  Recently, an opinion dynamics model has been proposed to describe a network
of individuals discussing a set of logically interdependent topics. For each
individual, the set of topics and the logical interdependencies between the
topics (captured by a logic matrix) form a belief system. We investigate the
role the logic matrix and its structure plays in determining the final
opinions, including existence of the limiting opinions, of a strongly connected
network of individuals. We provide a set of results that, given a set of
individuals' belief systems, allow a systematic determination of which topics
will reach a consensus, and which topics will disagreement in arise. For
irreducible logic matrices, each topic reaches a consensus. For reducible logic
matrices, which indicates a cascade interdependence relationship, conditions
are given on whether a topic will reach a consensus or not. It turns out that
heterogeneity among the individuals' logic matrices, including especially
differences in the signs of the off-diagonal entries, can be a key determining
factor. This paper thus attributes, for the first time, a strong diversity of
limiting opinions to heterogeneity of belief systems in influence networks, in
addition to the more typical explanation that strong diversity arises from
individual stubbornness.

Social and Information Networks;Multiagent Systems;Systems and Control;Dynamical Systems