Reciprocal characterization from multivariate time series to multilayer complex networks

Various transformations from time series to complex networks have recently gained significant attention. These transformations provide an alternative perspective to better investigate complex systems. We present a transformation from multivariate time series to multilayer networks for their reciprocal characterization. This transformation ensures that the underlying geometrical features of time series are preserved in their network counterparts. We identify underlying dynamical transitions of the time series through statistics of the structure of the corresponding networks. Meanwhile, this allows us to propose the concept of interlayer entropy to measure the coupling strength between the layers of a network. Specifically, we prove that under mild conditions, for the given transformation method, the application of interlayer entropy in networks is equivalent to transfer entropy in time series. Interlayer entropy is utilized to describe the information flow in a multilayer network.