  This article is concerned with a general class of stochastic spatial models
for the dynamics of opinions. Like in the voter model, individuals are located
on the vertex set of a connected graph and update their opinion at a constant
rate based on the opinion of their neighbors. However, unlike in the voter
model, the set of opinions is represented by the set of vertices of another
connected graph that we call the opinion graph: when an individual interacts
with a neighbor, she imitates this neighbor if and only if the distance between
their opinions, defined as the graph distance induced by the opinion graph,
does not exceed a certain confidence threshold. When the confidence threshold
is at least equal to the radius of the opinion graph, we prove that the
one-dimensional process fluctuates and clusters and give a universal lower
bound for the probability of consensus of the process on finite connected
graphs. We also establish a general sufficient condition for fixation of the
infinite system based on the structure of the opinion graph, which we then
significantly improve for opinion graphs which are distance-regular. Our
general results are used to understand the dynamics of the system for various
examples of opinion graphs: paths and stars, which are not distance-regular,
and cycles, hypercubes and the five Platonic solids, which are
distance-regular.
