Limit theorems for the number of crossings and stress in projections of the random geometric graph

2024/08/06 by Hanna Döring, Döring, Hanna, Lianne de Jonge +1 · 1 citation
Computer Science · Mathematics · #60D05 #60F05 #68R10 #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2408.03218

openalex publication_date 2024/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set W⊂ ℝ^d, d≥ 3, onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich-Rubinstein distance. We further show a multivariate central limit theorem between the number of crossings and a second variable called the stress that holds when the expected vertex degree in the random geometric graph converges to a positive finite value.

Cited by

• Existence of a plane without edge crossings in projections of the random geometric graph

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• Crossing Numbers and Stress of Random Graphs
• Existence of a plane without edge crossings in projections of the random geometric graph
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• Poisson fluctuations for edge counts in high-dimensional random geometric graphs
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