Majority-vote model on spatially embedded networks: Crossover from mean-field to Ising universality classes.

We study through Monte Carlo simulations and finite-size scaling analysis the nonequilibrium phase transitions of the majority-vote model taking place on spatially embedded networks. These structures are built from an underlying regular lattice over which directed long-range connections are randomly added according to the probability P_{ij}∼r^{-α}, where r_{ij} is the Manhattan distance between nodes i and j, and the exponent α is a controlling parameter [J. M. Kleinberg, Nature (London) 406, 845 (2000)NATUAS0028-083610.1038/35022643]. Our results show that the collective behavior of this system exhibits a continuous order-disorder phase transition at a critical parameter, which is a decreasing function of the exponent α. Precisely, considering the scaling functions and the critical exponents calculated, we conclude that the system undergoes a crossover among distinct universality classes. For α≤3 the critical behavior is described by mean-field exponents, while for α≥4 it belongs to the Ising universality class. Finally, in the region where the crossover occurs, 3<α<4, the critical exponents are dependent on α.

Scaling functions for systems with finite range of interaction.

We present a numerical determination of the scaling functions of the magnetization, the susceptibility, and the Binder's cumulant for two nonequilibrium model systems with varying range of interactions. We consider Monte Carlo simulations of the block voter model (BVM) on square lattices and of the majority-vote model (MVM) on random graphs. In both cases, the satisfactory data collapse obtained for several system sizes and interaction ranges supports the hypothesis that these functions are universal. Our analysis yields an accurate estimation of the long-range exponents, which govern the decay of the critical amplitudes with the range of interaction, and is consistent with the assumption that the static exponents are Ising-like for the BVM and classical for the MVM.

Block voter model: phase diagram and critical behavior.

We introduce and study the block voter model with noise on two-dimensional square lattices using Monte Carlo simulations and finite-size scaling techniques. The model is defined by an outflow dynamics where a central set of N(PCS) spins, here denoted by persuasive cluster spins (PCS), tries to influence the opinion of their neighboring counterparts. We consider the collective behavior of the entire system with varying PCS size. When N(PCS)>2, the system exhibits an order-disorder phase transition at a critical noise parameter q(c) which is a monotonically increasing function of the size of the persuasive cluster. We conclude that a larger PCS has more power of persuasion, when compared to a smaller one. It also seems that the resulting critical behavior is Ising-like independent of the range of interaction.