  Nature, technology and society are full of complexity arising from the
intricate web of the interactions among the units of the related systems (e.g.,
proteins, computers, people). Consequently, one of the most successful recent
approaches to capturing the fundamental features of the structure and dynamics
of complex systems has been the investigation of the networks associated with
the above units (nodes) together with their relations (edges). Most complex
systems have an inherently hierarchical organization and, correspondingly, the
networks behind them also exhibit hierarchical features. Indeed, several papers
have been devoted to describing this essential aspect of networks, however,
without resulting in a widely accepted, converging concept concerning the
quantitative characterization of the level of their hierarchy. Here we develop
an approach and propose a quantity (measure) which is simple enough to be
widely applicable, reveals a number of universal features of the organization
of real-world networks and, as we demonstrate, is capable of capturing the
essential features of the structure and the degree of hierarchy in a complex
network. The measure we introduce is based on a generalization of the m-reach
centrality, which we first extend to directed/partially directed graphs. Then,
we define the global reaching centrality (GRC), which is the difference between
the maximum and the average value of the generalized reach centralities over
the network. We investigate the behavior of the GRC considering both a
synthetic model with an adjustable level of hierarchy and real networks.
Results for real networks show that our hierarchy measure is related to the
controllability of the given system. We also propose a visualization procedure
for large complex networks that can be used to obtain an overall qualitative
picture about the nature of their hierarchical structure.
