  The interactions among the elementary components of many complex systems can
be qualitatively different. Such systems are therefore naturally described in
terms of multiplex or multi-layer networks, i.e. networks where each layer
stands for a different type of interaction between the same set of nodes. There
is today a growing interest in understanding when and why a description in
terms of a multiplex network is necessary and more informative than a
single-layer projection. Here, we contribute to this debate by presenting a
comprehensive study of correlations in multiplex networks. Correlations in node
properties, especially degree-degree correlations, have been thoroughly studied
in single-layer networks. Here we extend this idea to investigate and
characterize correlations between the different layers of a multiplex network.
Such correlations are intrinsically multiplex, and we first study them
empirically by constructing and analyzing several multiplex networks from the
real-world. In particular, we introduce various measures to characterize
correlations in the activity of the nodes and in their degree at the different
layers, and between activities and degrees. We show that real-world networks
exhibit indeed non-trivial multiplex correlations. For instance, we find cases
where two layers of the same multiplex network are positively correlated in
terms of node degrees, while other two layers are negatively correlated. We
then focus on constructing synthetic multiplex networks, proposing a series of
models to reproduce the correlations observed empirically and/or to assess
their relevance.
