  We analyze the critical behavior of a class of discrete opinion models in the
presence of disorder. Within this class, each agent opinion takes a discrete
value ($\pm 1$ or 0) and its time evolution is ruled by two terms, one
representing agent-agent interactions and the other the degree of conviction or
persuasion (a self-interaction). The mean-field limit, where each agent can
interact evenly with any other, is considered. Disorder is introduced in the
strength of both interactions, with either quenched or annealed random
variables. With probability $p$ (1-$p$), a pairwise interaction reflects a
negative (positive) coupling, while the degree of conviction also follows a
binary probability distribution (two different discrete probability
distributions are considered). Numerical simulations show that a
non-equilibrium continuous phase transition, from a disordered state to a state
with a prevailing opinion, occurs at a critical point $p_{c}$ that depends on
the distribution of the convictions, the transition being spoiled in some
cases. We also show how the critical line, for each model, is affected by the
update scheme (either parallel or sequential) as well as by the kind of
disorder (either quenched or annealed).

Physics and Society;Statistical Mechanics;Social and Information Networks