Effect of social behaviors in the opinion dynamics $q$-voter model

Order-disorder phase transition is one of the interesting features in the opinion dynamics model of sociophysics. This phenomenon is caused by noise parameters, which can be treated as social responses such as anticonformity and independence (nonconformity) in a social context. This paper explores the effect of nonconformity behavior on changes in the macroscopic state of the system. A probability $p$ represents anticonformity and independence, and the model is defined on the complete graph and the two-dimensional (2D) square lattice. We also introduce a skepticism parameter represented by $s$, which describes a voter's propensity to behave nonconformist. Our results analytically and numerically show that the model undergoes a continuous and discontinuous phase for all values of non-zero $s$ and at specific $q$-sized agent. No phase transition is observed in the model with independent voters on the 2D square lattice, and the model with anticonformist voters undergoes a continuous phase transition. We also analyze the effect of the probability $p$ and skepticism level $s$ on the outcome of the system's opinion.