Vertex Degree of Random Geometric Graph on Exponentially Distributed Points

2006/09/07 by Bhupendra Gupta, B. M. Gupta, Gupta, Bhupendra
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Mathematics #Graph Theory and Algorithms #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.math/0609193

arxiv created 2006/09/07 · openalex publication_date 2006/09/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X₁,X₂,... be an infinite sequence of i.i.d. random vectors distributed exponentially with parameter \lam . For each y and n≥ 1, form a graph G_n(y) with vertex set V_n = \X₁,...,X_n\, two vertices are connected if and only if edge distance between them is greater then y, i.e, ‖X_i-X_j‖ ≤ 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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