Phase transition in the majority-vote model on the Archimedean lattices

Critical short-time behavior of majority-vote model on scale-free networks

We discuss the short-time behavior of the majority vote dynamics on scale-free networks at the critical threshold. A heterogeneous mean-field theory on the critical short-time behavior of the majority-vote model on scale-free networks is introduced. In addition, the heterogeneous mean-field predictions are compared with extensive Monte Carlo simulations of the short-time dependencies of the order parameter and the susceptibility. Closed expressions of the dynamical exponent z and the time correlation exponent ν_{∥} are obtained. The short-time scaling is compatible with a nonuniversal critical behavior for 5/2<γ<7/2. However, for γ≥7/2, we have the mean-field Ising criticality with additional logarithmic corrections for γ=7/2, the same as the stationary scaling.

Droplet finite-size scaling of the majority-vote model on scale-free networks

We discuss the majority vote model coupled with scale-free networks and investigate its critical behavior. Previous studies point to a nonuniversal behavior of the majority vote model, where the critical exponents depend on the connectivity. At the same time, the effective dimension D_{eff} is unity for a degree distribution exponent 5/2<γ<7/2. We introduce 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. Our finite-size approach has two sources of size dependence: an external field representing the influence of the mass media on consensus formation and the scale-free network cutoff. The critical exponents are nonuniversal, dependent on the degree distribution exponent, precisely when 5/2<γ<7/2. For γ≥7/2, the model is in the same universality class as the majority vote model on Erdős-Rényi random graphs. However, for γ=7/2, the critical behavior includes additional logarithmic corrections.

Diffusive majority-vote model

We define a stochastic reaction-diffusion process that describes a consensus formation in a nonsedentary population. The process is a diffusive version of the majority-vote model, where the state update follows two stages: In the first stage, spins are allowed to jump to a random neighbor node with probabilities D_{+} and D_{-} for the respective spin orientations, and in the second stage, the spins in the same node can change its values according to the majority-vote update rule. The model presents a consensus formation phase when the concentration is greater than a threshold value and a paramagnetic phase on the converse for equal diffusion probabilities, i.e., maintaining the inversion symmetry. Setting unequal diffusion probabilities for the respective spin orientations is the same as applying an external magnetic field. The system undergoes a discontinuous phase transition for concentrations higher than the critical threshold on the external field. The individuals that diffuse more dominate the stationary collective opinion.

Critical phenomena and strategy ordering with hub centrality approach in the aspiration-based coordination game

We study the coordination game with an aspiration-driven update rule in regular graphs and scale-free networks. We prove that the model coincides exactly with the Ising model and shows a phase transition at the critical selection noise when the aspiration level is zero. It is found that the critical selection noise decreases with clustering in random regular graphs. With a non-zero aspiration level, the model also exhibits a phase transition as long as the aspiration level is smaller than the degree of graphs. We also show that the critical exponents are independent of clustering and aspiration level to confirm that the coordination game belongs to the Ising universality class. As for scale-free networks, the effect of aspiration level on the order parameter at a low selection noise is examined. In model networks (the Barabási-Albert network and the Holme-Kim network), the order parameter abruptly decreases when the aspiration level is the same as the average degree of the network. In contrast, in real-world networks, the order parameter decreases gradually. We explain this difference by proposing the concepts of hub centrality and local hub. The histogram of hub centrality of real-world networks separates into two parts unlike model networks, and local hubs exist only in real-world networks. We conclude that the difference of network structures in model and real-world networks induces qualitatively different behavior in the coordination game.

Majority-vote model with degree-weighted influence on complex networks

We study the phase transition of the degree-weighted majority vote (DWMV) model on Erdős-Rényi networks (ERNs) and scale-free networks (SFNs). In this model, a weight parameter α adjusts the level of influence of each node on its connected neighbors. Through the Monte Carlo simulations and the finite-size scaling analysis, we find that the DWMV model on ERNs and SFNs with degree exponents λ>5 belongs to the mean-field Ising universality class, regardless of α. On SFNs with 3<λ<5 the model belongs to the Ising universality class only when α=0. For α>0 we find that the critical exponents continuously change as α increases from α=0. However, on SFNs with λ<3 we find that the model undergoes a continuous transition only for α=0, but the critical exponents significantly deviate from those for the mean-field Ising model. For α>0 on SFNs with λ<3 the model is always in the disordered phase.

