稀疏蓝牙网络的连通性
Connectivity of sparse Bluetooth networks
摘要：
考虑定义在$d$维单位环面中均匀分布的$n$顶点上的随机几何图。如果两个顶点的距离小于“可见半径”$r_n$，则它们是连接的。我们考虑{ \sl蓝牙网络}是局部稀疏的随机几何图。每个顶点在随机几何图中随机选择其邻居中的$c$，并且仅连接到所选择的点。我们表明，如果可见性半径至少是$n^{-(1-δ)/d}$的顺序为一些$δ> 0$，那么一个恒定的值$c$是足够的图被连接，具有很高的概率。对于任何阳性$ε$，取$c \ge \sqrt{(1+ε)/δ} + K$就足够了，其中$K$是仅取决于$d$的常数。另一方面，使用$c\le \sqrt{(1-ε)/δ}$，图是断开的，具有高概率。
英文摘要：
Consider a random geometric graph defined on $n$ vertices uniformly distributed in the $d$-dimensional unit torus. Two vertices are connected if their distance is less than a "visibility radius" $r_n$. We consider {\sl Bluetooth networks} that are locally sparsified random geometric graphs. Each vertex selects $c$ of its neighbors in the random geometric graph at random and connects only to the selected points. We show that if the visibility radius is at least of the order of $n^{-(1-δ)/d}$ for some $δ> 0$, then a constant value of $c$ is sufficient for the graph to be connected, with high probability. It suffices to take $c \ge \sqrt{(1+ε)/δ} + K$ for any positive $ε$ where $K$ is a constant depending on $d$ only. On the other hand, with $c\le \sqrt{(1-ε)/δ}$, the graph is disconnected, with high probability.
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