  Many natural, engineered, and social systems can be represented using the
framework of a layered network, where each layer captures a different type of
interaction between the same set of nodes. The study of such multiplex networks
is a vibrant area of research. Yet, understanding how to quantify the
correlations present between pairs of layers, and more so present in their
co-evolution, is lacking. Such methods would enable us to address fundamental
questions involving issues such as function, redundancy and potential
disruptions. Here we show first how the edge-set of a multiplex network can be
used to construct an estimator of a joint probability distribution describing
edge existence over all layers. We then adapt an information-theoretic measure
of general correlation called the conditional mutual information, which uses
the estimated joint probability distribution, to quantify the pairwise
correlations present between layers. The pairwise comparisons can also be
temporal, allowing us to identify if knowledge of a certain layer can provide
additional information about the evolution of another layer.
  We analyze datasets from three distinct domains---economic, political, and
airline networks---to demonstrate how pairwise correlation in structure and
dynamical evolution between layers can be identified and show that anomalies
can serve as potential indicators of major events such as shocks.
