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Extensive numerical simulations reveal universal spectral and eigenfunction properties in Erdős-Rényi random networks. These findings hold for fixed average degree and characterize transitions in network connectivity.

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Area of Science:

  • Network science
  • Statistical physics
  • Graph theory

Background:

  • Erdős-Rényi (ER) networks are fundamental models in network science.
  • Understanding spectral and eigenfunction properties is crucial for characterizing network behavior.
  • Universality in complex systems suggests predictable patterns despite underlying randomness.

Purpose of the Study:

  • To investigate the universality of spectral and eigenfunction properties in ER random networks.
  • To characterize the energy-level spacing distribution and eigenfunction localization length.
  • To explore these properties in networks with varying connectivity and disorder.

Main Methods:

  • Extensive numerical simulations were employed.
  • Analysis focused on adjacency matrices of Erdős-Rényi networks.
  • Key metrics analyzed include nearest-neighbor energy-level spacing distribution P(s) and entropic eigenfunction localization length.
  • The Brody distribution was used to characterize P(s).

Main Results:

  • Universality of P(s) and eigenfunction localization length was demonstrated for fixed average degree (ξ).
  • The Brody distribution accurately describes P(s) across network connectivity transitions (α=0 to α=1).
  • Universality persists in ER networks with diagonal disorder, unlike standard ER networks.
  • Spectral and eigenfunction properties of small-world networks were also discussed.

Conclusions:

  • The spectral and eigenfunction properties of ER random networks exhibit universality for a fixed average degree.
  • Network connectivity and disorder influence these universal properties, with the Brody distribution providing a robust characterization.
  • These findings contribute to a deeper understanding of random network behavior and its deviations.