半径依赖于位置的非均匀随机几何图
Nonuniform random geometric graphs with location-dependent radii
摘要：
我们提出了一种无分布的方法来研究随机几何图。顶点的分布遵循具有强度函数$nf(\cdot)$的泊松点过程，其中$n\in \mathbb{N}$和$f$是$\mathbb{R}^d$上的概率密度函数。位于$x$处的顶点经由有向边连接到在截止距离$r_n(x)$内的其他顶点。我们证明了强法律的结果（i）临界截止函数，使几乎肯定，图不包含任何节点出度为零足够大的$n$和（ii）的最大和最小顶点度。我们还提供了一个特征的截止函数的节点数与出度为零收敛分布的泊松随机变量。我们说明了这一结果的一类密度与紧凑的支持，在最多项式衰减率为零。最后，我们陈述了一个充分条件，使上述图的增强版本最终几乎必然连通。
英文摘要：
We propose a distribution-free approach to the study of random geometric graphs. The distribution of vertices follows a Poisson point process with intensity function $nf(\cdot)$, where $n\in \mathbb{N}$, and $f$ is a probability density function on $\mathbb{R}^d$. A vertex located at $x$ connects via directed edges to other vertices that are within a cut-off distance $r_n(x)$. We prove strong law results for (i) the critical cut-off function so that almost surely, the graph does not contain any node with out-degree zero for sufficiently large $n$ and (ii) the maximum and minimum vertex degrees. We also provide a characterization of the cut-off function for which the number of nodes with out-degree zero converges in distribution to a Poisson random variable. We illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero. Finally, we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually.
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