Hypothesis testing for the uniformity of random geometric graph

2025-10-15

AI Summary

This paper addresses the problem of testing spatial uniformity of node distributions in random geometric graphs. To overcome the absence of existing methods, we propose, for the first time, an asymptotically normal testing framework based on degenerate U-statistics: a kernel function dependent on the number of nodes is constructed, enabling efficient computation of the test statistic directly from the adjacency matrix; under the null hypothesis of uniformity, the statistic converges in distribution to a standard normal, and an analytical expression for the test’s power is derived. Crucially, the method bypasses reliance on explicit node coordinates or density estimation, substantially improving computational efficiency and applicability. Simulation studies and empirical analyses demonstrate high statistical power, robust discrimination between uniform and non-uniform configurations, and superior performance on real-world network data.

Technology Category

Reasoning under Uncertainty: Graphical ModelsMachine Learning: Graph-based Machine LearningKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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 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.

Problem

• Testing uniformity of node distribution in random geometric graphs
• Addressing edge dependency challenges in geometric graph hypothesis testing
• Developing statistical methods for non-uniform network distribution identification

Innovation

• Proposes first statistical test for geometric graph uniformity
• Uses degenerate U-statistics with node-dependent kernel function
• Calculates test statistic directly from adjacency matrix