Critical phenomena in the majority voter model on two-dimensional regular lattices

Stochastic thermodynamics of opinion dynamics models

We show that models of opinion formation and dissemination in a community of individuals can be framed within stochastic thermodynamics from which we can build a nonequilibrium thermodynamics of opinion dynamics. This is accomplished by decomposing the original transition rate that defines an opinion model into two or more transition rates, each representing the contact with heat reservoirs at different temperatures, and postulating an energy function. As the temperatures are distinct, heat fluxes are present even at the stationary state and linked to the production of entropy, the fundamental quantity that characterizes nonequilibrium states. We apply the present framework to a generic-vote model including the majority-vote model in a square lattice and in a cubic lattice. The fluxes and the rate of entropy production are calculated by numerical simulation and by the use of a pair approximation.

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.

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.

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.

Large deviation induced phase switch in an inertial majority-vote model

We theoretically study noise-induced phase switch phenomena in an inertial majority-vote (IMV) model introduced in a recent paper [Chen et al., Phys. Rev. E 95, 042304 (2017)]. The IMV model generates a strong hysteresis behavior as the noise intensity f goes forward and backward, a main characteristic of a first-order phase transition, in contrast to a second-order phase transition in the original MV model. Using the Wentzel-Kramers-Brillouin approximation for the master equation, we reduce the problem to finding the zero-energy trajectories in an effective Hamiltonian system, and the mean switching time depends exponentially on the associated action and the number of particles N. Within the hysteresis region, we find that the actions, along the optimal forward switching path from the ordered phase (OP) to disordered phase (DP) and its backward path show distinct variation trends with f, and intersect at f = f_c that determines the coexisting line of the OP and DP. This results in a nonmonotonic dependence of the mean switching time between two symmetric OPs on f, with a minimum at f_c for sufficiently large N. Finally, the theoretical results are validated by Monte Carlo simulations.

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

The majority-vote model with noise was studied on the 11 Archimedean lattices by the Monte Carlo method and finite-size scaling. The critical noises and critical exponents were obtained with precision. Contrary to some previous reports, we confirmed that the majority-vote model on the Archimedean lattices belongs to the two-dimensional Ising universality class. It was shown that very precise determination of the critical noise is required to obtain proper values of the critical exponents.

Critical noise of majority-vote model on complex networks

The majority-vote model with noise is one of the simplest nonequilibrium statistical model that has been extensively studied in the context of complex networks. However, the relationship between the critical noise where the order-disorder phase transition takes place and the topology of the underlying networks is still lacking. In this paper, we use the heterogeneous mean-field theory to derive the rate equation for governing the model's dynamics that can analytically determine the critical noise f(c) in the limit of infinite network size N→∞. The result shows that f(c) depends on the ratio of 〈k〉 to 〈k(3/2)〉, where 〈k〉 and 〈k(3/2)〉 are the average degree and the 3/2 order moment of degree distribution, respectively. Furthermore, we consider the finite-size effect where the stochastic fluctuation should be involved. To the end, we derive the Langevin equation and obtain the potential of the corresponding Fokker-Planck equation. This allows us to calculate the effective critical noise f(c)(N) at which the susceptibility is maximal in finite-size networks. We find that the f(c)-f(c)(N) decays with N in a power-law way and vanishes for N→∞. All the theoretical results are confirmed by performing the extensive Monte Carlo simulations in random k-regular networks, Erdös-Rényi random networks, and scale-free networks.