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

TL;DR

This paper introduces a simple, model-free spectral method for accurately estimating the number of communities in various networks, including sparse and dense ones, without complex parameter tuning.

Contribution

It proposes a novel eigengap ratio-based spectral inference technique that works across different block models and network sparsity levels without explicit model fitting.

Findings

Method accurately estimates community number in simulations

Effective for both sparse and dense networks

Demonstrated usefulness on real-world network data

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.