  Many practical systems can be described by dynamic networks, for which modern
technique can measure their output signals, and accumulate extremely rich data.
Nevertheless, the network structures producing these data are often deeply
hidden in these data. Depicting network structures by analysing the available
data, i.e., the inverse problems turns to be of great significant. On one hand,
dynamics are often driven by various unknown facts, called noises. On the other
hand, network structures of practical systems are commonly nonlinear, and
different nonlinearities can provide rich dynamic features and meaningful
functions of realistic networks. So far, no method, both theoretically or
numerically, has been found to systematically treat the both difficulties
together. Here we propose to use high-order correlation computations (HOCC) to
treat nonlinear dynamics; use two-time correlations to treat noise effects; and
use suitable basis and correlator vectors to unifiedly depict all dynamic
nonlinearities and topological interaction links and noise statistical
structures. All the above theoretical frameworks are constructed in a closed
form and numerical simulations fully verify the validity of theoretical
predictions.