Non-Markovian majority-vote model

Non-Markovian dynamics pervades human activity and social networks and it induces memory effects and burstiness in a wide range of processes including interevent time distributions, duration of interactions in temporal networks, and human mobility. Here, we propose a non-Markovian majority-vote model (NMMV) that introduces non-Markovian effects in the standard (Markovian) majority-vote model (SMV). The SMV model is one of the simplest two-state stochastic models for studying opinion dynamics, and displays a continuous order-disorder phase transition at a critical noise. In the NMMV model we assume that the probability that an agent changes state is not only dependent on the majority state of his neighbors but it also depends on his age, i.e., how long the agent has been in his current state. The NMMV model has two regimes: the aging regime implies that the probability that an agent changes state is decreasing with his age, while in the antiaging regime the probability that an agent changes state is increasing with his age. Interestingly, we find that the critical noise at which we observe the order-disorder phase transition is a nonmonotonic function of the rate β of the aging (antiaging) process. In particular the critical noise in the aging regime displays a maximum as a function of β while in the antiaging regime displays a minimum. This implies that the aging/antiaging dynamics can retard/anticipate the transition and that there is an optimal rate β for maximally perturbing the value of the critical noise. The analytical results obtained in the framework of the heterogeneous mean-field approach are validated by extensive numerical simulations on a large variety of network topologies.

Three-State Majority-vote Model on Scale-Free Networks and the Unitary Relation for Critical Exponents

We investigate the three-state majority-vote model for opinion dynamics 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 different 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, a preferential attachment network with degree distribution k^−λ, where λ = 3 for a large number of nodes N. In this work, z is called the growth parameter. Using finite-size scaling analysis, we obtain that the critical exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta /\bar{\nu }$$\end{document}β/ν¯ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /\bar{\nu }$$\end{document}γ/ν¯ associated with the magnetization and the susceptibility, respectively. 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 exponents add up to unity when using a special volumetric scaling, regardless of the dimension of the network of interactions. We verify this result by obtaining the critical noise and the critical exponents for the two and three-state majority-vote model on cubic lattice networks.

Phase transitions in a multistate majority-vote model on complex networks

We generalize the original majority-vote (MV) model from two states to arbitrary p states and study the order-disorder phase transitions in such a p-state MV model on complex networks. By extensive Monte Carlo simulations and a mean-field theory, we show that for p≥3 the order of phase transition is essentially different from a continuous second-order phase transition in the original two-state MV model. Instead, for p≥3 the model displays a discontinuous first-order phase transition, which is manifested by the appearance of the hysteresis phenomenon near the phase transition. Within the hysteresis loop, the ordered phase and disordered phase are coexisting, and rare flips between the two phases can be observed due to the finite-size fluctuation. Moreover, we investigate the type of phase transition under a slightly modified dynamics [Melo et al., J. Stat. Mech. (2010) P110321742-546810.1088/1742-5468/2010/11/P11032]. We find that the order of phase transition in the three-state MV model depends on the degree heterogeneity of networks. For p≥4, both dynamics produce the first-order phase transitions.

Effect of Strong Opinions on the Dynamics of the Majority-Vote Model

We study how the presence of individuals with strong opinions affects a square lattice majority-vote model with noise. In a square lattice network we perform Monte-Carlo simulations and replace regular actors σ with strong actors μ in a random distribution. We find that the value of the critical noise parameter q_c is a decreasing function of the concentration r of strong actors in the social interaction network. We calculate the critical exponents β/ν, γ/ν, and 1/ν and find that the presence of strong actors does not change the Ising universality class of the isotropic majority-vote model.

Exact solution of the isotropic majority-vote model on complete graphs

The isotropic majority-vote (MV) model, which, apart from the one-dimensional case, is thought to be nonequilibrium and violating the detailed balance condition. We show that this is not true when the model is defined on a complete graph. In the stationary regime, the MV model on a fully connected graph fulfills the detailed balance and is equivalent to the modified Ehrenfest urn model. Using the master equation approach, we derive the exact expression for the probability distribution of finding the system in a given spin configuration. We show that it only depends on the absolute value of magnetization. Our theoretical predictions are validated by numerical simulations.