Community detection in multi-layer networks is a crucial problem in network analysis. In this paper, we analyze the performance of two spectral clustering algorithms for community detection within the multi-layer degree-corrected stochastic block model (MLDCSBM) framework. One algorithm is based on the sum of adjacency matrices, while the other utilizes the debiased sum of squared adjacency matrices. We establish consistency results for community detection using these methods under MLDCSBM as the size of the network and/or the number of layers increases. Our theorems demonstrate the advantages of utilizing multiple layers for community detection. Moreover, our analysis indicates that spectral clustering with the debiased sum of squared adjacency matrices is generally superior to spectral clustering with the sum of adjacency matrices. Numerical simulations confirm that our algorithm, employing the debiased sum of squared adjacency matrices, surpasses existing methods for community detection in multi-layer networks. Finally, the analysis of several real-world multi-layer networks yields meaningful insights.

翻译：多层网络中的社区检测是网络分析中的一个关键问题。本文分析了两种谱聚类算法在多层度修正随机块模型（MLDCSBM）框架下进行社区检测的性能。一种算法基于邻接矩阵之和，另一种则利用去偏的平方邻接矩阵之和。我们在MLDCSBM下证明了当网络规模和/或层数增加时，使用这些方法进行社区检测的一致性结果。我们的定理展示了利用多层网络进行社区检测的优势。此外，分析表明，基于去偏平方邻接矩阵之和的谱聚类通常优于基于邻接矩阵之和的谱聚类。数值模拟证实，我们采用去偏平方邻接矩阵之和的算法超越了现有的多层网络社区检测方法。最后，对几个真实世界多层网络的分析得出了有意义的见解。