Hypothesis testing for general network models

TL;DR

This paper introduces a spectral goodness-of-fit test for general network models that is robust to model misspecification, applicable to various popular models, and validated through simulations and real data.

Contribution

It develops a universal spectral test for network models based on random matrix theory, applicable to multiple models including stochastic block and latent space models.

Findings

Test statistic converges to standard normal distribution under null hypothesis.

Method performs well in simulations and real-world data.

Applicable to nearly all popular network models.

Abstract

The network data has attracted considerable attention in modern statistics. In research on complex network data, one key issue is finding its underlying connection structure given a network sample. The methods that have been proposed in literature usually assume that the underlying structure is a known model. In practice, however, the true model is usually unknown, and network learning procedures based on these methods may suffer from model misspecification. To handle this issue, based on the random matrix theory, we first give a spectral property of the normalized adjacency matrix under a mild condition. Further, we establish a general goodness-of-fit test procedure for the unweight and undirected network. We prove that the null distribution of the proposed statistic converges in distribution to the standard normal distribution. Theoretically, this testing procedure is suitable for nearly all popular network models, such as stochastic block models, and latent space models. Further, we apply the proposed method to the degree-corrected mixed membership model and give a sequential estimator of the number of communities. Both simulation studies and real-world data examples indicate that the proposed method works well.