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Using arborescences to estimate hierarchicalness in directed complex networks.

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We introduce an "arborescence score" to measure network hierarchy. This score quantifies how close a directed network is to a perfect hierarchy, offering a more intuitive and accurate assessment of complex system organization.

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Area of Science:

  • Complex systems science
  • Network science
  • Graph theory

Background:

  • Complex systems often exhibit hierarchical organization, with control flowing downwards through network levels.
  • Quantifying the degree of hierarchy in directed networks is crucial for understanding system structure and function.

Purpose of the Study:

  • To propose a novel structural measure for estimating the hierarchical nature of directed complex networks.
  • To introduce the "arborescence score" as a method for quantifying network hierarchy.

Main Methods:

  • Defining a perfect hierarchy as an arborescence, where all edges point away from a root node.
  • Proposing the arborescence score based on the number of edge deletions required to transform a network into an arborescence.
  • Comparing the arborescence score against existing hierarchy detection methods (agony, flow hierarchy, global reaching centrality) on synthetic and real-world networks.

Main Results:

  • The arborescence score is intuitive, visualizable, and effectively distinguishes hierarchical from non-hierarchical networks.
  • The score demonstrates strong agreement with established literature on the hierarchy of known complex systems.
  • The arborescence score provides a comprehensive scheme for visualizing the underlying hierarchy of directed networks.

Conclusions:

  • The arborescence score offers a robust and insightful new metric for assessing hierarchy in directed networks.
  • This method surpasses current state-of-the-art techniques in accuracy and interpretability for hierarchy detection.
  • The arborescence score facilitates a deeper understanding of complex system organization and control structures.