The model theory of geometric random graphs
Logic
2023-04-24 v2 Probability
Abstract
We study the logical properties of infinite geometric random graphs, introduced by Bonato and Janssen. These are graphs whose vertex set is a dense ``generic'' subset of a metric space, where two vertices are adjacent with probability provided the distance between them is bounded by some constant number. We prove that for a large class of metric spaces, including circles, spheres and the complete Urysohn space, almost all geometric random graphs on a given space are elementary equivalent. Moreover, their first-order theory can reveal geometric properties of the underlying metric space.
Cite
@article{arxiv.2303.11292,
title = {The model theory of geometric random graphs},
author = {Omer Ben-Neria and Itay Kaplan and Tingxiang Zou},
journal= {arXiv preprint arXiv:2303.11292},
year = {2023}
}
Comments
Fixed some typos, added reference to previous work by Ackerman, Freer, Kruckman and Patel