  Many network systems are composed of interdependent but distinct types of
interactions, which cannot be fully understood in isolation. These different
types of interactions are often represented as layers, attributes on the edges
or as a time-dependence of the network structure. Although they are crucial for
a more comprehensive scientific understanding, these representations offer
substantial challenges. Namely, it is an open problem how to precisely
characterize the large or mesoscale structure of network systems in relation to
these additional aspects. Furthermore, the direct incorporation of these
features invariably increases the effective dimension of the network
description, and hence aggravates the problem of overfitting, i.e. the use of
overly-complex characterizations that mistake purely random fluctuations for
actual structure. In this work, we propose a robust and principled method to
tackle these problems, by constructing generative models of modular network
structure, incorporating layered, attributed and time-varying properties, as
well as a nonparametric Bayesian methodology to infer the parameters from data
and select the most appropriate model according to statistical evidence. We
show that the method is capable of revealing hidden structure in layered,
edge-valued and time-varying networks, and that the most appropriate level of
granularity with respect to the additional dimensions can be reliably
identified. We illustrate our approach on a variety of empirical systems,
including a social network of physicians, the voting correlations of deputies
in the Brazilian national congress, the global airport network, and a proximity
network of high-school students.
