Hypothesis testing for the dimension of random geometric graph

2025-10-13

AI Summary

This paper addresses the fundamental challenge in random geometric graph (RGG) modeling that the true embedding dimension is unknown. We propose the first hypothesis-testing framework for dimension verification: given observed network data, we test the null hypothesis that the underlying dimension equals a prespecified value $ m $. Our method constructs a node-count-dependent kernel function and leverages degenerate U-statistic asymptotics to derive a test statistic that converges in distribution to a standard normal under the null and diverges in probability under alternatives. An efficient algorithm—based solely on the adjacency matrix—is developed for scalable computation. Simulation studies demonstrate high statistical power and accurate size control. Empirical analysis on real-world networks further confirms the method’s validity and practical utility. This work overcomes a critical limitation of existing network model validation approaches, which lack verifiable dimensionality assumptions.

Technology Category

Reasoning under Uncertainty: Graphical ModelsMachine Learning: Kernel MethodsData Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsSocial Networks and Social Media: Computational social science

Abstract

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.

Problem

• Testing if a random geometric graph's dimension matches a specified value
• Developing the first statistical test for geometric graph dimensionality
• Validating dimension hypotheses using asymptotic distribution theory

Innovation

• Statistical test for random geometric graph dimension
• Uses degenerate U-statistics with kernel functions
• Efficient computation based on adjacency matrix