标题： 随机几何图的维度假设检验 显示英文标题

标题： Hypothesis testing for the dimension of random geometric graph

摘要： 随机几何图（RGGs）为分析现实世界网络中的几何和依赖结构提供了一个强大的工具。 例如，已经观察到RGGs是蛋白质-蛋白质相互作用网络的一个良好模型。 在RGGs中，节点被随机分布在$m$维度的度量空间中，当且仅当节点之间的距离小于某个阈值时，节点之间就会连接。 在将RGGs拟合到现实世界网络时，第一步可能是输入或估计维度$m$。 然而，不清楚预设的维度是否等于真实的维度。 在本文中，我们使用假设检验来研究这个问题。 在原假设下，维度等于一个特定值，而备择假设则断言维度不等于该值。 我们提出了第一个统计检验。 在原假设下，所提出的检验统计量会以法律收敛到标准正态分布，在备择假设下，检验统计量在概率上是无界的。 我们通过利用核函数依赖于节点数量的退化U统计量的渐近理论，推导了渐近分布。 这种方法与在网络假设检验问题中常用的现有方法有显著不同。 此外，我们还提出了一种基于邻接矩阵计算检验统计量的高效方法。 模拟研究表明，所提出的检验表现良好。 我们还将所提出的检验应用于多个现实世界网络，以测试它们的维度。 显示英文摘要

摘要： Random geometric graphs (RGGs) offer a powerful tool for analyzing the geometric and dependence structures in real-world networks. For example, it has been observed that RGGs are a good model for protein-protein interaction networks. In RGGs, nodes are randomly distributed over an $m$-dimensional metric space, and edges connect the nodes if and only if their distance is less than some threshold. When fitting RGGs to real-world networks, the first step is probably to input or estimate the dimension $m$. However, it is not clear whether the prespecified dimension is equal to the true dimension. In this paper, we investigate this problem using hypothesis testing. Under the null hypothesis, the dimension is equal to a specific value, while the alternative hypothesis asserts the dimension is not equal to that value. We propose the first statistical test. Under the null hypothesis, the proposed test statistic converges in law to the standard normal distribution, and under the alternative hypothesis, the test statistic is unbounded in probability. We derive the asymptotic distribution by leveraging the asymptotic theory of degenerate U-statistics with kernel function dependent on the number of nodes. This approach differs significantly from prevailing methods used in network hypothesis testing problems. Moreover, we also propose an efficient approach to compute the test statistic based on the adjacency matrix. Simulation studies show that the proposed test performs well. We also apply the proposed test to multiple real-world networks to test their dimensions.