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.

翻译：我们探讨了两种随机图模型中的Wilks现象：β模型和Bradley-Terry模型。针对两种维度递增的原假设，包括指定型原假设$H_0: \beta_i=\beta_i^0$（$i=1,\ldots, r$）和齐次型原假设$H_0: \beta_1=\cdots=\beta_r$，我们揭示了高维Wilks现象：标准化对数似然比统计量$[2\{\ell(\widehat{\mathbf{\beta}}) - \ell(\widehat{\mathbf{\beta}}^0)\} -r]/(2r)^{1/2}$在$r$趋于无穷时依分布收敛到标准正态分布。其中，$\ell( \mathbf{\beta})$为模型参数$\mathbf{\beta}=(\beta_1, \ldots, \beta_n)^\top$的对数似然函数，$\widehat{\mathbf{\beta}}$为全参数空间下的最大似然估计，$\widehat{\mathbf{\beta}}^0$为原假设参数空间下的约束最大似然估计。对于固定$r$的齐次型原假设，我们建立了Wilks型定理：当参数总数$n$趋于无穷时，$2\{\ell(\widehat{\mathbf{\beta}}) - \ell(\widehat{\mathbf{\beta}}^0)\}$依分布收敛到自由度为$r-1$的卡方分布。在检验固定维度的指定型原假设时，我们发现β模型中其渐近零分布为卡方分布；然而，这一结论在Bradley-Terry模型中并未成立。通过发展若干新颖的渐近展开技术方法，我们以原则性方式探索了Wilks型结果；这些原则性方法应适用于β模型和Bradley-Terry模型之外的一类随机图模型。模拟研究和实际网络数据应用进一步验证了理论结果。