Vertex Degree of Random Geometric Graph on Exponentially Distributed Points

Abstract

Let $X_1,X_2,...$ be an infinite sequence of i.i.d. random vectors distributed exponentially with parameter $\lam .$ For each $y$ and $n\geq 1,$ form a graph $G_n(y)$ with vertex set $V_n = \{X_1,...,X_n\},$ two vertices are connected if and only if edge distance between them is greater then $y$, i.e, $\|X_i-X_j\| \leq y.$ Almost-sure asymptotic rates of convergence/divergence are obtained for the minimum and maximum vertex degree of the random geometric graph, as the number of vertices becomes large $n,$ and the edge distance varies with the number of vertices.