Exit probability of the one-dimensional q-voter model: analytical results and simulations for large networks

We discuss the exit probability of the one-dimensional q-voter model and present tools to obtain estimates about this probability, both through simulations in large networks (around 10(7) sites) and analytically in the limit where the network is infinitely large. We argue that the result E(ρ) = ρ(q)/ρ(q) + (1-ρ)(q), that was found in three previous works [F. Slanina, K. Sznajd-Weron, and P. Przybyła, Europhys. Lett. 82, 18006 (2008); R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008), for the case q = 2; and P. Przybyła, K. Sznajd-Weron, and M. Tabiszewski, Phys. Rev. E 84, 031117 (2011), for q > 2] using small networks (around 10(3) sites), is a good approximation, but there are noticeable deviations that appear even for small systems and that do not disappear when the system size is increased (with the notable exception of the case q = 2). We also show that, under some simple and intuitive hypotheses, the exit probability must obey the inequality ρ(q)/ρ(q) + (1-ρ) ≤ E(ρ) ≤ ρ/ρ + (1-ρ)(q) in the infinite size limit. We believe this settles in the negative the suggestion made [S. Galam and A. C. R. Martins, Europhys. Lett. 95, 48005 (2001)] that this result would be a finite size effect, with the exit probability actually being a step function. We also show how the result that the exit probability cannot be a step function can be reconciled with the Galam unified frame, which was also a source of controversy.