标题： 随机几何图的均匀性假设检验 显示英文标题

标题： Hypothesis testing for the uniformity of random geometric graph

摘要： 随机几何图被广泛用于建模网络中的几何结构和依赖关系。 在随机几何图中，节点是从度量空间上的某个概率分布$F$独立生成的，如果节点之间的距离小于某个阈值，则连接这些节点。 大多数研究假设分布$F$是均匀的。 然而，最近的研究表明，一些现实世界的网络可能更适合用非均匀分布$F$来建模。 此外，具有非均匀$F$的图与具有均匀$F$的图有显著不同的性质。 一个基本问题是：给定一个来自随机几何图的网络，分布$F$是否是均匀的？ 在本文中，我们通过假设检验来解决这个问题。 由于随机几何图中边之间存在固有的依赖性，这一问题特别具有挑战性，这是经典随机图所不具备的特性。 我们提出了第一个统计检验。 在原假设下，检验统计量收敛于标准正态分布。 渐近分布是通过使用核函数依赖于节点数的退化U统计量的渐近理论推导出来的。 这种方法不同于现有网络假设检验问题中的方法。 此外，我们提出了一种直接从邻接矩阵计算检验统计量的方法。 我们还分析地描述了所提出检验的功效。 模拟研究表明，所提出的均匀性检验具有高功效。 还提供了实际数据应用。 显示英文摘要

摘要： Random geometric graphs are widely used in modeling geometry and dependence structure in networks. In a random geometric graph, nodes are independently generated from some probability distribution $F$ over a metric space, and edges link nodes if their distance is less than some threshold. Most studies assume the distribution $F$ to be uniform. However, recent research shows that some real-world networks may be better modeled by nonuniform distribution $F$. Moreover, graphs with nonuniform $F$ have notably different properties from graphs with uniform $F$. A fundamental question is: given a network from a random geometric graph, is the distribution $F$ uniform or not? In this paper, we approach this question through hypothesis testing. This problem is particularly challenging due to the inherent dependencies among edges in random geometric graphs, a property not present in classic random graphs. We propose the first statistical test. Under the null hypothesis, the test statistic converges in distribution to the standard normal distribution. The asymptotic distribution is derived using the asymptotic theory of degenerate U-statistics with a kernel function dependent on the number of nodes. This technique is different from existing methods in network hypothesis testing problems. In addition, we present a method for efficiently calculating the test statistic directly from the adjacency matrix. We also analytically characterize the power of the proposed test. The simulation study shows that the proposed uniformity test has high power. Real data applications are also provided.