Phase transition and universality of the three-one spin interaction based on the majority-rule model

In this work, we study the opinion dynamics of majority-rule model on a complete graph with additional social behavior namely anticonformity. We consider four spins with three-one interaction; three spins persuade the fourth spin in the population. We perform analytical and numerical calculations to find the critical behavior of the system. From both, we obtained the agreement results, e.g. the system undergoes a second-order phase transition and the critical point of the system only depends on the population number. In addition, the critical point decays exponentially as the number population increases. For the infinite population, the obtained critical point is 1/6, which agrees well with that of the previous work. We also obtained the critical exponents beta approximate to 0.5,gamma approximate to 1, and nu approximate to 2 of the model, thus, the model is in the same universality class with the mean-field Ising.