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

Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias

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 TTc 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.

Unanimity, Coexistence, and Rigidity: Three Sides of Polarization

Political polarization is perceived as a threat to democracies. Using the Galam model of opinion dynamics deployed in a five-dimensional parameter space, I show that polarization is the byproduct of an essential hallmark of a vibrant democratic society, namely open and informal discussions among agents. Indeed, within a homogeneous social community with floaters, the dynamics lead gradually toward unanimity (zero entropy). Polarization can eventually appear as the juxtaposition of non-mixing social groups sharing different prejudices about the issue at stake. On the other hand, the inclusion of contrarian agents produces a polarization within a community that mixes when their proportion x is beyond a critical value xc=16≈0.167 for discussing groups of size three and four. Similarly, the presence of stubborn agents also produces a polarization of a community that mixes when the proportion of stubborn agents is greater than some critical value. For equal proportions of stubborn agents a along each opinion, ac=29≈0.22 for group size four against ac=14=0.25 for group size three. However, the evaluation of the proportion of individual opinion shifts at the attractor 12 and indicates that the polarization produced by contrarians is fluid with a good deal of agents who keep shifting between the two opposed blocks (high entropy). That favors a coexistence of opposite opinions in a divided community. In contrast, the polarization created by stubborn agents is found to be frozen with very few individuals shifting opinion between the two opinions (low entropy). That yields a basis for the emergence of hate between the frozen opposed blocks.