Connectivity of Soft Random Geometric Graphs Over Annuli

Alexander P. Giles, Orestis Georgiou, Carl P. Dettmann

Published 2015-02-18Version 1

We present analytic formulas for the connection probability of dense, soft random geometric graphs formed within annuli and spherical shells using a connection model based on the Rayleigh fading of radio-frequency data signals. In these graphs, $N$ nodes are randomly distributed within the domain to form a point pattern of stochastically linked pairs whose Euclidean separation $r_{ij}$ determines the probability $H\left(r_{ij}\right)$ that they connect, given mutual visibility. The connecting graphs are then analytically enumerated, quantifying for the first time the effect of obstacles on the connectivity of wireless ad hoc networks.