Convergence of Opinion Dynamics With Heterogeneous Asymmetric Saturation Levels and Time-Varying Delays

Saturation and delay constraints, which are inevitable in reality, cause severe uncertainty to the behaviors of network dynamics. Accordingly, this paper studies the combined saturation and delay effect in the convergence of opinion dynamics over a switching cooperative-antagonistic network with weak connectivity. In particular, the considered saturations are in a generic form that the saturation level of each agent is asymmetric and the saturation levels of different agents are non-identical. And the delays of different channels are considered heterogeneous and time-varying. It is first demonstrated that the typical bipartite consensus result associated with the opinion dynamics is of benign tolerance to the saturation and delay constraints given a uniformly jointly strongly connected and structurally balanced topology condition. Next, by relaxing the uniformly jointly strongly connected condition just to roots, it is shown that the bipartite consensus is still ensured by the structural balance given a uniform joint reachability condition for non-roots. Besides, if the structural balance of the non-roots is absent, their opinions are shown to be ultimately bounded by the polarized values of the roots. Finally, two examples are simulated to validate the built theoretical results.