  Community detection in multi-layer networks has emerged as a crucial area of
modern network analysis. However, conventional approaches often assume that
nodes belong exclusively to a single community, which fails to capture the
complex structure of real-world networks where nodes may belong to multiple
communities simultaneously. To address this limitation, we propose novel
spectral methods to estimate the common mixed memberships in the multi-layer
mixed membership stochastic block model. The proposed methods leverage the
eigen-decomposition of three aggregate matrices: the sum of adjacency matrices,
the debiased sum of squared adjacency matrices, and the sum of squared
adjacency matrices. We establish rigorous theoretical guarantees for the
consistency of our methods. Specifically, we derive per-node error rates under
mild conditions on network sparsity, demonstrating their consistency as the
number of nodes and/or layers increases under the multi-layer mixed membership
stochastic block model. Our theoretical results reveal that the method
leveraging the sum of adjacency matrices generally performs poorer than the
other two methods for mixed membership estimation in multi-layer networks. We
conduct extensive numerical experiments to empirically validate our theoretical
findings. For real-world multi-layer networks with unknown community
information, we introduce two novel modularity metrics to quantify the quality
of mixed membership community detection. Finally, we demonstrate the practical
applications of our algorithms and modularity metrics by applying them to
real-world multi-layer networks, demonstrating their effectiveness in
extracting meaningful community structures.
