# Advances in Probability and Mathematical StatisticsGeometric Random Graphs on Circles

Advances in Probability and Mathematical Statistics: Geometric Random Graphs on Circles [Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case when the underlying metric space is a circle of circumference L, and find a surprising dependency of behaviour on the rationality of L.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Advances in Probability and Mathematical StatisticsGeometric Random Graphs on Circles

Part of the Progress in Probability Book Series (volume 79)
Editors: Hernández‐Hernández, Daniel; Leonardi, Florencia; Mena, Ramsés H.; Pardo Millán, Juan Carlos
18 pages

# References (11)

Publisher
Springer International Publishing
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
ISBN
978-3-030-85324-2
Pages
23 –41
DOI
10.1007/978-3-030-85325-9_2
Publisher site
See Chapter on Publisher Site

### Abstract

[Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case when the underlying metric space is a circle of circumference L, and find a surprising dependency of behaviour on the rationality of L.]

Published: Aug 5, 2021

Keywords: Rado graph; Graph isomorphism; Geometric random graphs