  In network applications, it has become increasingly common to obtain datasets
in the form of multiple networks observed on the same set of subjects, where
each network is obtained in a related but different experiment condition or
application scenario. Such datasets can be modeled by multilayer networks where
each layer is a separate network itself while different layers are associated
and share some common information. The present paper studies community
detection in a stylized yet informative inhomogeneous multilayer network model.
In our model, layers are generated by different stochastic block models, the
community structures of which are (random) perturbations of a common global
structure while the connecting probabilities in different layers are not
related. Focusing on the symmetric two block case, we establish minimax rates
for both global estimation of the common structure and individualized
estimation of layer-wise community structures. Both minimax rates have sharp
exponents. In addition, we provide an efficient algorithm that is
simultaneously asymptotic minimax optimal for both estimation tasks under mild
conditions. The optimal rates depend on the parity of the number of most
informative layers, a phenomenon that is caused by inhomogeneity across layers.
The method is extended to handle multiple and potentially asymmetric community
cases. We demonstrate its effectiveness on both simulated examples and a real
multi-modal single-cell dataset.
