Abstract

A random geometric graph is defined as follows: for a set of random points in some set 𝐴 ⊆ ℝ^𝑑, fix 𝑟 > 0 and connect a pair of points 𝑋,𝑌, provided that‖ 𝑋 −𝑌 ‖< 𝑟. The graph generated with the set of points as the vertices and the random set of edges thus constructed, is called random geometric graph. Previous studies have been conducted under the assumption that the random set of points originate as an i.i.d. sample from a density function that is bounded away from zero on 𝐴. We restrict our attention to densities on [0,1] having a zero at the origin. Under this assumption, we study the asymptotic behaviour of connectivity distance of the random geometric graph. Our results show that the connectivity distance behaves differently as compared to the case of densities that are bounded away from zero.

Author: Anuradha

Received on: January, 2022

Accepted on: April, 2023