Block voter model: Phase diagram and critical behavior

Abstract

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.

Scaling functions for systems with finite range of interaction

“…The BVM is defined by an outflow dynamics where a central set of N PCS spins, denoted by persuasive cluster spins (PCS), tries to influence the opinion of their neighboring counterparts. It is shown that the effects of increasing the size of the persuasive cluster are the reduction of the critical amplitudes and the increment of the ordered region in the phase diagram [7]. Therefore, within the context of the present study, the range of interaction parameter is defined by the number of spins N PCS inside the persuasive cluster (that is, = N PCS ).…”

“…The phase diagram of the model in the q − N PCS parameter space was reported in Ref. [7]. From the results for q c , we simulate the BVM on regular square lattices of linear length L = 100,160,180,200, and 300, considering periodic boundary conditions and asynchronous update.…”

“…1(b)] versus N PCS . We consider β/ν = 0.125, γ /ν = 1.75, and ν = 1 (ν = 2), which are the nonclassical exponents for the BVM on the square lattice [7]. For every N PCS , we have five values of M L and χ L that are associated with the sizes of the lattices considered.…”

“…The MVM exhibits a continuous phase transition in a two-dimensional parameter space defined by the noise parameter q (the probability that a spin adopts a state contrary of the state of the majority of its neighbors) and the strength of the range of the interaction . A general conclusion from these studies [7][8][9][10] is that the transition occurs at a critical noise q c , which is an increasing function of the parameter . Moreover, it should also be emphasized that the critical amplitudes of relevant thermodynamical quantities become reduced as the range of the interactions increases.…”

“…For instance, for inflow dynamics of spin systems [11] defined on regular lattices, we may define = R eff [12], the maximum effective distance for the central spin to be influenced by its neighbors. In a recent paper [7], we consider the collective behavior of the block voter model (BVM), which introduces long-ranged interactions in the system. The BVM is defined by an outflow dynamics where a central set of N PCS spins, denoted by persuasive cluster spins (PCS), tries to influence the opinion of their neighboring counterparts.…”