  Structural balance theory has been developed in sociology and psychology to
explain how interacting agents, e.g., countries, political parties, opinionated
individuals, with mixed trust and mistrust relationships evolve into polarized
camps. Recent results have shown that structural balance is necessary for
polarization in networks with fixed, strongly connected neighbor relationships
when the opinion dynamics are described by DeGroot-type averaging rules. We
develop this line of research in this paper in two steps. First, we consider
fixed, not necessarily strongly connected, neighbor relationships. It is shown
that if the network includes a strongly connected subnetwork containing
mistrust, which influences the rest of the network, then no opinion clustering
is possible when that subnetwork is not structurally balanced; all the opinions
become neutralized in the end. In contrast, it is shown that when that
subnetwork is indeed structurally balanced, the agents of the subnetwork evolve
into two polarized camps and the opinions of all other agents in the network
spread between these two polarized opinions. Second, we consider time-varying
neighbor relationships. We show that the opinion separation criteria carry over
if the conditions for fixed graphs are extended to joint graphs. The results
are developed for both discrete-time and continuous-time models.
