Forecasting synchronizability of complex networks from data.

Given a complex networked system whose topology and dynamical equations are unknown, is it possible to foresee that a certain type of collective dynamics can potentially emerge in the system, provided that only time-series measurements are available? We address this question by focusing on a commonly studied type of collective dynamics, namely, synchronization in coupled dynamical networks. We demonstrate that, using the compressive-sensing paradigm, even when the coupling strength is not uniform so that the network is effectively weighted, the full topology, the coupling weights, and the nodal dynamical equations can all be uncovered accurately. The reconstruction accuracy and data requirement are systematically analyzed, in a process that includes a validation of the reconstructed eigenvalue spectrum of the underlying coupling matrix. A master stability function (MSF), the fundamental quantity determining the network synchronizability, can then be calculated based on the reconstructed dynamical system, the accuracy of which can be assessed as well. With the coupling matrix and MSF fully uncovered, the emergence of synchronous dynamics in the network can be anticipated and controlled. To forecast the collective dynamics on complex networks is an extremely challenging problem with significant applications in many disciplines, and our work represents an initial step in this important area.