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:βi=βi0 for i=1,…,r and a homogenous null H0:β1=⋯=βr, we reveal high dimensional Wilks’ phenomena that the normalized log-likelihood ratio statistic, [2{l(β̂)−l(β̂0)}−r]/(2r)1/2, converges in distribution to the standard normal distribution as <i>r</i> goes to infinity. Here, l(β) is the log-likelihood function on the model parameter β=(β1,…,βn)⊤, β̂ is its maximum likelihood estimator (MLE) under the full parameter space, and β̂0 is the restricted MLE under the null parameter space. For the homogenous null with a fixed <i>r</i>, we establish Wilks-type theorems that 2{l(β̂)−l(β̂0)} converges in distribution to a chi-square distribution with r−1 degrees of freedom, as the total number of parameters, <i>n</i>, 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.