Properties of metrics and infinite geometric graphs

TL;DR

This paper investigates the isomorphism properties of infinite random geometric graphs across various metrics, revealing conditions under which such graphs are unique or non-unique up to isomorphism, especially in norm-induced metric spaces.

Contribution

It generalizes previous results to a broad class of metrics, introducing the property of truncating and showing non-uniqueness of infinite geometric graphs in $L_p$ spaces for all finite p > 1.

Findings

In $ ext{R}^n$ with $L_{ ext{infty}}$-metric, the infinite random geometric graph is almost surely unique.

In $L_2$-metric for $n=2$, the graph is not unique, contrasting with the $L_{ ext{infty}}$ case.

In norm-induced metric spaces with the truncating property, the infinite geometric graph is not unique for all finite $p > 1$.

Abstract

We consider isomorphism properties of infinite random geometric graphs defined over a variety of metrics. In previous work, it was shown that for $\mathbb{R}^n$ with the $L_{\infty}$-metric, the infinite random geometric graph is, with probability 1, unique up to isomorphism. However, in the case $n=2$ this is false with either of the $L_2$-metric. We generalize this result to a large family of metrics induced by norms. Within this class of metric spaces, we show that the infinite geometric graph is not unique up to isomorphism if it has the new property which we name truncating: each step-isometry from a dense set to itself is an isometry. As a corollary, we derive that the infinite random geometric graph defined in $L_p$ space is not unique up to isomorphism with probability 1 for all finite $p>1.$