Random Geometric Graphs in Reflexive Banach Spaces

TL;DR

This paper explores the properties of random geometric graphs in Banach spaces, identifying conditions under which dense sets produce unique graphs and demonstrating the rarity of Rado sets in certain spaces.

Contribution

It characterizes when step-isometries are true isometries in Banach spaces and shows most dense sets in strictly convex reflexive spaces are non-Rado, with specific exceptions.

Findings

Almost all dense sets in strictly convex reflexive spaces are non-Rado.

Existence of Rado sets in ll_2.

Construction of Banach spaces where all dense sets are non-Rado.

Abstract

We investigate a random geometric graph model introduced by Bonato and Janssen. The vertices are the points of a countable dense set $S$ in a (necessarily separable) normed vector space $X$, and each pair of points are joined independently with some fixed probability $p$ (with $0<p<1$) if they are less than distance $1$ apart. A countable dense set $S$ in a normed space is Rado, if the resulting graph is almost surely unique up to isomorphism: that is any two such graphs are, almost surely, isomorphic. Not surprisingly, understanding which sets are Rado is closely related to the geometry of the underlying normed space. It turns out that a key question is in which spaces must step-isometries (maps that preserve the integer parts of distances) on dense subsets necessarily be isometries. We answer this question for a large class of Banach spaces including all strictly convex reflexive spaces. In the process we prove results on the interplay between the norm topology and weak topology that may be of independent interest. As a consequence of these Banach space results we show that almost all countable dense sets in strictly convex reflexive spaces are strongly non-Rado (that is, any two graphs are almost surely non-isomorphic). However, we show that there do exist Rado sets even in $\ell_2$. Finally we construct a Banach spaces in which all countable dense set are strongly non-Rado.