  Although much of the focus of statistical works on networks has been on
static networks, multiple networks are currently becoming more common among
network data sets. Usually, a number of network data sets, which share some
form of connection between each other are known as multiple or multi-layer
networks. We consider the problem of identifying the common community
structures for multiple networks. We consider extensions of the spectral
clustering methods for the multiple sparse networks, and give theoretical
guarantee that the spectral clustering methods produce consistent community
detection in case of both multiple stochastic block model and multiple
degree-corrected block models. The methods are shown to work under sufficiently
mild conditions on the number of multiple networks to detect associative
community structures, even if all the individual networks are sparse and most
of the individual networks are below community detectability threshold. We
reinforce the validity of the theoretical results via simulations too.
