A large number of complex systems, naturally emerging in various domains, are well described by directed networks, resulting in numerous interesting features that are absent from their undirected counterparts. Among these properties is a strong non-normality, inherited by a strong asymmetry, which stands out as a universal signature that characterizes such systems and guides their underlying hierarchy. In this talk, we consider an extensive collection of empirical networks and analyze their structural properties by using tools such as the entropy rate borrowed from information theory. A ubiquitous feature is observed amongst such systems as the level of non-normality increases. Several highly directed substructures aiming towards a terminal (sink or source) node, namely a node with only incoming or only outgoing edges, uniformly emerges. Such spontaneous occurrence is characterized by a threshold in the non-normality which yields terminal nodes, denoted here as leaders, detected by a sudden collapse of the entropy measure. Based on the networks’ structural analysis, we develop a null model to capture features such as the aforementioned transition. Furthermore, depending on the relative number of leader nodes, the empirical networks’ domains may be classified from more anarchic to more autocratic. Also, we show that the nodes’ role at the pinnacle of the hierarchy is crucial in driving the dynamical processes in these systems. Taken together, these results pave the way for a deeper understanding of the architecture of hierarchical empirical complex systems and the processes taking place on them.
