Abstract
Complex networks are a useful tool for the understanding of complex systems. One of the emerging properties of such systems is their tendency to form hierarchies: networks can be organized in levels, with nodes in each level exerting control on the ones beneath them. In this paper, we focus on the problem of estimating how hierarchical a directed network is. We propose a structural argument: a network has a strong top-down organization if we need to delete only few edges to reduce it to a perfect hierarchy—an arborescence. In an arborescence, all edges point away from the root and there are no horizontal connections, both characteristics we desire in our idealization of what a perfect hierarchy requires. We test our arborescence score in synthetic and real-world directed networks against the current state of the art in hierarchy detection: agony, flow hierarchy and global reaching centrality. These tests highlight that our arborescence score is intuitive and we can visualize it; it is able to better distinguish between networks with and without a hierarchical structure; it agrees the most with the literature about the hierarchy of well-studied complex systems; and it is not just a score, but it provides an overall scheme of the underlying hierarchy of any directed complex network.
Highlights
Complex networks are a powerful analytical model, used to understand the emergence of complex phenomena arising from the interaction of many different parts in real world systems [1,2,3]
Nested hierarchy is about finding higher-order structures that fully contain lower order structures, at different levels ending in nodes
flow hierarchy” (FH) is implemented in the current version of the networkx Python package, while global reaching centrality” (GRC)’s function is written and scheduled for version 2.0
Summary
Complex networks are a powerful analytical model, used to understand the emergence of complex phenomena arising from the interaction of many different parts in real world systems [1,2,3]. Hierarchies are one of the many possible properties of real world complex networks. Many complex systems tend to organize themselves in different levels, increasingly centralized [4, 5]. Previous works have identified three different types of hierarchy: order, nested and flow [6, 7]. Order hierarchy is equivalent to node ranking: each node is associated with a score calculated in a given way, and nodes are sorted according to this score (e.g. PageRank [8] and HITS [9]). Nested hierarchy is about finding higher-order structures that fully contain lower order structures, at different levels ending in nodes.
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