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We investigate the three-state majority-vote model with noise on scale-freeand regular networks. In this model, an individual selects an opinion equal tothe opinion of the majority of its neighbors with probability $1 - q$ andopposite to it with probability $q$. The parameter $q$ is called the noiseparameter of the model. We build a network of interactions where $z$ neighborsare selected by each added site in the system, yielding a preferentialattachment network with degree distribution $k^{-\lambda}$, where $\lambda \sim3$. In this work, $z$ is called growth parameter. Using finite-size scalinganalysis, we show that the critical exponents associated with the magnetizationand magnetic susceptibility add up to unity when a volumetric scaling is used,regardless of the dimension of the network of interactions. Using Monte Carlosimulations, 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. Wefind that the critical noise is an increasing function of the growth parameter$z$, and we define and verify numerically the unitary relation $\upsilon$ forthe 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 thecritical noise and the critical exponents for the two and three-statemajority-vote model on cubic lattices networks where we illustrate theapplication of the unitary relation with a volumetric scaling.

Three-state Majority-Vote Model on Barab\'asi-Albert and Cubic Networks and the Unitary Relation for Critical Exponents