Geometric Random Graphs on Circles

Omer Angel, Yinon Spinka

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

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.

Original languageEnglish
Title of host publicationProgress in Probability
Pages23-41
Number of pages19
DOIs
StatePublished - 2021
Externally publishedYes

Publication series

NameProgress in Probability
Volume79

Keywords

  • Geometric random graphs
  • Graph isomorphism
  • Rado graph

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Applied Mathematics
  • Mathematical Physics
  • Mathematics (miscellaneous)

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