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

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 $r_n^{\ast }=\frac{\ln n}{n}$ in the literature. In this paper, we prove that a threshold of $r_n$ for the absence of isolated node is $\frac{\ln n}{2n}$ (i.e., a half of the threshold $r_n^{\ast }$). 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.