NONUNIFORM RANDOM GEOMETRIC GRAPHS WITH LOCATION-DEPENDENT RADII

作者：Iyer Srikanth K^*; Thacker Debleena

来源：Annals of Applied Probability, 2012, 22(5): 2048-2066.

DOI：10.1214/11-AAP823

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

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.

• 出版日期2012-10