球面上随机交图的边数
Number of Edges in Random Intersection Graph on Surface of a Sphere
摘要：
在这篇文章中，我们认为“$N$”球冠面积$4πp$均匀分布在一个单位球的表面上。我们研究了由这些帽构造的随机交图$G_N$。证明了对于$p = \frac{c}{N^{\al}},\:c >0$和$\al >2,$，图$G_N$的边数服从Poisson分布.我们还得到了关于$G_N$中孤立点个数的强律结果：对于$p = \frac{c}{N^{\al}},\:c >0$对于$\al < 1,$，几乎必然在$G_N$中没有孤立点，即，在$G_N$中至少存在$N/2$边并且对于$\al >3,$，$G_N$中的每个顶点是孤立的即，边集合$\cE_N.$中没有边
英文摘要：
In this article, we consider `$N$'spherical caps of area $4πp$ 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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