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 \(E(\rho) = \frac{\rho^q}{\rho^q + (1-\rho)^q}\), 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, \(\frac{\rho^q}{\rho^q + (1-\rho)} \leq E(\rho) \leq \frac{\rho}{\rho + (1-\rho)^q}\), 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.