Ordering-disordering dynamics of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -voter model under random external bias

We investigate a variant of the two-state q-voter model in which agents update their states under a random external field (which points upward with probability s and downward with probability 1-s) with probability p or adopt the unanimous opinion of q randomly selected neighbors with probability 1-p. Using mean-field analysis and Monte Carlo simulations, we identify an order-disorder transition at p_{c} when s=1/2. Notably, in the regime of p>p_{c}, we estimate the time for systems to reach disordered state from consensus state and find the logarithmic scaling T_{dis}∼BlnN, with B=1/(2p) for q=1, while for q>1, B depends on both p>p_{c} and q. We observe that disordering dynamics slow down significantly for nonlinear strengths q between 2 and 3, independent of the probability p. However, when s=0 or s=1, the system is bound to reach consensus, with the consensus time scaling logarithmically with system size as T_{con}∼BlnN, where B=1/p for q=1 and B=1 for q>1. Furthermore, in the limit of p=0, we derive a closed-form exit probability valid for arbitrary values of q and demonstrate a finite-size scaling collapse. These results clarify how external cues and peer conformity jointly control ordering and disordering in binary opinion dynamics.