Structural balance theory has been developed in sociology and psychology toexplain how interacting agents, e.g., countries, political parties, opinionatedindividuals, with mixed trust and mistrust relationships evolve into polarizedcamps. Recent results have shown that structural balance is necessary forpolarization in networks with fixed, strongly connected neighbor relationshipswhen the opinion dynamics are described by DeGroot-type averaging rules. Wedevelop this line of research in this paper in two steps. First, we considerfixed, not necessarily strongly connected, neighbor relationships. It is shownthat if the network includes a strongly connected subnetwork containingmistrust, which influences the rest of the network, then no opinion clusteringis possible when that subnetwork is not structurally balanced; all the opinionsbecome neutralized in the end. In contrast, it is shown that when thatsubnetwork is indeed structurally balanced, the agents of the subnetwork evolveinto two polarized camps and the opinions of all other agents in the networkspread between these two polarized opinions. Second, we consider time-varyingneighbor relationships. We show that the opinion separation criteria carry overif the conditions for fixed graphs are extended to joint graphs. The resultsare developed for both discrete-time and continuous-time models.
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