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On Graphs with Bounded and Unbounded Convergence Times in Social Hegselmann-Krause Dynamics

We address the problem of identifying physical connectivity graphs that guarantee a finite upper bound on the time required for the associated social Hegselmann-Krause dynamics to ϵ-converge to the steady state. We handle the cases of consensus as well as non-consensus steady states, and for each case, we provide sufficient conditions for a physical connectivity graph to have unbounded ϵ-convergence time. We then show that every complete r-partite graph on n vertices has a finite maximum ϵ-convergence time, regardless of the values of r and n. Finally, we show that enhancing the connectivity of agents may not always speed up convergence to the steady state, even when the steady state is a consensus.