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Graph hierarchy: a novel framework to analyse hierarchical structures in complex networks.

Giannis Moutsinas1, Choudhry Shuaib2, Weisi Guo3

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This study introduces a new hierarchical framework for analyzing any simple graph, extending beyond graphs with basal vertices. This framework offers novel metrics for network analysis and reveals insights into epidemic dynamics and economic systems.

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Area of Science:

  • Graph theory
  • Network analysis
  • Systems biology

Background:

  • Trophic coherence measures graph hierarchy, linked to stability and dynamics.
  • Trophic levels reveal vertex function but require basal vertices, limiting analysis.
  • Existing methods restrict graph analysis to specific network structures.

Purpose of the Study:

  • To develop a general hierarchical framework applicable to any simple graph.
  • To introduce new metrics for network analysis: hierarchical levels, influence centrality, and democracy coefficient.
  • To explore the relationship between network hierarchy, epidemic spread, and economic implications.

Main Methods:

  • Introduction of a generalized hierarchical framework for simple graphs.
  • Development of novel metrics: hierarchical levels, influence centrality, and democracy coefficient.
  • Application of the framework to analyze SIS epidemic models and economic systems.

Main Results:

  • The new framework successfully generalizes trophic analysis to all simple graphs.
  • Novel metrics provide deeper insights into network topology and dynamics.
  • Demonstrated correlation between network hierarchical structure and epidemic incidence rates.

Conclusions:

  • The generalized hierarchical framework expands the scope of network analysis.
  • New metrics offer valuable tools for understanding complex systems.
  • Network hierarchy has significant implications for epidemic modeling and economic analysis.