Hegselmann-Krause Dynamics: An Upper Bound on Termination Time

We discuss the Hegselmann-Krause model for opinion dynamics in discrete-time for a collection of homogeneous agents. Each agent opinion is modeled by a scalar variable, and the neighbors of an agent are defined by a symmetric confidence level. We provide an upper bound for the termination time of the dynamics by using a Lyapunov type approach. Specifically, through the use of a properly defined adjoint dynamics, we construct a Lyapunov comparison function that decreases along the trajectory of the opinion dynamics. Using this function, we develop a novel upper bound on the termination time that has order m in terms of the spread in the initial opinion profile.