Dynamic critical behavior of theXYmodel in small-world networks

Abstract

The critical behavior of the XY model on small-world network is investigated by means of dynamic Monte Carlo simulations. We use the short-time relaxation scheme, i.e., the critical behavior is studied from the nonequilibrium relaxation to equilibrium. Static and dynamic critical exponents are extracted through the use of the dynamic finite-size scaling analysis. It is concluded that the dynamic universality class at the transition is of the mean-field nature. We also confirm numerically that the value of dynamic critical exponent is independent of the rewiring probability P for P > ∼ 0.03.

Evolving small-world networks with geographical attachment preference

“…[10,12,13,22], the spread and percolation properties were investigated, dealing with the spread of information and disease along the shortest path in the graph or the spread along the spanning tree. Recently, researchers have also focused their attention on other different aspects, characterizing many properties of small-world networks [14,15,16,17,18,19,20].…”

How Crucial Is Small World Connectivity for Dynamics?

“…an infinite dimensional case [4]. For the XY model, Medvedyeva et al conjecture that the critical exponents are the same as for the mean field case [5]. They have confirmed it for p ≥ 0.03 and there is good reason to believe that it is true for any p > 0 (The obvious difficulty is that one needs to simulate larger and larger lattices at small p.) Similar conclusions are reached for the Ising model on small world networks as well [6].…”

Hamiltonian mean-field model: effect of temporal perturbation in coupling matrix

“…It has been reported that the critical energy depends on the network parameter in such a system. [21][22][23] Some recent studies have shown that spectral properties of coupling matrix plays significant role in the synchronization of this system. 24,25 HMF model on Erdös-Renyi networks was studied in Ref.…”