一维软随机几何图中孤立节点数的分布
The Distribution of the Number of Isolated Nodes in the 1-Dimensional Soft Random Geometric Graph
AI总结：
本研究针对一维环面上的软随机几何图，证明其孤立节点数在合适标度下全变差收敛于Poisson随机变量，并由此得到图连通概率的上界。
摘要：
我们研究软随机几何图中的孤立节点数量，该图的顶点构成长度为L的环面（即带周期边界条件的线段[0,L]）上的一个Poisson过程，且两节点间存在边的概率取决于它们之间的距离。不同节点对之间的边相互独立。在合适的标度区域下，我们证明孤立节点数在全变差意义下收敛于一个Poisson随机变量。该结果给出了该随机图连通概率的一个上界。
英文摘要：
We study the number of isolated nodes in a soft random geometric graph whose vertices constitute a Poisson process on the torus of length L (the line segment [0,L] with periodic boundary conditions), and where an edge is present between two nodes with a probability which depends on the distance between them. Edges between distinct pairs of nodes are mutually independent. In a suitable scaling regime, we show that the number of isolated nodes converges in total variation to a Poisson random variable. The result implies an upper bound on the probability that the random graph is connected.
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