The external field effect on the opinion formation based on the majority rule and the <i>q</i>-voter models on the complete graph

Abstract

We investigate the external field effect on opinion formation based on the majority rule and [Formula: see text]-voter models on a complete graph. The external field can be considered as the mass media in the social system, with the probability [Formula: see text] agents following the mass media opinion. Based on our Monte Carlo simulation, the mass media effect is not strong enough to make the system reach a homogeneous state (complete consensus) with the magnetization [Formula: see text] for all values of [Formula: see text], indicating the existence of a usual phase transition for all values of [Formula: see text]. In the [Formula: see text]-voter model, the mass media eliminates the usual phase transition at [Formula: see text]. We obtain the model’s critical point and scaling parameters using the finite-size scaling analysis and obtain that both models have the same scaling parameters. The external field effect decreases both models’ relaxation time and the relaxation time following the power-law relation such as [Formula: see text], where [Formula: see text] is the population size and [Formula: see text] depends on the probability [Formula: see text]. In the majority rule model, [Formula: see text] follows a linear relation, and in the q-voter model, [Formula: see text] follows a power-law relation.