We introduce a non-linear variant of the voter model, the q-voter model, in
which q neighbors (with possible repetition) are consulted for a voter to
change opinion. If the q neighbors agree, the voter takes their opinion; if
they do not have an unanimous opinion, still a voter can flip its state with
probability Ï”. We solve the model on a fully connected network (i.e.
in mean-field) and compute the exit probability as well as the average time to
reach consensus. We analyze the results in the perspective of a recently
proposed Langevin equation aimed at describing generic phase transitions in
systems with two (Z2â symmetric) absorbing states. We find that in mean-field
the q-voter model exhibits a disordered phase for high Ï” and an
ordered one for low Ï” with three possible ways to go from one to the
other: (i) a unique (generalized voter-like) transition, (ii) a series of two
consecutive Ising-like and directed percolation transition, and (iii) a series
of two transitions, including an intermediate regime in which the final state
depends on initial conditions. This third (so far unexplored) scenario, in
which a new type of ordering dynamics emerges, is rationalized and found to be
specific of mean-field, i.e. fluctuations are explicitly shown to wash it out
in spatially extended systems.Comment: 9 pages, 7 figure
