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arXiv · 2610.15624

Maximum degree for random geometric graphs in the connectivity regime and beyond

Junpei Otsuka
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Abstract

A random geometric graph is a random graph model whose vertex set is a binomial point process, where two vertices are connected by an edge if each lies within the ball of radius $r$ centered at the other. In this paper, we investigate the asymptotic fluctuations of the threshold radius required for the maximum degree of the graph to reach a target value. Assuming this target value grows with the number of vertices $n$ at an order of at least $\log n$ but strictly less than $(\log n)^d$, we show that these fluctuations follow a Gumbel distribution. We also investigate the spatial configurations of vertices with extremely large degrees, and show that the point process marked by their degrees converges in distribution to a Poisson point process.

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Junpei Otsuka. 2026-10-11. Maximum degree for random geometric graphs in the connectivity regime and beyond. https://arxiv.org/abs/2610.15624

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