An analytical expression for the exit probability of the q-voter model in one dimension

We present in this paper an approximation that is able to give an analytical expression for the exit probability of the $q$-voter model in one dimension. This expression gives a better fit for the more recent data about simulations in large networks, and as such, departs from the expression $\frac{\rho^q}{\rho^q + (1-\rho)^q}$ found in papers that investigated small networks only. The approximation consists in assuming a large separation on the time scales at which active groups of agents convince inactive ones and the time taken in the competition between active groups. Some interesting findings are that for $q=2$ we still have $\frac{\rho^2}{\rho^2 + (1-\rho)^2}$ as the exit probability and for large values of $q$ the difference between the result and $\frac{\rho^q}{\rho^q + (1-\rho)^q}$ becomes negligible (the difference is maximum for $q=5$ and 6)