Introducing Small-World Network Effect to Critical Dynamics

TL;DR

This paper analytically explores how small-world network structures influence the critical dynamics of Gaussian and Ising models, revealing mean-field effects, critical point shifts, and unchanged dynamic critical exponents.

Contribution

It introduces a systematic analytical approach to study critical dynamics on small-world networks and confirms mean-field behavior and critical exponent invariance.

Findings

Mean-field-like influence on spin dynamics

Analytical p-dependence of the critical point

Invariance of the dynamic critical exponent z=2

Abstract

We analytically investigate the kinetic Gaussian model and the one-dimensional kinetic Ising model on two typical small-world networks (SWN), the adding-type and the rewiring-type. The general approaches and some basic equations are systematically formulated. The rigorous investigation of the Glauber-type kinetic Gaussian model shows the mean-field-like global influence on the dynamic evolution of the individual spins. Accordingly a simplified method is presented and tested, and believed to be a good choice for the mean-field transition widely (in fact, without exception so far) observed on SWN. It yields the evolving equation of the Kawasaki-type Gaussian model. In the one-dimensional Ising model, the p-dependence of the critical point is analytically obtained and the inexistence of such a threshold p_c, for a finite temperature transition, is confirmed. The static critical exponents, gamma and beta are in accordance with the results of the recent Monte Carlo simulations, and also with the mean-field critical behavior of the system. We also prove that the SWN effect does not change the dynamic critical exponent, z=2, for this model. The observed influence of the long-range randomness on the critical point indicates two obviously different hidden mechanisms.