Hierarchy Depth in Directed Networks.

Krzysztof Suchecki,Janusz A Hołyst

doi:10.3390/e24020252

Abstract

In this study, we explore the depth measures for flow hierarchy in directed networks. Two simple measures are defined—rooted depth and relative depth—and their properties are discussed. The method of loop collapse is introduced, allowing investigation of networks containing directed cycles. The behavior of the two depth measures is investigated in Erdös-Rényi random graphs, directed Barabási-Albert networks, and in Gnutella p2p share network. A clear distinction in the behavior between non-hierarchical and hierarchical networks is found, with random graphs featuring unimodal distribution of depths dependent on arc density, while for hierarchical systems the distributions are similar for different network densities. Relative depth shows the same behavior as existing trophic level measure for tree-like networks, but is only statistically correlated for more complex topologies, including acyclic directed graphs.

Highlights

We have studied the behavior of the model on two synthetic network models: ErdösRényi random graphs (E-R) and directed Barabási-Albert (B-A) scale-free network, as well as the Gnutella file sharing network
We have tested some of the properties of the depth measures in synthetic and real networks—in Erdös-Rényi (E-R) random graphs, Barabási-Albert (B-A) scale-free networks, and the Gnutella file sharing network [26]
The simple rooted depth is defined as the shortest path from one of the network’s roots, while the more complex relative depth is defined through the relations between vertices and the difference in depths is effectively equal to the longest path between vertices