Collective Relaxation Dynamics of Small-World Networks

TL;DR

This paper develops a mean-field theory to analytically describe the spectral properties of small-world networks, covering from regular to highly randomized topologies, and validates predictions with numerical checks.

Contribution

It introduces a unified analytic framework for network spectra across different topologies, enhancing understanding of collective relaxation in complex networks.

Findings

Derived a formula for network spectra from regular to random topologies.

Validated predictions with numerical simulations across small-world regimes.

Applied random matrix theory for highly randomized networks.

Abstract

Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph Laplacian or a similar linear operator. The structure of networks with regular, small-world and random connectivities are reasonably well understood, but their collective dynamical properties remain largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies in Watts-Strogatz networks, explaining the simultaneous dependencies on network size N, average degree k and topological randomness q. We present simplified analytic predictions for the second largest and smallest eigenvalue, and numerical checks confirm our theoretical predictions for zero, small and moderate topological randomness q, including the entire small-world regime. For large q of the order of one, we apply standard random matrix theory thereby overarching the full range from regular to randomized network topologies. These results may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network dynamical systems.