  In complex systems, the network of interactions we observe between system's
components is the aggregate of the interactions that occur through different
mechanisms or layers. Recent studies reveal that the existence of multiple
interaction layers can have a dramatic impact in the dynamical processes
occurring on these systems. However, these studies assume that the interactions
between systems components in each one of the layers are known, while typically
for real-world systems we do not have that information. Here, we address the
issue of uncovering the different interaction layers from aggregate data by
introducing multilayer stochastic block models (SBMs), a generalization of
single-layer SBMs that considers different mechanisms of layer aggregation.
First, we find the complete probabilistic solution to the problem of finding
the optimal multilayer SBM for a given aggregate observed network. Because this
solution is computationally intractable, we propose an approximation that
enables us to verify that multilayer SBMs are more predictive of network
structure in real-world complex systems.
