Strong law for number of edges on surface of a unit sphere

In this article, we consider `N' spherical caps of area 4¿p were uniformly distributed over the surface of a unit sphere. We study the random intersection graph G
_N
constructed by these caps. we derive the strong law results for the number of isolated vertices in G
_N
: for p = c/N
^r
, c > 0 for r < 1, there is no isolated vertex in G
_N
almost surely i.e., there are atleast N/2 edges in G
_N
and for r > 3, every vertex in G
_N
is isolated i.e., there is no edge in edge set ¿
_N
.