On the exit probability of the one dimensional q-voter model. Analytical results and simulations for large networks

Abstract

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 analyticaly in the limit where the network is infinetely large. We argue that the result $E(ρ) = \frac{ρ^q}{ρ^q + (1-ρ)^q}$, that was found in 3 previous works (2008 EPL 82 18006 and 2008 EPL 82 18007, for the case $q=2$ and 2011 PRE 84 031117, for $q>2$) using small networks (around $10^3$ sites), is a good approximation, but there are noticeable deviations for larger system sizes. We also show that, under some simple and intuitive hypothesis, the exit probability must obey the inequality, $\frac{ρ^q}{ρ^q + (1-ρ)} \leq E(ρ) \leq \fracρ{ρ+ (1-ρ)^q}$, in the infinite size limit. We believe this settles in the negative the suggestion made (2011 EPL 95 48005) that this result would be a finite size effect, with the exit probability actualy 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.