  Reverse engineering of complex dynamical networks is important for a variety
of fields where uncovering the full topology of unknown networks and estimating
parameters characterizing the network structure and dynamical processes are of
interest. We consider complex oscillator networks with time-delayed
interactions in a noisy environment, and develop an effective method to infer
the full topology of the network and evaluate the amount of time delay based
solely on noise- contaminated time series. In particular, we develop an
analytic theory establishing that the dynamical correlation matrix, which can
be constructed purely from time series, can be manipulated to yield both the
network topology and the amount of time delay simultaneously. Extensive
numerical support is provided to validate the method. While our method provides
a viable solution to the network inverse problem, significant difficulties,
limitations, and challenges still remain, and these are discussed thoroughly.
