  It has been recognized that many complex dynamical systems in the real world
require a description in terms of multiplex networks, where a set of common,
mutually connected nodes belong to distinct network layers and play a different
role in each layer. In spite of recent progress towards data based inference of
single-layer networks, to reconstruct complex systems with a multiplex
structure remains largely open. We articulate a mean-field based maximum
likelihood estimation framework to solve this outstanding and challenging
problem. We demonstrate the power of the reconstruction framework and
characterize its performance using binary time series from a class of
prototypical duplex network systems that host two distinct types of spreading
dynamics. In addition to validating the framework using synthetic and
real-world multiplex networks, we carry out a detailed analysis to elucidate
the impacts of structural and dynamical parameters as well as noise on the
reconstruction accuracy and robustness.
