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.

翻译：随机几何图广泛用于建模网络中的几何结构与依赖关系。在随机几何图中，节点依据度量空间上的概率分布$F$独立生成，若节点间距离小于给定阈值则连边。现有研究大多假设分布$F$服从均匀分布。然而，最新研究表明，某些现实网络可能更适合用非均匀分布$F$建模。此外，非均匀分布$F$生成的图与均匀分布$F$生成的图具有显著不同的性质。一个根本性问题是：给定来自随机几何图的网络，其分布$F$是否均匀？本文通过假设检验方法探讨该问题。由于随机几何图中边之间存在经典随机图所不具备的内在依赖性，该问题具有特殊挑战性。我们提出了首个统计检验方法。在原假设下，检验统计量依分布收敛于标准正态分布。该渐近分布通过采用核函数依赖节点数的退化U统计量渐近理论推导得出，此项技术与现有网络假设检验方法存在本质差异。此外，我们提出了一种直接通过邻接矩阵高效计算检验统计量的方法，并从理论上解析刻画了所提检验的功效。仿真研究表明，所提出的均匀性检验具有较高功效。本文同时提供了实际数据应用案例。