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Non-Markovian dynamics pervades human activity and social networks and it induces memory effects and burstiness in a wide range of processes including inter-event 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 {em 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 anti-aging 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 non-monotonic function of the rate $beta$ of the aging (anti-aging) process. In particular the critical noise in the aging regime displays a maximum as a function of $beta$ while in the anti-aging regime displays a minimum. This implies that the aging/anti-aging dynamics can retard/anticipate the transition and that there is an optimal rate $beta$ 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.
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Through Monte Carlo Simulation, the well-known majority-vote model has been studied with noise on directed random graphs. In order to characterize completely the observed order-disorder phase transition, the critical noise parameter $q_c$, as well as the critical exponents $beta/nu$, $gamma/nu$ and $1/nu$ have been calculated as a function of the connectivity $z$ of the random graph.
In this paper, we generalize the original majority-vote (MV) model with noise from two states to arbitrary $q$ states, where $q$ is an integer no less than two. The main emphasis is paid to the comparison on the nature of phase transitions between the two-state MV (MV2) model and the three-state MV (MV3) model. By extensive Monte Carlo simulation and mean-field analysis, we find that the MV3 model undergoes a discontinuous order-disorder phase transition, in contrast to a continuous phase transition in the MV2 model. A central feature of such a discontinuous transition is a strong hysteresis behavior as noise intensity goes forward and backward. Within the hysteresis region, the disordered phase and ordered phase are coexisting.
The majority-vote (MV) model is one of the simplest nonequilibrium Ising-like model that exhibits a continuous order-disorder phase transition at a critical noise. In this paper, we present a quenched mean-field theory for the dynamics of the MV model on networks. We analytically derive the critical noise on arbitrary quenched unweighted networks, which is determined by the largest eigenvalue of a modified network adjacency matrix. By performing extensive Monte Carlo simulations on synthetic and real networks, we find that the performance of the quenched mean-field theory is superior to a heterogeneous mean-field theory proposed in a previous paper [Chen emph{et al.}, Phys. Rev. E 91, 022816 (2015)], especially for directed networks.
The stationary critical properties of the isotropic majority vote model on random lattices with quenched connectivity disorder are calculated by using Monte Carlo simulations and finite size analysis. The critical exponents $gamma$ and $beta$ are found to be different from those of the Ising and majority vote on the square lattice model and the critical noise parameter is found to be $q_{c}=0.117pm0.005$.
On Archimedean lattices, the Ising model exhibits spontaneous ordering. Three examples of these lattices of the majority-vote model with noise are considered and studied through extensive Monte Carlo simulations. The order/disorder phase transition is observed in this system. The calculated values of the critical noise parameter are q_c=0.089(5), q_c=0.078(3), and q_c=0.114(2) for honeycomb, Kagome and triangular lattices, respectively. The critical exponents beta/nu, gamma/nu and 1/nu for this model are 0.15(5), 1.64(5), and 0.87(5); 0.14(3), 1.64(3), and 0.86(6); 0.12(4), 1.59(5), and 1.08(6) for honeycomb, Kagome and triangular lattices, respectively. These results differs from the usual Ising model results and the majority-vote model on so-far studied regular lattices or complex networks. The effective dimensionalities of the system D_{eff}= 1.96(5) (honeycomb), D_{eff} =1.92(4) (Kagome), and D_{eff}= 1.83(5) (triangular) for these networks are just compatible to the embedding dimension two.
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