Testing degree heterogeneity in directed networks

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.