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Index statistical properties of sparse random graphs.

F L Metz1,2, Daniel A Stariolo2

  • 1Departamento de Física, Universidade Federal de Santa Maria, 97105-900 Santa Maria, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 14, 2015
PubMed
Summary
This summary is machine-generated.

We developed a replica method to analyze eigenvalue distributions in large sparse random graphs. This method characterizes fluctuations and reveals linear scaling of index variance for localized eigenvectors.

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Area of Science:

  • Statistical mechanics
  • Network science
  • Random matrix theory

Background:

  • Understanding eigenvalue distributions in large random graphs is crucial for characterizing their structural and dynamical properties.
  • Sparse random graphs exhibit complex behaviors distinct from dense or rotationally invariant models.
  • Eigenvector localization impacts the statistical properties of eigenvalues.

Purpose of the Study:

  • To develop an analytical method for computing the characteristic function of eigenvalue distributions in large sparse random graphs.
  • To characterize sample-to-sample fluctuations of eigenvalue counts.
  • To investigate the scaling behavior of index variance for localized eigenvectors.

Main Methods:

  • Utilizing the replica method to derive an analytical approach.
  • Computing the characteristic function for the probability P(N)(K,λ) of K eigenvalues below threshold λ.
  • Analyzing the index variance for sparse random graph models.

Main Results:

  • The replica method allows for the computation of all moments of P(N)(K,λ), characterizing fluctuations.
  • For localized eigenvectors, index variance scales linearly with N (N≫1) for |λ|>0.
  • Prefactors for Erdös-Rényi and regular random graphs show nonmonotonic behavior with λ, contrasting with rotationally invariant matrices.
  • Numerical results confirm the analytical approach and suggest Gaussian index fluctuations.

Conclusions:

  • The replica method provides an exact analytical tool for studying eigenvalue statistics in sparse random graphs.
  • The linear scaling of index variance highlights unique properties of sparse graphs with localized eigenvectors.
  • Findings offer insights into the differences between sparse random graphs and rotationally invariant random matrices.