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Recent advances in signal processing and information theory are boosting thedevelopment of new approaches for the data-driven modelling of complex networksystems. In the fields of Network Physiology and Network Neuroscience where thesignals of interest are often rich of oscillatory content, the spectralrepresentation of network systems is essential to ascribe the analyzedinteractions to specific oscillations with physiological meaning. In thiscontext, the present work formalizes a coherent framework which integratesseveral information dynamics approaches to quantify node-specific, pairwise andhigher-order interactions in network systems. The framework establishes ahierarchical organization of interactions of different order using measures ofentropy rate, mutual information rate and O-information rate, to quantifyrespectively the dynamics of individual nodes, the links between pairs ofnodes, and the redundant/synergistic hyperlinks between groups of nodes. Allmeasures are formulated in the time domain, and then expanded to the spectraldomain to obtain frequency-specific information. The practical computation ofall measures is favored presenting a toolbox that implements their parametricand non-parametric estimation, and includes approaches to assess theirstatistical significance. The framework is illustrated first using theoreticalexamples where the properties of the measures are displayed in benchmarksimulated network systems, and then applied to representative examples ofmultivariate time series in the context of Network Neuroscience and NetworkPhysiology.

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