Dynamic communicability and flow to describe complex network dynamics with linear feedback

Graph theory has become a widely used toolbox to investigate complex networks estimated from real-life data. A specific direction of research has used collective dynamics obtained for propagation-like processes, such as oscillators and random walks, in order to characterize intricate topologies. However, the study of complex network dynamics lacks a dedicated theoretical framework. Here we derive a rigorous framework based on the network response (i.e,. the Green function) to study interactions between nodes across time -quantified by dynamic communicability- in a multivariate Ornstein-Uhlenbeck process, which describes stable and non-conservative dynamics. Our framework also describes how the properties of external inputs contribute to the network activity, by defining the measure of flow. We illustrate the interplay between network connectivity and input properties with several classical network examples, such as the small-world ring network and the hierarchical network. Our theory defines a comprehensive framework to study and compare directed weighted networks, from pairwise interactions to global properties (e.g., small-worldness) and community detection.