  We explore the Wilks phenomena in two random graph models: the $\beta$-model
and the Bradley-Terry model. For two increasing dimensional null hypotheses,
including a specified null $H_0: \beta_i=\beta_i^0$ for $i=1,\ldots, r$ and a
homogenous null $H_0: \beta_1=\cdots=\beta_r$, we reveal high dimensional
Wilks' phenomena that the normalized log-likelihood ratio statistic,
$[2\{\ell(\widehat{\mathbf{\beta}}) - \ell(\widehat{\mathbf{\beta}}^0)\}
-r]/(2r)^{1/2}$, converges in distribution to the standard normal distribution
as $r$ goes to infinity. Here, $\ell( \mathbf{\beta})$ is the log-likelihood
function on the model parameter $\mathbf{\beta}=(\beta_1, \ldots,
\beta_n)^\top$, $\widehat{\mathbf{\beta}}$ is its maximum likelihood estimator
(MLE) under the full parameter space, and $\widehat{\mathbf{\beta}}^0$ is the
restricted MLE under the null parameter space. For the homogenous null with a
fixed $r$, we establish Wilks-type theorems that
$2\{\ell(\widehat{\mathbf{\beta}}) - \ell(\widehat{\mathbf{\beta}}^0)\}$
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 $\beta$-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 $\beta$-model and the
Bradley-Terry model. Simulation studies and real network data applications
further demonstrate the theoretical results.
