Geometric random graphs and Rado sets in sequence spaces

TL;DR

This paper studies random geometric graphs in infinite-dimensional sequence spaces, showing that under certain conditions, almost all dense countable sets are Rado, leading to unique graph isomorphism classes that can recover the underlying space.

Contribution

It extends the concept of Rado sets and geometric random graphs from finite to infinite-dimensional spaces, specifically sequence spaces like c and c0, demonstrating almost sure isomorphism and space recoverability.

Findings

Almost all dense sets in c and c0 are Rado under certain measures.

Graphs from these sets are almost surely isomorphic.

The resulting graph classes are non-isomorphic to those from finite-dimensional spaces.

Abstract

We consider a random geometric graph model, where pairs of vertices are points in a metric space and edges are formed independently with fixed probability $p$ between pairs within threshold distance $\delta $. A countable dense set in a metric space is {\sl Rado} if this random model gives, with probability 1, a graph that is unique up to isomorphism. In earlier work, the first two authors proved that in finite dimensional spaces $\mathbb{R}^n$ equipped with the $\ell_{\infty}$ norm, all countable dense set satisfying a mild non-integrality condition are Rado. In this paper, we extend this result to infinite-dimensional spaces. If the underlying metric space is a separable Banach space, then we show in some cases that we can almost surely recover the Banach space from such a geometric random graph. More precisely, we show that in the sequence spaces $c$ and $c_0$, for measures $\mu$ satisfying certain conditions, $\mu^\N$-almost all countable sets are Rado. Moreover, with probability 1, in $c$ as in $c_0$, all graphs obtained from the random geometric model with a randomly chosen dense countable vertex set are isomorphic to each other. Finally, we show that representatives of the isomorphism classes obtained in this way from $c$ and $c_0$ are non-isomorphic to each other, and also non-isomorphic to their counterparts obtained from finite dimensional spaces.