  Characterizing large-scale organization in networks, including multilayer
networks, is one of the most prominent topics in network science and is
important for many applications. One type of mesoscale feature is community
structure, in which sets of nodes are densely connected internally but sparsely
connected to other dense sets of nodes. Two of the most popular approaches for
community detection are to maximize an objective function called "modularity"
and to perform statistical inference using stochastic block models.
Generalizing work by Newman on monolayer networks (Physical Review E 94,
052315), we show in multilayer networks that maximizing modularity is
equivalent, under certain conditions, to maximizing the posterior probability
of community assignments under a suitably chosen stochastic block model. We
derive versions of this equivalence for various types of multilayer structure,
including temporal, multiplex, and multilevel networks. We consider cases in
which the key parameters are constant, as well as ones in which they vary
across layers; in the latter case, this yields a novel, layer-weighted version
of the modularity function. Our results also help address a longstanding
difficulty of multilayer modularity-maximization algorithms, which require the
specification of two sets of tuning parameters that have been difficult to
choose in practice. We show how to perform this parameter selection in a
statistically-grounded way, and we demonstrate the effectiveness of our
approach on both synthetic and empirical networks.
