Likelihood Ratio Tests in Random Graph Models with Increasing Dimensions

We explore the Wilks phenomena in two random graph models: the 𝛽-model and the Bradley–Terry model. For two increasing dimensional null hypotheses, including a specified null 𝐻₀ : 𝛽_𝑖=𝛽⁰_𝑖 for 𝑖 = 1,…,𝑟 and a homogenous null 𝐻₀ : 𝛽₁ = ⋯ = 𝛽_𝑟, we reveal high dimensional Wilks’ phenomena that the normalized log-likelihood ratio statistic, [2⁢{𝓁⁡(ˆ𝜷)−𝓁⁡(ˆ𝜷⁰)}−𝑟]/(2⁢𝑟)^1/2, converges in distribution to the standard normal distribution as r goes to infinity. Here, 𝓁⁡(𝜷) is the log-likelihood function on the model parameter 𝜷=(𝛽₁,…,𝛽_𝑛)^⊤,ˆ𝜷 is its maximum likelihood estimator (MLE) under the full parameter space, andˆ𝜷⁰ is the restricted MLE under the null parameter space. For the homogenous null with a fixed r, we establish Wilks-type theorems that 2⁢{𝓁⁡(ˆ𝜷)−𝓁⁡(ˆ𝜷⁰)} converges in distribution to a chi-square distribution with 𝑟−1 degrees of freedom, as the total number of parameters, n, goes to infinity. When testing the fixed dimensional specified null, we find that its asymptotic null distribution is a chi-square distribution in the 𝛽-model. However, unexpectedly, this is not true in the Bradley–Terry model. By developing several novel technical methods for asymptotic expansion, we explore Wilks-type results in a principled manner; these principled methods should be applicable to a class of random graph models beyond the 𝛽-model and the Bradley–Terry model. Simulation studies and real network data applications further demonstrate the theoretical results. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. © 2025 American Statistical Association