Threshold $q$-voter model

We introduce the threshold $q$-voter opinion dynamics where an agent, facing a binary choice, can change its mind when at least $q_0$ amongst $q$ neighbors share the opposite opinion. Otherwise, the agent can still change its mind with a certain probability $\varepsilon$. This threshold dynamics contemplates the possibility of persuasion by an influence group even when there is not full agreement among its members. In fact, individuals can follow their peers not only when there is unanimity ($q_0=q$) in the lobby group, as assumed in the $q$-voter model, but, depending on the circumstances, also when there is simple majority ($q_0>q/2$), Byzantine consensus ($q_0>2q/3$), or any minimal number $q_0$ amongst $q$. This realistic threshold gives place to emerging collective states and phase transitions which are not observed in the standard $q$-voter. The threshold $q_0$, together with the stochasticity introduced by $\varepsilon$, yields a phenomenology that mimics as particular cases the $q$-voter with stochastic drivings such as nonconformity and independence. In particular, nonconsensus majority states are possible, as well as mixed phases. Continuous and discontinuous phase transitions can occur, but also transitions from fluctuating phases into absorbing states.