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

We investigate the external field effect on opinion formation based on the majority rule and q-voter models on a complete graph. The external field can be considered as the mass media in the social system, with the probability p 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 m=1 for all values of p, indicating the existence of a usual phase transition for all values of p. In the q-voter model, the mass media eliminates the usual phase transition at p approximate to 0.21. 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 tau similar to N-beta, where N is the population size and beta depends on the probability p. In the majority rule model, beta follows a linear relation, and in the q-voter model, beta follows a power-law relation.