  Multilayer networks are a useful data structure for simultaneously capturing
multiple types of relationships between a set of nodes. In such networks, each
relational definition gives rise to a layer. While each layer provides its own
set of information, community structure across layers can be collectively
utilized to discover and quantify underlying relational patterns between nodes.
To concisely extract information from a multilayer network, we propose to
identify and combine sets of layers with meaningful similarities in community
structure. In this paper, we describe the "strata multilayer stochastic block
model'' (sMLSBM), a probabilistic model for multilayer community structure. The
central extension of the model is that there exist groups of layers, called
"strata'', which are defined such that all layers in a given stratum have
community structure described by a common stochastic block model (SBM). That
is, layers in a stratum exhibit similar node-to-community assignments and SBM
probability parameters. Fitting the sMLSBM to a multilayer network provides a
joint clustering that yields node-to-community and layer-to-stratum
assignments, which cooperatively aid one another during inference. We describe
an algorithm for separating layers into their appropriate strata and an
inference technique for estimating the SBM parameters for each stratum. We
demonstrate our method using synthetic networks and a multilayer network
inferred from data collected in the Human Microbiome Project.
