Disagreement of hierarchical opinion dynamics with changing antagonisms

For opinion dynamics in social networks with antagonisms, distributed processes lead to challenging difficulties to characterize them, especially under changing network topologies. There may appear polarization, or neutralization, or fluctuation due to the simultaneous existence of cooperative and antagonistic interactions. In this paper, a leader-follower hierarchical framework is first established for opinion dynamics under the representation of signed digraphs and then a class of disagreement behaviors is identified by introducing a new rule to distinguish leaders and followers. It is shown that for opinion dynamics under switching signed digraphs, the agents in the leader layer always converge, whereas the agents in the follower layer generally may not converge but fluctuate within a symmetric interval spanned by the converged opinions of all leaders. Furthermore, the convergence of leaders is divided into clusters, and for each cluster, the opinions of leaders are ensured to polarize or neutralize, which are correspondingly tied to the structural balance or unbalance of the associated switching signed digraph with the cluster. Illustrative examples are presented to demonstrate the disagreement behaviors observed for opinion dynamics.