Contrarian Voter Model under the Influence of an Oscillating Propaganda: Consensus, Bimodal Behavior and Stochastic Resonance

Abstract

We study the contrarian voter model for opinion formation in a society under the influence of an external oscillating propaganda and stochastic noise. Each agent of the population can hold one of two possible opinions on a given issue—against or in favor—and interacts with its neighbors following either an imitation dynamics (voter behavior) or an anti-alignment dynamics (contrarian behavior): each agent adopts the opinion of a random neighbor with a time-dependent probability p(t), or takes the opposite opinion with probability 1−p(t). The imitation probability p(t) is controlled by the social temperature T, and varies in time according to a periodic field that mimics the influence of an external propaganda, so that a voter is more prone to adopt an opinion aligned with the field. We simulate the model in complete graph and in lattices, and find that the system exhibits a rich variety of behaviors as T is varied: opinion consensus for T=0, a bimodal behavior for T<Tc, an oscillatory behavior where the mean opinion oscillates in time with the field for T>Tc, and full disorder for T≫1. The transition temperature Tc vanishes with the population size N as Tc≃2/lnN in complete graph. In addition, the distribution of residence times tr in the bimodal phase decays approximately as tr−3/2. Within the oscillatory regime, we find a stochastic resonance-like phenomenon at a given temperature T*. Furthermore, mean-field analytical results show that the opinion oscillations reach a maximum amplitude at an intermediate temperature, and that exhibit a lag with respect to the field that decreases with T.

Keywords: noise; opinion dynamics; periodic field; stochastic resonance; voter model.

Figure 1

(a) Time evolution of the magnetization m in a single realization of the dynamics for a system of $N=1000$ agents on a complete graph, subject to a periodic field $H\left(t\right)=H_{0}sin\left(\omega t\right)$ of amplitude $H_0=0.1$ and period $\tau =512$ ($\omega =2\pi /\tau$), and an external noise associated to a temperature T. Each curve corresponds to a different temperature, as indicated in the legend. Four behaviors are observed: $-1$ consensus for $T=0$, bimodal behavior for $T=0.25<T_c$, oscillations for $T=1.0>T_c$, and full disorder for $T=100\gg 1$. Here, $T_c\simeq 0.2895$ is the transition temperature. The field $H\left(t\right)$ is also plotted for reference. (b) Evolution of m in a single realization for the same parameters as in panel (a), on a $1D$-lattice of $N=8192$ sites (top-left), a $2D$ square lattice of size $N=64\times 64$ (top-right), a $2D$ triangular lattice of size $N=64\times 64$ (bottom-left), and a $2D$ hexagonal lattice of size $N=64\times 64$ (bottom-right).

Figure 2

(a) Histogram of m for the same system and parameters as those in Figure 1, and for the temperatures indicated in the legends. At the transition point $T_c\simeq 0.2895$, the distribution $P\left(m\right)$ is uniform (top-right). (b) Normalized histograms of the residence time $t_r$ for the same system and parameters of panel. (a) Each plot corresponds to a different temperature T, as indicated in the legends. The bottom-right plot is in linear-log scale, while the other plots are in double logarithmic scale. The dashed line in the top-left plot has slope $-3/2$. Each histogram was obtained by running a single realization up to a time ${10}^8$.

Figure 3

Signal-to-noise ratio $SNR$ vs. temperature T on the five interaction topologies indicated in the legend. The system sizes are $N=1024$ for the $1D$ lattice and $CG$, and $N=32\times 32$ for the $2D$ lattices (hexagonal, square and triangular). The amplitude of the external field is $H_0=0.1$. Each panel corresponds to a different period $\tau =128$, 256, 512 and 1024.

Figure 4

Top panels: signal-to-noise ratio $SNR$ as a function of the amplitude of the external field $H_0$, for temperatures $T=0.2,0.3,0.5,1.3$ and $2.3$. Bottom panels: $SNR$ vs. T for $H_0=0.6,0.7,0.8,0.9$ and $1.0$. The period of the field is $\tau =128$. Left, central and right panels correspond to simulations on a complete graph (CG) of $N=1024$ nodes, a $2D$ square lattice of $N=32\times 32$ sites and a $1D$ lattice of $N=1024$ sites, respectively.

Figure 5

Total response $\mathcal{R}$ vs. field amplitude $H_0$, for the topologies indicated in the legend. The period of the field is $\tau =128$, and the system sizes are $N=1024$ for $CG$ and $1D$ lattice, and $N=32\times 32$ for $2D$ lattices. Inset: $\mathcal{R}$ vs. $H_0$ on a log-log scale. The dashed line has slope 2.

Figure 6

(a) Time evolution of the average magnetization $\langle m\rangle$ for temperatures $T=0.2$ (squares), $T=0.5$ (circles), $T=1.0$ (diamonds), $T=2.0$ (up triangles) and $T=5.0$ (down triangles), on a CG of size $N=100$, with a field of amplitude $H_0=0.1$ and period $\tau =512$. The average was done over ${10}^5$ independent realizations of the dynamics. Solid lines are the analytical approximation from Equation (8). (b) Lag $\mathcal{L}$ respect to the period $\tau$ vs. temperature T for field periods $\tau =256$ (squares), $\tau =512$ (circles) and $\tau =1024$ (diamonds), and amplitude $H_0=0.1$, for the same topology of panel (a). Solid lines are the approximation from Equation (15). Inset: Amplitude A vs. T for the same parameter values as in the main panel. Solid lines are the analytical approximation from Equation (13).

Figure 7

(a) Time evolution of the average magnetization $\langle m\rangle$ over ${10}^5$ realizations of the dynamics on a CG of size $N=100$, with temperature $T=0.5$, under a field of period $\tau =512$ and amplitudes $H_0=0.02$ (squares), $H_0=0.1$ (circles) and $H_0=0.5$ (diamonds). The solid pink line corresponds to the numerical integration of Equations (A1) and (A2), while the other solid lines are the analytical approximation from Equation (8). (b) Normalized lag $\mathcal{L}/\tau$ vs. temperature T for a field of period $\tau =512$ and amplitudes $H_0$ indicated in the legend. The solid line is the approximation from Equation (15). Inset: A vs. T for the same parameter values as in the main panel. Solid lines are the approximation from Equation (13).