Three-state Majority-Vote Model on Barabási-Albert and Cubic Networks and the Unitary Relation for Critical Exponents

Abstract

We investigate the three-state majority-vote model with noise on scale-free and regular networks. In this model, an individual selects an opinion equal to the opinion of the majority of its neighbors with probability $1-q$ and opposite to it with probability $q$. The parameter $q$ is called the noise parameter of the model. We build a network of interactions where $z$ neighbors are selected by each added site in the system, yielding a preferential attachment network with degree distribution $k^{-\lambda }$, where $\lambda \sim 3$. In this work, $z$ is called growth parameter. Using finite-size scaling analysis, we show that the critical exponents associated with the magnetization and magnetic susceptibility add up to unity when a volumetric scaling is used, regardless of the dimension of the network of interactions. Using Monte Carlo simulations, we calculate the critical noise parameter $q_c$ as a function of $z$ for the scale-free networks and obtain the phase diagram of the model. We find that the critical noise is an increasing function of the growth parameter $z$, and we define and verify numerically the unitary relation $\upsilon$ for the critical exponents by calculating $\beta /\bar{\nu }$, $\gamma /\bar{\nu }$ and $1/\bar{\nu }$ for several values of the network parameter $z$. We also obtain the critical noise and the critical exponents for the two and three-state majority-vote model on cubic lattices networks where we illustrate the application of the unitary relation with a volumetric scaling.

arXiv comment11 pages, 10 figures, regular paper

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