Reconstructing network topology and coupling strengths in directed networks of discrete-time dynamics

Reconstructing network connection topology and interaction strengths solely from measurement of the dynamics of the nodes is a challenging inverse problem of broad applicability in various areas of science and engineering. For a discrete-time step network under noises whose noise-free dynamics is stationary, we derive general analytic results relating the weighted connection matrix of the network to the correlation functions obtained from time-series measurements of the nodes for networks with one-dimensional intrinsic node dynamics. Information about the intrinsic node dynamics and the noise strengths acting on the nodes can also be obtained. Based on these results, we develop a scheme that can reconstruct the above information of the network using only the time-series measurements of node dynamics as input. Reconstruction formulas for higher-dimensional node dynamics are also derived and illustrated with a two-dimensional node dynamics network system. Furthermore, we extend our results and obtain a reconstruction scheme even for the cases when the noise-free dynamics is periodic. We demonstrate that our method can give accurate reconstruction results for weighted directed networks with linear or nonlinear node dynamics of various connection topologies, and with linear or nonlinear couplings.