单位球上的随机几何图直径
Random Geometric Graph Diameter in the Unit Ball
摘要：
单位球随机几何图$G=G^d_p(λ,n)$的顶点$n$是在$d$维单位球上独立均匀分布的点，两个顶点相邻当且仅当它们的$l_p$ -距离至多为$λ$ .像它的表亲Erdos-Renyi随机图一样，$G$有一个连通性阈值：$λ$的一个渐近值，以$n$表示，高于该值时$G$是连通的，低于该值时$G$是不连通的（实际上在大多数情况下有孤立的顶点）。在连通区域中，我们确定了$G$的图直径的上界和下界。具体而言，几乎总是$\diam_p(\mathbf{B})(1-o(1))/λ\leq \diam(G) \leq \diam_p(\mathbf{B})(1+O((\ln \ln n/\ln n)^{1/d}))/λ$，其中$\diam_p(\mathbf{B})$是单位球$\mathbf{B}$的$\ell_p$ -直径。我们采用的方法从概率组合学和随机几何的组合。
英文摘要：
The unit ball random geometric graph $G=G^d_p(λ,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most $λ$. Like its cousin the Erdos-Renyi random graph, $G$ has a connectivity threshold: an asymptotic value for $λ$ in terms of $n$, above which $G$ is connected and below which $G$ is disconnected (and in fact has isolated vertices in most cases). In the connected zone, we determine upper and lower bounds for the graph diameter of $G$. Specifically, almost always, $\diam_p(\mathbf{B})(1-o(1))/λ\leq \diam(G) \leq \diam_p(\mathbf{B})(1+O((\ln \ln n/\ln n)^{1/d}))/λ$, where $\diam_p(\mathbf{B})$ is the $\ell_p$-diameter of the unit ball $\mathbf{B}$. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.
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