Vertex Degree of Random Geometric Graph on Exponentially Distributed Points

Details Vertex Degree of Random Geometric Graph on Exponentially Distributed Points Vertex Degree of Random Geometric Graph on Exponentially Distributed Points

2006-09-07

arxiv.org/abs/math/0609193v1

Mathematics

Probability

Scientific paper

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.

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