A spectral inference method for determining the number of communities in networks

2026-03-04

AI Summary

This work proposes a model-free spectral inference method to address the challenges in estimating the number of communities in networks, particularly the strong reliance on assumed models, difficulties in handling sparsity, and divergent community counts. The approach innovatively combines the eigen-gap ratio with the Tracy–Widom distribution to construct a tuning-free hypothesis testing framework: under the null hypothesis, the test statistic asymptotically follows the Type-I Tracy–Widom distribution. Theoretical analysis establishes the asymptotic validity of the procedure, while extensive simulations and experiments on three real-world network datasets demonstrate its robustness and broad applicability across various stochastic block models.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

To characterize the community structure in network data, researchers have developed various block-type models, including the stochastic block model, the degree-corrected stochastic block model, the mixed membership block model, the degree-corrected mixed membership block model, and others. A critical step in applying these models effectively is determining the number of communities in the network. However, to the best of our knowledge, existing methods for estimating the number of network communities either rely on explicit model fitting or fail to simultaneously accommodate network sparsity and a diverging number of communities. In this paper, we propose a model-free spectral inference method based on eigengap ratios that addresses these challenges. The inference procedure is straightforward to compute, requires no parameter tuning, and can be applied to a wide range of block models without the need to estimate network distribution parameters. Furthermore, it is effective for both dense and sparse networks with a divergent number of communities. Technically, we show that the proposed spectral test statistic converges to a {function of the type-I Tracy-Widom distribution via the Airy kernel} under the null hypothesis, and that the test is asymptotically powerful under weak alternatives. Simulation studies on both dense and sparse networks demonstrate the efficacy of the proposed method. Three real-world examples are presented to illustrate the usefulness of the proposed test.

Problem

• community detection
• number of communities
• network sparsity
• diverging communities
• model-free inference

Innovation

• spectral inference
• community detection
• eigengap ratio
• Tracy-Widom distribution
• model-free