Measuring hierarchically-organized interactions in dynamic networks through spectral entropy rates: Theory, estimation, and illustrative application to physiological networks

Recent advances in signal processing and information theory are boosting the development of new approaches for the data-driven modeling of complex network systems. In the fields of Network Physiology and Network Neuroscience where the signals of interest are rich of oscillatory content, the spectral representation of network systems is essential to ascribe interactions to specific oscillations with physiological meaning. The present work introduces a coherent framework integrating several information dynamics approaches to quantify node-specific, pairwise and higher-order interactions in network systems. A hierarchical organization of interactions of different order is established using measures of entropy rate, mutual information rate and O-information rate to quantify the dynamics of individual nodes, the links between pairs of nodes, and the redundant/synergistic hyperlinks in groups of nodes. All measures are formulated in the time domain and expanded to the spectral domain to obtain frequency-specific information. The practical computation of all measures is favored presenting a toolbox that implements parametric and non-parametric estimators and includes statistical validation approaches. The framework is illustrated using theoretical examples where the properties of the measures are displayed in benchmark simulated network systems, and representative multivariate time series in the context of Network Neuroscience and Network Physiology.