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

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

2008-09-06

arxiv.org/abs/0809.1143v1

Mathematics

Probability

Scientific paper

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 $\cE_N.$

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