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Maximum degree for random geometric graphs in the connectivity regime and beyond
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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BibTeX
@misc{otsuka2026maximum,
title = {{Maximum degree for random geometric graphs in the connectivity regime and beyond}},
author = {Junpei Otsuka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15624},
eprint = {2610.15624},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15624},
}
RIS
TY - UNPB
TI - Maximum degree for random geometric graphs in the connectivity regime and beyond
AU - Junpei Otsuka
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15624
UR - https://arxiv.org/abs/2610.15624
AB - 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.
ER -