  We study the problem of community detection in multi-layer networks, where
pairs of nodes can be related in multiple modalities. We introduce a general
framework, i.e., mixture multi-layer stochastic block model (MMSBM), which
includes many earlier models as special cases. We propose a tensor-based
algorithm (TWIST) to reveal both global/local memberships of nodes, and
memberships of layers. We show that the TWIST procedure can accurately detect
the communities with small misclassification error as the number of nodes
and/or the number of layers increases. Numerical studies confirm our
theoretical findings. To our best knowledge, this is the first systematic study
on the mixture multi-layer networks using tensor decomposition. The method is
applied to two real datasets: worldwide trading networks and malaria parasite
genes networks, yielding new and interesting findings.
