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This paper is concerned with the dynamics evolution of opinions in thepresence of both cooperations and antagonisms. The class of Laplacian flows isaddressed through signed digraphs subject to switching topologies. Further, astate transition matrix-based approach is developed for the analysis of opiniondynamics, regardless of any assumptions on connectivity, structural balance ordigon sign-symmetry of signed digraphs. It is shown that based on theseparation of cooperations and antagonisms, a relationship can be bridgedbetween opinion dynamics under signed digraphs and under conventional digraphs.This helps to solve convergence problems for opinion dynamics. In particular,bipartite consensus (or stability) emerges if and only if the associatedswitching signed digraph is simultaneously structurally balanced (orunbalanced), which generalizes the use of structural balance theory in opiniondynamics to the case study of changing network topologies.

A State Transition Matrix-Based Approach to Separation of Cooperations and Antagonisms in Opinion Dynamics