The longest edge of the one-dimensional soft random geometric graph with boundaries

The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been stud...

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Bibliographic Details
Published in:Stochastic models 2024-04, Vol.40 (2), p.399-416
Main Authors: Rousselle, Arnaud, Sönmez, Ercan
Format: Article
Language:English
Subjects:
Apexes
Asymptotic properties
Decay
Extreme value theory
Extreme values
Graph theory
Mathematics
maximum edge-length
Phase transitions
Poisson approximation
Polynomials
Primary: 05C80
random graphs
Secondary: 60F05
soft random geometric graph
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Summary:The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this article, we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant α, which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result, we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power α and we recover a phase transition which coincides with exactly the same phases in Benjamini and Berger [ 2 ] .
ISSN:1532-6349
1532-4214
DOI:10.1080/15326349.2023.2256825