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 \(p\), indicates that 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 \(p \approx 0.21\). 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 \(\tau \sim N^{\beta}\), 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.