  We are concerned here with the likelihood ratio statistics in two exponential
random graph models -- the $\beta$-model and the Bradley-Terry model, in which
the degree sequence on an undirected graph and the out-degree sequence on a
weighted directed graph are the exclusively sufficient statistics in the
exponential-family distributions on graphs, respectively. We prove the Wilks
type of theorems for some fixed and growing dimensional hypothesis testing
problems. More specifically, under two fixed dimensional null hypotheses $H_0:
\beta_i=\beta_i^0$ for $i=1,\ldots, r$ and $H_0: \beta_1=\ldots=\beta_r$, we
show that $2[\ell(\widehat{\boldsymbol{\beta}}) -
\ell(\widehat{\boldsymbol{\beta}}^0)]$ converges in distribution to a
Chi-square distribution with the respective degrees of freedoms, $r$ and $r-1$,
as the dimension $n$ of the full parameter space goes to infinity. Here,
$\ell(\boldsymbol{\beta})$ is the log-likelihood function on the parameter
$\boldsymbol{\beta}$, $\widehat{\boldsymbol{\beta}}$ is the MLE under the full
parameter space, and $\widehat{\boldsymbol{\beta}}^0$ is the restricted MLE
under the null parameter space. For two increasing dimensional null hypotheses
$H_0: \beta_i = \beta_i^0$ for $i=1, \ldots, n$ and $H_0:
\beta_1=\ldots=\beta_r$ with $r/n \ge c$, we show that the normalized
log-likelihood ratio statistics, $(2[\ell(\widehat{\boldsymbol{\beta}}) -
\ell(\boldsymbol{\beta}^0)] -n)/(2n)^{1/2}$ and
$(2[\ell(\widehat{\boldsymbol{\beta}}) - \ell(\widehat{\boldsymbol{\beta}}^0)]
-r)/(2r)^{1/2}$, both converge in distribution to the standard normal
distribution. Simulation studies and an application to NBA data illustrate the
theoretical results.
