随机点上哪些连通空间网络具有线性路径-路径？
Which Connected Spatial Networks on Random Points have Linear Route-Lengths?
摘要：
在平面上任意点上的连通网络模型中，人们期望相距$r$的顶点之间的最短路径的平均长度仅随着$O(r)$和$r \to \infty$的增长而增长，但这并不总是容易验证的。在Poisson点过程的情形下，给出了这种线性的一般充分条件。在一个$L \times L$正方形中，定义一个子网$\GG_L$，它的边是存在的，而不管正方形外部的配置如何;条件是$\GG_L$的最大组成部分应该包含一定比例的顶点$1 - o(1)$，如$L \to \infty$。证明是通过与定向渗流的比较。我们表明，一般的结果适用于相对邻域图，并建立该网络的线性性质立即意味着它的一个大家庭的邻近图。
英文摘要：
In a model of a connected network on random points in the plane, one expects that the mean length of the shortest route between vertices at distance $r$ apart should grow only as $O(r)$ as $r \to \infty$, but this is not always easy to verify. We give a general sufficient condition for such linearity, in the setting of a Poisson point process. In a $L \times L$ square, define a subnetwork $\GG_L$ to have the edges which are present regardless of the configuration outside the square; the condition is that the largest component of $\GG_L$ should contain a proportion $1 - o(1)$ of the vertices, as $L \to \infty$. The proof is by comparison with oriented percolation. We show that the general result applies to the relative neighborhood graph, and establishing the linearity property for this network immediately implies it for a large family of proximity graphs.
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