Infinite random geometric graphs

TL;DR

This paper introduces a new class of infinite random geometric graphs in metric spaces, characterizes their isomorphism types in certain settings, and contrasts properties under different metrics.

Contribution

It defines and analyzes infinite random geometric graphs in metric spaces, establishing conditions for unique isomorphism types and providing a deterministic construction.

Findings

Infinite random geometric graphs in R^n with L_infinity norm have a unique isomorphism type with probability 1.

In R^2 with Euclidean metric, infinite random geometric graphs are not necessarily isomorphic.

The paper characterizes the isomorphism type via a geometric analogue of the existentially closed property.

Abstract

We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given threshold. If the vertex set is a countable dense set in R^n equipped with the metric derived from the L_{\infty}-norm, then it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GR_n, is characterized by a geometric analogue of the existentially closed adjacency property, and we give a deterministic construction of GR_n. In contrast, we show that infinite random geometric graphs in R^2 with the Euclidean metric are not necessarily isomorphic.