Collective relaxation dynamics of small-world networks

Area of Science:

• Complex Systems
• Network Science
• Mathematical Physics

Background:

• Collective dynamics in complex networks, such as synchronization and relaxation, are crucial but not fully understood.
• Network structures (regular, small-world, random) are known, yet their dynamical properties remain largely elusive.
• Asymptotic dynamics are typically described by linear operators like the graph Laplacian.

Purpose of the Study:

• To develop a unified analytic theory for network spectra across diverse topologies.
• To derive expressions for spectral properties dependent on network size, average degree, and randomness.
• To provide a mechanistic understanding of collective relaxation phenomena in dynamical systems.

Main Methods:

• A two-stage mean-field theory is introduced to derive analytic expressions for network spectra.
• A single formula is presented that spans regular to randomized network topologies (Watts-Strogatz networks).
• Standard random matrix theory is applied for highly randomized networks (large topological randomness q).

Main Results:

• Analytic expressions for network spectra are derived, covering regular, small-world, and random topologies.
• A unified formula explains dependencies on network size (N), average degree (k), and topological randomness (q).
• Predictions for key eigenvalues are confirmed numerically across various randomness levels, including the small-world regime.

Conclusions:

• The developed theory provides a unified framework for understanding network spectra and collective dynamics.
• The findings offer analytic insights into relaxation processes in complex dynamical systems.
• This work bridges the gap between network structure and emergent dynamical behaviors.