Nonlinear $q$-voter model involving nonconformity on networks

The order-disorder phase transition is a fascinating phenomenon in opinion dynamics models within sociophysics. This transition emerges due to noise parameters, interpreted as social behaviors such as anticonformity and independence (nonconformity) in a social context. In this study, we examine the impact of nonconformist behaviors on the macroscopic states of the system. Both anticonformity and independence are parameterized by a probability $p$, with the model implemented on a complete graph and a scale-free network. Furthermore, we introduce a skepticism parameter $s$, which quantifies a voter’s propensity for nonconformity. Our analytical and simulation results reveal that the model exhibits continuous and discontinuous phase transitions for nonzero values of $s$ at specific values of $q$. We estimate the critical exponents using finite-size scaling analysis to classify the model’s universality. The findings suggest that the model on the complete graph and the scale-free network share the same universality class as the mean-field Ising model. Additionally, we explore the scaling behavior associated with variations in $s$ and assess the influence of $p$ and $s$ on the system’s opinion dynamics.