  One of the fundamental problems in network analysis is detecting community
structure in multi-layer networks, of which each layer represents one type of
edge information among the nodes. We propose integrative spectral clustering
approaches based on effective convex layer aggregations. Our aggregation
methods are strongly motivated by a delicate asymptotic analysis of the
spectral embedding of weighted adjacency matrices and the downstream $k$-means
clustering, in a challenging regime where community detection consistency is
impossible. In fact, the methods are shown to estimate the optimal convex
aggregation, which minimizes the mis-clustering error under some specialized
multi-layer network models. Our analysis further suggests that clustering using
Gaussian mixture models is generally superior to the commonly used $k$-means in
spectral clustering. Extensive numerical studies demonstrate that our adaptive
aggregation techniques, together with Gaussian mixture model clustering, make
the new spectral clustering remarkably competitive compared to several
popularly used methods.
