  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.
