  We introduce a three-state model to study the effects of a neutral party on
opinion spreading, in which the tendency of agents to agree with their
neighbors can be tuned to favor either the neutral party or two oppositely
polarized parties, and can be disrupted by social agitation mimicked as
temperature. We study the equilibrium phase diagram and the non-equilibrium
stochastic dynamics of the model with various analytical approaches and with
Monte Carlo simulations on different substrates: the fully-connected (FC)
graph, the one-dimensional (1D) chain, and Erd\"os-R\'enyi (ER) random graphs.
We show that, in the mean-field approximation, the phase boundary between the
disordered and polarized phases is characterized by a tricritical point. On the
FC graph, in the absence of social agitation, kinetic barriers prevent the
system from reaching optimal consensus. On the 1D chain, the main result is
that the dynamics is governed by the growth of opinion clusters. Finally, for
the ER ensemble a phase transition analogous to that of the FC graph takes
place, but now the system is able to reach optimal consensus at low
temperatures, except when the average connectivity is low, in which case
dynamical traps arise from local frozen configurations.
