Fast Monte Carlo Algorithms on Re-Wired Small-World Spin Models

Small-world networks have become an important model for understanding many complex phenomena in science and in sociological contexts. One tool for exploring the critical and phase transitional behaviour in small-world systems is to simulate known models on grids and networks that have been re-wired using the small-world approach. A well known model for studying phase transitions is the Ising spin model. This system is usually studied using an artificial Monte Carlo dynamics scheme and for the study of large model systems fast update algorithms based on whole cluster updating are necessary. These algorithms can not be applied to arbitrarily re-wired models. In this paper we apply a special pair-wise re-wiring that conserves the number of connected neighbours of each spin site and to which fast cluster update algorithms can be applied. We describe the algorithm and its implementation and present some convergence results which are compared with those from more traditional Monte Carlo approaches.