Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias

Abstract

We study the Galam majority rule dynamics with contrarian behavior and an oscillating external propaganda in a population of agents that can adopt one of two possible opinions. In an iteration step, a random agent interacts with three other random agents and takes the majority opinion among the agents with probability p(t) (majority behavior) or the opposite opinion with probability 1-p(t) (contrarian behavior). The probability of following the majority rule p(t) varies with the temperature T and is coupled to a time-dependent oscillating field that mimics a mass media propaganda, in a way that agents are more likely to adopt the majority opinion when it is aligned with the sign of the field. We investigate the dynamics of this model on a complete graph and find various regimes as T is varied. A transition temperature Tc separates a bimodal oscillatory regime for T<Tc, where the population's mean opinion m oscillates around a positive or a negative value from a unimodal oscillatory regime for T>Tc in which m oscillates around zero. These regimes are characterized by the distribution of residence times that exhibit a unique peak for a resonance temperature T*, where the response of the system is maximum. An insight into these results is given by a mean-field approach, which also shows that T* and Tc are closely related.

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

Figure 1

(a) Time evolution of the average value of the absolute magnetization $\left|m\right|$ in a population of $N={10}^3$ agents, zero field $H=0$ and various values of majority probability $p={(1+e^{-2/T})}^{-1}$, as indicated in the legend. (b) Stationary value of $\langle \left|m\right|\rangle$ vs. temperature T for constant fields $H=0.0$ (circles), $H=0.1$ (squares) and $H=0.5$ (diamonds). The solid line is the analytical expression from Equation (11), while the dashed lines are the numerical integration of Equation (8). The averages were performed over ${10}^3$ independent realizations starting from a symmetric condition $m_0=0$.

Figure 2

Time evolution of the magnetization m in a single realization for a population of $N=1024$ agents under the influence of an oscillating field with period $\tau =256$ and amplitudes $H_0=0.1$ and $0.5$, panels (a) and (b), respectively, and the temperatures indicated in the legends. Solid lines correspond to MC simulations, while dashed lines in panel (a) represent the numerical integration of Equation (8).

Figure 3

Normalized histograms of the residence time $t_r$ in a system of $N=1025$ agents under a field of amplitude $H_0=0.1$, period $\tau =256$ (a) and $\tau =1024$ (b), and the temperatures indicated in the legends. The bottom-right panels are on a linear-log scale.

Figure 4

(a) Response $\mathcal{A}$ as a function of the temperature T for a field of amplitude $H_0=0.1$ and period $\tau$ indicated in the legend. (b) Resonance temperature $T^{*}$ [maximum of $\mathcal{A}$ vs. T curves from (a)] and transition temperature $T_c$ vs. period $\tau$.

Figure 5

(a) Time evolution of the magnetization m from Equation (8) for a field of amplitude $H_0=0.1$, period $\tau =256$ and the temperatures indicated in the legend. Horizontal dashed lines represent the time average value of m, $\overline{m}$, in the interval $t\in (0,1000\tau )$. (b) Time average of the magnetization, $\overline{m}$, vs. temperature T for the field’s periods indicated in the legend. The inset shows a closer look around the transition values $T_c$.