Ordering-disordering dynamics of the $q$-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 $s=\frac{1}{2}$. 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 $T_{\text{dis}}\sim \mathcal{B}\ln N$, with $\mathcal{B}=1\mathrm{/}(2p)$ for $q=1$, while for $q>1$, $\mathcal{B}$ 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 $p$. On the other hand, when $s=0$ or $s=1$, the system is bound to reach consensus, with the consensus time scaling logarithmically with system size as $T_{\text{con}}\sim \mathcal{B}\ln N$, where $\mathcal{B}=1\mathrm{/}p$ for $q=1$ and $\mathcal{B}=1$ 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.