The paper discusses a statistical problem related to testing for differences between two sparse networks with community structures. The community-wise edge probability matrices have entries of order $O(n^{-1}/\log n)$, where $n$ represents the size of the network. The authors propose a test statistic that combines a method proposed by Wu et al. \cite{WuTwoSampleSBM2022} and a resampling process. They derive the asymptotic null distribution of the test statistic and provide a guarantee of asymptotic power against the alternative hypothesis. To evaluate the performance of the proposed test statistic, the authors conduct simulations and provide real data examples. The results indicate that the proposed test statistic performs well in practice.

翻译：本文探讨了针对具有社区结构的两个稀疏网络之间差异进行检验的统计问题。社区级别的边概率矩阵元素阶为 $O(n^{-1}/\log n)$，其中 $n$ 表示网络规模。作者提出了一种结合Wu等人 \cite{WuTwoSampleSBM2022} 的方法与重采样过程的检验统计量。他们推导了该检验统计量的渐近零分布，并给出了针对备择假设的渐近功效保证。为评估所提检验统计量的性能，作者进行了模拟实验并提供了真实数据案例。结果表明，所提检验统计量在实践中表现良好。