How to quantify hierarchy?

Although hierarchical organization is highly prevalent in natural, technological and social systems, we are far from a general quantitative understanding of the essence of hierarchy and its generative mechanisms. A first step towards such a theory is to be able to measure the level of hierarchical organization of a system. Based on the framework of directed networks, I will discuss previous attempts to construct such a measure relying on different intuitive ideas, and then I present a possible axiomatic framework to classify these measures and to understand their relation to each other. Finally, a new, computationally efficient measure is introduced based on random walks on the network, being the first one satisfying all of the axioms.

D. Czege, G. Palla, ArXiv 1508.07732, Sci Rep (2015)