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This study introduces spectral methods to reveal hidden structures in directed networks by connecting node reordering to random graph models. It compares two algorithms for identifying periodic and linear hierarchies, aiding in network analysis.

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Area of Science:

  • Network Science
  • Graph Theory
  • Data Analysis

Background:

  • Directed networks contain complex structures that are often hidden.
  • Spectral methods offer powerful tools for uncovering these structures.
  • Existing methods focus on minimizing frustration or trophic incoherence.

Purpose of the Study:

  • To establish and exploit connections between node reordering and random graph models.
  • To compare two spectral algorithms for uncovering network structures.
  • To quantify the likelihood of periodic versus linear hierarchies in networks.

Main Methods:

  • Utilizing spectral methods based on Laplacian-style matrices.
  • Minimizing objective functions related to network frustration and trophic incoherence.
  • Associating node reordering with new classes of directed random graph models.

Main Results:

  • Demonstrated that node reordering via spectral methods relates to specific random graph models.
  • Developed a framework to compare algorithms for identifying periodic and linear hierarchies.
  • Successfully applied the approach to both synthetic and real-world networks.

Conclusions:

  • Spectral methods provide a robust framework for analyzing hidden structures in directed networks.
  • The proposed random graph setting enables quantitative comparison of different structural uncovering algorithms.
  • The study offers practical insights for implementing these methods in network analysis.