The absence of isolated node in geometric random graphs

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 τ_n. These graphs have received much interest and been used in various applications including wireless networks. A threshold of τ_n for connectivity is known as τ*_n = ln n/n in the literature. In this paper, we prove that a threshold of τ_n for the absence of isolated node is ln n/2n (i.e., a half of the threshold τ*_n). Our result shows there is a gap between thresholds of connectivity and the absence of isolated node in one-dimensional geometric random graphs; in particular, when τ_n equals c ln n/n for a constant c ∈ (1/2, 1), a one-dimensional geometric random graph has no isolated node but is not connected. This 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, Erdös-Rényi graphs, and random intersection graphs, all of which in the asymptotic sense become connected as soon as there is no isolated node.