Hierarchy Depth in Directed Networks

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.

God (≡ Elohim), The First Small World Network

“…These links are interpreted as the specific connections of ("spiritual") information flow. In the present case, the flow of information goes according to the node number hierarchy; such a type of directed flow consideration has been recently studied [60] in a different context.…”

God ($\equiv Elohim$), the first small world network

“…These links are interpreted as the specific connections of ("spiritual") information flow. In the present case, the flow of information goes according to the node number hierarchy; such a type of directed flow consideration has been recently studied in [60] in a different context.…”