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Modeling the Functional Network for Spatial Navigation in the Human Brain
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Published on: October 13, 2023

Random matrix analysis of complex networks.

Sarika Jalan1, Jayendra N Bandyopadhyay

  • 1Max-Planck Institute for the Physics of Complex Systems, Nöthnitzerstrasse 38, D-01187 Dresden, Germany. sarika@mpipks-dresden.mpg.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2007
PubMed
Summary
This summary is machine-generated.

Complex network analysis using random matrix theory (RMT) reveals eigenvalue distributions matching RMT predictions. Spectral rigidity in random and scale-free networks aligns with RMT, while small-world networks show deviations at larger scales.

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Area of Science:

  • Complex Networks
  • Statistical Physics
  • Random Matrix Theory

Background:

  • Complex networks are ubiquitous in nature and technology.
  • Understanding the spectral properties of network adjacency matrices is crucial.
  • Random Matrix Theory (RMT) provides a framework for analyzing eigenvalue statistics.

Purpose of the Study:

  • To investigate the applicability of Random Matrix Theory (RMT) to various complex network models.
  • To analyze the nearest-neighbor and next-nearest-neighbor spacing distributions of eigenvalues.
  • To probe long-range correlations in eigenvalues using spectral rigidity.

Main Methods:

  • Analysis of adjacency matrix eigenvalues for random, scale-free, and small-world networks.
  • Calculation of nearest-neighbor and next-nearest-neighbor spacing distributions.
  • Application of the Delta_3 statistic to assess spectral rigidity.

Main Results:

  • Eigenvalue spacing distributions for all studied networks conform to Gaussian Orthogonal Ensemble (GOE) statistics of RMT.
  • Spectral rigidity, measured by the Delta_3 statistic, exhibits linear behavior in semilogarithmic scale, consistent with RMT predictions (slope ≈ 1/π²).
  • Random and scale-free networks closely follow RMT predictions for large scales; small-world networks show adherence for sufficiently large scales, but to a lesser extent.

Conclusions:

  • RMT provides a robust framework for characterizing the spectral properties of complex networks.
  • The findings suggest universal behavior in eigenvalue statistics across different network topologies.
  • Deviations in small-world networks highlight their unique structural characteristics impacting spectral properties.