  An important problem in many areas of science is that of recovering
interaction networks from simultaneous time-series of many interacting
dynamical processes. A common approach is to use the elements of the
correlation matrix or its inverse as proxies of the interaction strengths, but
the reconstructed networks are necessarily undirected. Transfer entropy methods
have been proposed to reconstruct directed networks but the reconstructed
network lacks information about interaction strengths. We propose a network
reconstruction method that inherits the best of the two approaches by
reconstructing a directed weighted network from noisy data under the assumption
that the network is sparse and the dynamics are governed by a linear (or
weakly-nonlinear) stochastic dynamical system. The two steps of our method are
i) constructing an (infinite) family of candidate networks by solving the
covariance matrix Lyapunov equation for the state matrix and ii) using
L1-regularization to select a sparse solution. We further show how to use prior
information on the (non)existence of a few directed edges to drastically
improve the quality of the reconstruction.
