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.

翻译：$β$模型被广泛用于刻画网络中的度异质性，其中每个节点被赋予一个唯一的参数。本文考虑$β$模型中两个节点$i$和$j$具有相同节点参数的假设检验问题。我们证明所提出统计量的原分布依分布收敛于标准正态分布。进一步，我们通过结合个体$p$值来聚合多个检验的微小效应，研究$β$模型的同质性检验。模拟研究与实际数据案例均表明，所提出的方法效果良好。