Dynamic Transitions in Small World Networks: Approach to Equilibrium

TL;DR

This paper investigates how the transition to phase synchronization in small world networks depends on the rate of driving, revealing that slower driving causes the transition point to approach zero, similar to equilibrium systems.

Contribution

It demonstrates that the transition point in a small world network model shifts towards zero as the driving rate decreases, linking nonequilibrium dynamics to equilibrium behavior.

Findings

Transition point decreases with slower driving rate

Transition approaches zero in the slow driving limit

Behavior resembles equilibrium network models

Abstract

We study the transition to phase synchronization in a model for the spread of infection defined on a small world network. It was shown (Phys. Rev. Lett. {\bf 86} (2001) 2909) that the transition occurs at a finite degree of disorder $p$, unlike equilibrium models where systems behave as random networks even at infinitesimal $p$ in the infinite size limit. We examine this system under variation of a parameter determining the driving rate, and show that the transition point decreases as we drive the system more slowly. Thus it appears that the transition moves to $p=0$ in the very slow driving limit, just as in the equilibrium case.