New Challenges in Cellular Automata Due to Network Geometry - Ferromagnetic Transition Study

Self-organization to ferromagnetic phase transition in cellular automata of spins governed by stochastic majority rule and when topology between spins is changed, is investigated numerically. Three types of edge rewiring are considered. The algorithm of Watts and Strogatz is applied at each time step to establish a network evolving stochastically (the first network). The preference functions are defined to amplify the role of strongly connected vertices (the second network). Demand to preserve graph connectivity provides the third way of edge rewiring. Each of these processes yields the different network: small-world, scattered nodes with one strongly connected component, and scale-free network when rewiring is properly adjusted. The stochastic majority rule applied to these networks can lead or not to ferromagnetic transition. Classical mean-field transition in case of the first and third network is observed. Scale-free ferromagnetic transition can be observed when rewiring is properly adjusted.