geometric graphs with location-dependent radii

Citation
K. Iyer, Srikanth et Thacker, Debleena, Nonuniform random geometric graphs with location-dependent radii, Annals of applied probability , 22(5), 2012, pp. 2048-2066
ISSN journal
10505164
Volume
22
Issue
5
Year of publication
2012
Pages
2048 - 2066
Database
ACNP
SICI code
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(.), where n.N, and f is a probability density function on Rd. A vertex located at x connects via directed edges to other vertices that are within a cut-off distance rn(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.