Analyzing contrarian behavior using nonlinear biased q -voter model

We investigate the role of contrarians in a recently proposed weighted-influence variant of the q-voter model. In this framework, nonunanimous influence groups affect the focal agent through weighted contributions governed by a bias parameter p. We extend this setting by introducing a fraction α (α>0) of contrarians, defined as agents who systematically oppose the prevailing influence irrespective of whether the group is unanimous or divided. Analytical mean-field calculations and Monte Carlo simulations reveal that the final states of the system are governed by simple phase boundaries: regions of positive and negative majority separated by the lines p=1/2 and α=1/2, with equally-mixed states confined to these boundaries. While low contrarian densities are insufficient to overturn the bias, higher values of α systematically drive the system closer to a balanced coexistence of opinions, though exact parity is prevented by the presence of bias p. We further analyze the temporal relaxation of opinions and extract the characteristic time scales of convergence. Our findings highlight how contrarians, acting as structured nonconformists, can suppress consensus and maintain opinion diversity, while internal biases ultimately hinder a perfectly even split.