Introducing small-world network effects to critical dynamics

We analytically investigate the kinetic Gaussian model and the one-dimensional kinetic Ising model of 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, which is believed to be a good choice for the mean-field transition widely (in fact, without exception so far) observed for 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 nonexistence 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.