NONUNIFORM RANDOM GEOMETRIC GRAPHS WITH LOCATION-DEPENDENT RADII

• 成果类型: Article
• 署名作者: Iyer, Srikanth K.; Thacker, Debleena
• 署名单位: Indian Institute of Science (IISC) - Bangalore; Indian Statistical Institute; Indian Statistical Institute Delhi
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP823
• 发表日期: 2012
• 页码: 2048-2066
• 关键词: nearest-neighbor link; sensor networks; connectivity; extremes; points

摘要

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.