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This study analyzes heterogeneous random networks with power-law degree distributions. We prove that fluctuations in hub behavior are small over long timescales, explaining observed phenomena in complex systems.

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Area of Science:

  • Complex systems
  • Network science
  • Statistical physics

Background:

  • Heterogeneous random networks exhibit power-law degree distributions.
  • Node dynamics involve random dynamical systems with random coupling.
  • Hub behavior in such networks is influenced by mean-field interactions and fluctuations.

Purpose of the Study:

  • To characterize hub behavior in heterogeneous random networks.
  • To analyze the impact of statistically controlled fluctuations on network dynamics.
  • To provide theoretical explanations for observed numerical phenomena.

Main Methods:

  • Mathematical analysis of random dynamical systems.
  • Mean-field approximation with controlled fluctuations.
  • Application of Berry-Esseen estimates for fluctuation statistics.

Main Results:

  • Hub behavior fluctuations are small over exponentially long time scales.
  • Berry-Esseen estimates quantify fluctuation statistics at fixed times.
  • Theoretical explanation for scaling relations, desynchronization, and Gaussian fluctuation behavior.

Conclusions:

  • The study provides a rigorous framework for understanding hub behavior in complex networks.
  • Results reconcile theoretical predictions with numerical observations in heterogeneous random networks.
  • The findings contribute to the broader understanding of emergent phenomena in large-scale dynamical systems.