A curious gap in one-dimensional geometric random graphs between connectivity and the absence of isolated node

TL;DR

This paper reveals a surprising gap in one-dimensional geometric random graphs where the absence of isolated nodes occurs at a lower threshold than connectivity, contrasting with other random graph models.

Contribution

It establishes that in 1D geometric random graphs, the thresholds for absence of isolated nodes and connectivity differ, with the former being at half the value of the latter.

Findings

Threshold for absence of isolated nodes is rac{\u2212} ln n}{2 n}

Graphs with radius crac{ ln n}{n} are isolated-node free but not connected

Contrasts with higher-dimensional and other random graph models

Abstract

One-dimensional geometric random graphs are constructed by distributing $n$ nodes uniformly and independently on a unit interval and then assigning an undirected edge between any two nodes that have a distance at most $r_n$. These graphs have received much interest and been used in various applications including wireless networks. A threshold of $r_n$ for connectivity is known as $r_n^{*} = \frac{\ln n}{n}$ in the literature. In this paper, we prove that a threshold of $r_n$ for the absence of isolated node is $\frac{\ln n}{2 n}$ (i.e., a half of the threshold $r_n^{*}$). Our result shows there is a curious gap between thresholds of connectivity and the absence of isolated node in one-dimensional geometric random graphs; in particular, when $r_n$ equals $\frac{c\ln n}{ n}$ for a constant $c \in( \frac{1}{2}, 1)$, a one-dimensional geometric random graph has no isolated node but is not connected. This curious gap in one-dimensional geometric random graphs is in sharp contrast to the prevalent phenomenon in many other random graphs such as two-dimensional geometric random graphs, Erd\H{o}s-R\'enyi graphs, and random intersection graphs, all of which in the asymptotic sense become connected as soon as there is no isolated node.