Inflexibility and independence: Phase transitions in the majority-rule model

In this work we study opinion formation in a population participating in a public debate with two distinct choices. We consider three distinct mechanisms of social interactions and individuals' behavior: conformity, nonconformity, and inflexibility. The conformity is ruled by the majority-rule dynamics, whereas the nonconformity is introduced in the population as an independent behavior, implying the failure of attempted group influence. Finally, the inflexible agents are introduced in the population with a given density. These individuals present a singular behavior, in a way that their stubbornness makes them reluctant to change their opinions. We consider these effects separately and all together, with the aim to analyze the critical behavior of the system. We perform numerical simulations in some lattice structures and for distinct population sizes. Our results suggest that the different formulations of the model undergo order-disorder phase transitions in the same universality class as the Ising model. Some of our results are complemented by analytical calculations.