Testing degree heterogeneity in directed networks

TL;DR

This paper develops likelihood ratio tests for degree heterogeneity in directed networks, establishing their asymptotic distributions in various high-dimensional settings, and verifies results through simulations and real data analysis.

Contribution

It extends Wilks-type theorems to directed graphs in high-dimensional regimes, involving complex asymptotic expansions and new matrix approximations.

Findings

Normalized likelihood ratio statistic converges to normal distribution in increasing dimensions.

Chi-square distribution convergence for fixed dimensions as network size grows.

Simulation and real data confirm theoretical asymptotic results.

Abstract

In this study, we focus on the likelihood ratio tests in the $p_0$ model for testing degree heterogeneity in directed networks, which is an exponential family distribution on directed graphs with the bi-degree sequence as the naturally sufficient statistic. For testing the homogeneous null hypotheses $H_0: \alpha_1 = \cdots = \alpha_r$, we establish Wilks-type results in both increasing-dimensional and fixed-dimensional settings. For increasing dimensions, the normalized log-likelihood ratio statistic $[2\{\ell(\widehat{\mathbf{\theta}})-\ell(\widehat{\mathbf{\theta}}^0)\}-r]/(2r)^{1/2}$ converges in distribution to a standard normal distribution. For fixed dimensions, $2\{\ell(\widehat{\mathbf{\theta}})-\ell(\widehat{\mathbf{\theta}}^0)\}$ converges in distribution to a chi-square distribution with $r-1$ degrees of freedom as $n\rightarrow \infty$, independent of the nuisance parameters. Additionally, we present a Wilks-type theorem for the specified null $H_0: \alpha_i=\alpha_i^0$, $i=1,\ldots, r$ in high-dimensional settings, where the normalized log-likelihood ratio statistic also converges in distribution to a standard normal distribution. These results extend the work of \cite{yan2025likelihood} to directed graphs in a highly non-trivial way, where we need to analyze much more expansion terms in the fourth-order asymptotic expansions of the likelihood function and develop new approximate inverse matrices under the null restricted parameter spaces for approximating the inverse of the Fisher information matrices in the $p_0$ model. Simulation studies and real data analyses are presented to verify our theoretical results.