Diffusive majority-vote model

J. R. S. Lima, F. W. S. Lima, T. F. A. Alves, G. A. Alves, and A. Macedo-Filho

Phys. Rev. E 105, 034105 – Published 3 March 2022

Abstract

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.

Received 12 September 2021
Accepted 17 February 2022

DOI:https://doi.org/10.1103/PhysRevE.105.034105

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics

Nonlinear DynamicsInterdisciplinary PhysicsStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

J. R. S. Lima¹, F. W. S. Lima ¹, T. F. A. Alves¹, G. A. Alves², and A. Macedo-Filho ²

¹Departamento de Física, Universidade Federal do Piauí, 57072-970, Teresina - PI, Brazil
²Departamento de Física, Universidade Estadual do Piauí, 64002-150, Teresina - PI, Brazil

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Vol. 105, Iss. 3 — March 2022

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Images

Figure 1
Results of the averages given in Eq. (6) for the DMV model on a square lattice, with periodic boundary conditions, diffusion probabilities $D_{+} = D_{-} = 0.5$ , and noise parameter $q = 0.1$ . In (a), (c), and (e), we show our numerical data for the Binder cumulant $U$ , the order parameter $M$ , and susceptibility $χ$ for different lattice sizes. In (b), (d), and (f) we show the respective data collapses following Eq. (7), and the critical exponents given in Table 1 for two dimensions (2D). The estimated value of the critical threshold for the square lattice is $ρ_{c} = 3.862 (5)$ . Error bars are smaller than symbols and are not shown.
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Figure 2
The same as Fig. 1 for the simple cubic lattice with periodic boundary conditions. The estimated value of the critical threshold for the simple cubic lattice is $ρ_{c} = 2.692 (5)$ . The data collapses follow Eq. (7), and the critical exponents given in Table 1 for three dimensions (3D). Error bars are smaller than symbols and are not shown.
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Figure 3
Snapshots of DMV dynamics on a square lattice with periodic boundary conditions, with diffusion probabilities $D_{+} = D_{-} = 0.5$ , and noise parameter $q = 0.1$ . A node is black if most of its spins have the value 1, red for the opposite, and white for no local majority. We simulated the dynamics for the value of the susceptibility maxima $ρ = 3.81 (5)$ . Note the presence of clusters, where walkers with the same opinion state tend to stay closer.
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Figure 4
We show the phase diagrams for the DMV model's constant noise parameter $q = 0.1$ on the square and cubic lattices in (a) and (b), respectively. The critical threshold increases when decreasing the diffusion probability $D_{+} = D_{-} = D$ and presents an approximately linear dependence in the slow diffusion (small $D$ ) regime. The curve gives the critical thresholds $ρ_{c}$ as functions of $1 / D$ for the continuous phase transitions between the paramagnetic phase (shown in yellow) to the ferromagnetic phase (shown in blue). In the blue region (ferromagnetic phase), for a constant $ρ > ρ_{c}$ , the system undergoes a discontinuous phase transition as a function of $D_{+} - D_{-}$ , analogous to the Ising model isotherms.
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Figure 5
In (a), we show magnetizations for the DMV model on a square lattice of size $L = 100$ with periodic boundary conditions. We consider the case $D_{+} = 0.5$ , $D_{-} = 0.5 + h$ , and noise parameter $q = 0.1$ , where the external field is identified as $h = | D_{+} - D_{-} |$ . In (b), we show the respective data collapse following the FSS relation given in Eq. (8), the critical threshold $ρ_{c} = 3.862 (5)$ , and the critical isotherm exponent $δ$ for 2D, given in Table 1. In (c) and (d), we show the same of (a) and (b), respectively, for a cubic lattice of size $L = 24$ with periodic boundary conditions where the critical threshold is $ρ_{c} = 2.692 (5)$ . Error bars are smaller than the symbols and are not shown.
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Figure 6
In (a), we show two histograms of the time series of $m$ for the stationary evolution of DMV on a square lattice with $D_{+} = 0.55$ , and $D_{-} = 0.5$ (black circles), and with $D_{+} = 0.5$ , and $D_{-} = 0.55$ (red circles). For both data sets, the noise parameter is $q = 0.1$ , $L = 100$ , and the concentration is $ρ = 1$ . In (b), we have the same parameters as (a) except the concentration, which is $ρ = 1.5$ . In (c), we show the histogram of a time series for a concentration $ρ = 4 > ρ_{c}$ . Note that the sign of the most probable value of $m$ follows one of the most diffusing particles. The most probable value of $m$ increases with the concentration, which can be seen from the spacing between the maxima of the histograms in (a) and (b). Eventually, for $ρ > ρ_{c}$ , the system undergoes a discontinuous phase transition on the external field $h$ , where the histogram presents a metastable state and a bimodal shape.
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