一维几何随机图中连通性与无孤立点之间的一个奇异缺口
A curious gap in one-dimensional geometric random graphs between connectivity and the absence of isolated node
摘要：
一维几何随机图是通过在单位间隔上均匀且独立地分布$n$节点，然后在具有至多$r_n$的距离的任何两个节点之间分配无向边来构造的。这些图已经收到了很大的兴趣，并被用于各种应用，包括无线网络。用于连接性的$r_n$的阈值在文献中被称为$r_n^{*} = \frac{\ln n}{n}$。在本文中，我们证明了不存在孤立节点的$r_n$的阈值是$\frac{\ln n}{2 n}$（即，阈值$r_n^{*}$的一半）。我们的结果表明，在一维几何随机图中，连通性阈值和孤立节点的不存在之间存在一个奇怪的差距;特别是，当$r_n$等于$\frac{c\ln n}{ n}$时，对于常数$c \in( \frac{1}{2}, 1)$，一维几何随机图没有孤立节点，但不连通。一维几何随机图中的这个奇怪的间隙与许多其他随机图中的普遍现象形成鲜明对比，如二维几何随机图、Erdens-Rényi图和随机相交图，所有这些图在渐近意义上只要没有孤立节点就变得连通。
英文摘要：
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, Erdős-Rényi graphs, and random intersection graphs, all of which in the asymptotic sense become connected as soon as there is no isolated node.
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