  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.
