  This paper gives lower bounds for the probability of consensus for two
spatially explicit stochastic opinion models. Both processes are characterized
by two finite connected graphs, that we call respectively the spatial graph and
the opinion graph. The former represents the social network describing how
individuals interact, while the latter represents the topological structure of
the opinion space. The representation of the opinions as a graph induces a
distance between opinions which we use to measure disagreements. Individuals
can only interact with their neighbors on the spatial graph, and each
interaction results in a local change of opinion only if the two interacting
individuals do not disagree too much, which is quantified using a confidence
threshold. In the first model, called the imitation process, an update results
in both neighbors having the exact same opinion, whereas in the second model,
called the attraction process, an update results in the neighbors' opinions
getting one unit closer. For both models, we derive a lower bound for the
probability of consensus that holds for any finite connected spatial graph. For
the imitation process, the lower bound for the probability of consensus also
holds for any finite connected opinion graph, whereas for the attraction
process, the lower bound only holds for a certain class of opinion graphs that
includes finite integer lattices, regular trees and star-like graphs.
