  Multilayer and multiplex networks are becoming common network data sets in
recent times. We consider the problem of identifying the common community
structure for a special type of multilayer networks called multi-relational
networks. We consider extensions of the spectral clustering methods for
multi-relational networks and give theoretical guarantees that the spectral
clustering methods recover community structure consistently for
multi-relational networks generated from multilayer versions of both stochastic
and degree-corrected block models even with dependence between network layers.
The methods are shown to work under optimal conditions on the degree parameter
of the networks to detect both assortative and disassortative community
structures with vanishing error proportions even if individual layers of the
multi-relational network has the network structures below community
detectability threshold. We reinforce the validity of the theoretical results
via simulations too.
