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Spectral partitioning in equitable graphs.

Paolo Barucca1

  • 1Department of Banking and Finance, University of Zurich, Zurich, Switzerland and London Institute for Mathematical Sciences, London W1K 2XF, United Kingdom.

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Summary
This summary is machine-generated.

This study analyzes equitable graphs, offering an exact solution for graph partitioning in two-community structures. Findings suggest efficient recovery is unlikely in these complex systems.

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Area of Science:

  • Complex systems analysis
  • Network science
  • Graph theory

Background:

  • Graph partitioning is crucial for analyzing complex systems like biological and financial networks.
  • Rigorous analysis is limited to specific graph ensembles.
  • Equitable graphs with block-regular structures offer a tractable model.

Purpose of the Study:

  • To analyze an ensemble of equitable graphs.
  • To compute the spectral density for modular and bipartite structures.
  • To propose and verify an exact solution for graph partitioning in two equal-sized communities.

Main Methods:

  • Analytical computation of spectral density.
  • Application of Kesten-McKay's law to modular and bipartite structures.
  • Numerical verification of the proposed graph partitioning solution.

Main Results:

  • Exact computation of spectral density for modular and bipartite equitable graphs.
  • Demonstration that Kesten-McKay's law extends to homogeneous blocks in these structures.
  • An exact partitioning solution for two equal-sized communities was proposed and numerically validated.

Conclusions:

  • The study provides an exact solution for graph partitioning in specific equitable graph structures.
  • A conjecture regarding the absence of efficient recovery detectability transitions in equitable graphs was proposed.
  • Results offer insights into detectability thresholds and resolution limits in stochastic block models.