Number of Edges in Random Intersection Graph on Surface of a Sphere

TL;DR

This paper analyzes the properties of a random intersection graph formed by spherical caps on a sphere, deriving distributional results for edges and isolated vertices based on the cap area scaling.

Contribution

It establishes the Poisson distribution of edges and strong law results for isolated vertices in the graph under different scaling regimes of cap area.

Findings

Number of edges follows a Poisson distribution for certain parameters.

Almost surely no isolated vertices when the area scales as c/N^α with α<1.

All vertices are isolated when the area scales as c/N^α with α>3.

Abstract

In this article, we consider `$N$'spherical caps of area $4\pi p$ were uniformly distributed over the surface of a unit sphere. We study the random intersection graph $G_N$ constructed by these caps. We prove that for $p = \frac{c}{N^{\al}},\:c >0$ and $\al >2,$ the number of edges in graph $G_N$ follow the Poisson distribution. Also we derive the strong law results for the number of isolated vertices in $G_N$: for $p = \frac{c}{N^{\al}},\:c >0$ for $\al < 1,$ there is no isolated vertex in $G_N$ almost surely i.e., there are atleast $N/2$ edges in $G_N$ and for $\al >3,$ every vertex in $G_N$ is isolated i.e., there is no edge in edge set $\cE_N.$