Hypothesis testing for homogenous of nodes in $\beta$-models

The $\beta$-model has been extensively utilized to model degree heterogeneity in networks, wherein each node is assigned a unique parameter. In this article, we consider the hypothesis testing problem that two nodes $i$ and $j$ of a $\beta$-model have the same node parameter. We prove that the null distribution of the proposed statistic converges in distribution to the standard normal distribution. Further, we investigate the homogeneous test for $\beta$-model by combining individual $p$-values to aggregate small effects of multiple tests. Both simulation studies and real-world data examples indicate that the proposed method works well.