Droplet Finite-Size Scaling of the Majority Vote Model on Quenched Scale-Free Networks

Abstract

We consider the Majority Vote model coupled with scale-free networks. Recent works point to a non-universal behavior of the Majority Vote model, where the critical exponents depend on the connectivity while the network's effective dimension $D_\mathrm{eff}$ is unity for a degree distribution exponent $5/2<\gamma<7/2$. We present a finite-size theory of the Majority Vote Model for uncorrelated networks and present generalized scaling relations with good agreement with Monte-Carlo simulation results. The presented finite-size theory has two main sources of size dependence. The first source is an external field describing a mass media influence on the consensus formation and the second source is the scale-free network cutoff. The model indeed presents non-universal critical behavior where the critical exponents depend on the degree distribution exponent $5/2<\gamma<7/2$. For $\gamma \geq 7/2$, the model is on the same universality class of the Majority Vote model on Erd\"os-Renyi random graphs, while for $\gamma=7/2$, the critical behavior presents additional logarithmic corrections.