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Register a new userExplosive Phase Transition in a Majority-Vote Model with Inertia
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We generalize the original majority-vote model by incorporating an inertia into the microscopic dynamics of the spin flipping, where the spin-flip probability of any individual depends not only on the states of its neighbors, but also on its own state. Surprisingly, the order-disorder phase transition is changed from a usual continuous type to a discontinuous or an explosive one when the inertia is above an appropriate level. A central feature of such an explosive transition is a strong hysteresis behavior as noise intensity goes forward and backward. Within the hysteresis region, a disordered phase and two symmetric ordered phases are coexisting and transition rates between these phases are numerically calculated by a rare-event sampling method. A mean-field theory is developed to analytically reveal the property of this phase transition.
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We study a nonequilibrium model with up-down symmetry and a noise parameter $q$ known as majority-vote model of M.J. Oliveira 1992 with heterogeneous agents on square lattice. By Monte Carlo simulations and finite-size scaling relations the critical exponents $beta/ u$, $gamma/ u$, and $1/ u$ and points $q_{c}$ and $U^*$ are obtained. After extensive simulations, we obtain $beta/ u=0.35(1)$, $gamma/ u=1.23(8)$, and $1/ u=1.05(5)$. The calculated values of the critical noise parameter and Binder cumulant are $q_{c}=0.1589(4)$ and $U^*=0.604(7)$. Within the error bars, the exponents obey the relation $2beta/ u+gamma/ u=2$ and the results presented here demonstrate that the majority-vote model heterogeneous agents belongs to a different universality class than the nonequilibrium majority-vote models with homogeneous agents on square lattice.
The dynamics of opinion formation in a society is a complex phenomenon where many variables play an important role. Recently, the influence of algorithms to filter which content is fed to social networks users has come under scrutiny. Supposedly, the algorithms promote marketing strategies, but can also facilitate the formation of filters bubbles in which a user is most likely exposed to opinions that conform to their own. In the two-state majority-vote model an individual adopts an opinion contrary to the majority of its neighbors with probability $q$, defined as the noise parameter. Here, we introduce a visibility parameter $V$ in the dynamics of the majority-vote model, which equals the probability of an individual ignoring the opinion of each one of its neighbors. For $V=0.5$ each individual will, on average, ignore the opinion of half of its neighboring nodes. We employ Monte Carlo simulations to calculate the critical noise parameter as a function of the visibility $q_c(V)$ and obtain the phase diagram of the model. We find that the critical noise is an increasing function of the visibility parameter, such that a lower value of $V$ favors dissensus. Via finite-size scaling analysis we obtain the critical exponents of the model, which are visibility-independent, and show that the model belongs to the Ising universality class. We compare our results to the case of a network submitted to a static site dilution, and find that the limited visibility model is a more subtle way of inducing opinion polarization in a social network.
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 the paper, we use the heterogeneous mean-field theory to derive the rate equation for governing the models dynamics that can analytically determine the critical noise $f_c$ in the limit of infinite network size $Nrightarrow infty$. The result shows that $f_c$ depends on the ratio of ${leftlangle k rightrangle }$ to ${leftlangle k^{3/2} rightrangle }$, where ${leftlangle k rightrangle }$ and ${leftlangle k^{3/2} rightrangle }$ 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 $Nrightarrow infty$. All the theoretical results are confirmed by performing the extensive Monte Carlo simulations in random $k$-regular networks, Erdos-Renyi random networks and scale-free networks.
We consider two consensus formation models coupled to Barabasi-Albert networks, namely the Majority Vote model and Biswas-Chatterjee-Sen model. Recent works point to a non-universal behavior of the Majority Vote model, where the critical exponents have a dependence on the connectivity while the effective dimension $D_mathrm{eff} = 2beta/ u + gamma/ u$ of the lattice is unity. We considered a generalization of the scaling relations in order to include logarithmic corrections. We obtained the leading critical exponent ratios $1/ u$, $beta/ u$, and $gamma/ u$ by finite size scaling data collapses, as well as the logarithmic correction pseudo-exponents $widehat{lambda}$, $widehat{beta}+betawidehat{lambda}$, and $widehat{gamma}-gammawidehat{lambda}$. By comparing the scaling behaviors of the Majority Vote and Biswas-Chatterjee-Sen models, we argue that the exponents of Majority Vote model, in fact, are universal. Therefore, they do not depend on network connectivity. In addition, the critical exponents and the universality class are the same of Biswas-Chatterjee-Sen model, as seen for periodic and random graphs. However, the Majority Vote model has logarithmic corrections on its scaling properties, while Biswas-Chatterjee-Sen model follows usual scaling relations without logarithmic corrections.
On ($3,12^2$), ($4,6,12$) and ($4,8^2$) Archimedean lattices, the critical properties of majority-vote model are considered and studied using the Glauber transition rate proposed by Kwak {it et all.} [Phys. Rev. E, {bf 75}, 061110 (2007)] rather than the traditional majority-vote with noise [Jose Mario de Oliveira, J. Stat. Phys. {bf 66}, 273 (1992)]. The critical temperature and the critical exponents for this transition rate are obtained from extensive Monte Carlo simulations and with a finite size scaling analysis. The calculated values of the critical temperatures Binder cumulant are $T_c=0.363(2)$ and $U_4^*=0.577(4)$; $T_c=0.651(3)$ and $U_4^*=0.612(5)$; and $T_c=0.667(2)$ and $U_4^*=0.613(5)$ for ($3,12^2$), ($4,6,12$) and ($4,8^2$) lattices, respectively. The critical exponents $beta/ u$, $gamma/ u$ and $1/ u$ for this model are $0.237(6)$, $0.73(10)$, and $ 0.83(5)$; $0.105(8)$, $1.28(11)$, and $1.16(5)$; $0.113(2)$, $1.60(4)$, and $0.84(6)$ for ($3,12^2$), ($4,6,12$) and ($4,8^2$) lattices, respectively. These results differ from the usual Ising model results and the majority-vote model on so-far studied regular lattices or complex networks.
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