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Hierarchy Depth in Directed Networks.

Krzysztof Suchecki1, Janusz A Hołyst1,2

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This study introduces rooted and relative depth measures for analyzing flow hierarchy in directed networks, distinguishing hierarchical from non-hierarchical systems. These depth measures offer insights into network structure and organization.

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Area of Science:

  • Network Science
  • Graph Theory
  • Complex Systems Analysis

Background:

  • Directed networks are prevalent in various systems, from biological to technological.
  • Understanding flow hierarchy is crucial for characterizing network organization and function.
  • Existing depth measures may not adequately capture hierarchy in networks with cycles.

Purpose of the Study:

  • To define and investigate novel depth measures for flow hierarchy in directed networks.
  • To analyze the properties of these measures, particularly in the presence of directed cycles.
  • To differentiate between hierarchical and non-hierarchical network structures using these measures.

Main Methods:

  • Definition of rooted depth and relative depth measures.
  • Introduction of the loop collapse method for handling directed cycles.
  • Empirical analysis on Erdös-Rényi random graphs, directed Barabási-Albert networks, and the Gnutella peer-to-peer network.

Main Results:

  • Random graphs exhibit unimodal depth distributions dependent on arc density.
  • Hierarchical systems show similar depth distributions across varying network densities.
  • Relative depth correlates with existing trophic level measures in tree-like networks and complex topologies.

Conclusions:

  • The proposed depth measures effectively distinguish between hierarchical and non-hierarchical directed networks.
  • The loop collapse method facilitates the analysis of networks with cyclic structures.
  • Relative depth provides a robust measure for hierarchy across diverse network types.