On the termination time of the one-sided asymmetric Hegselmann-Krause dynamics

In this paper, we provide a novel upper bound for the termination time of the one-dimensional asymmetric Hegselmann-Krause dynamics, when the asymmetry is one-sided. In addition to the number of agents, our upper bound depends on the ratio of asymmetry and the confidence range in the opinions of agents, and recovers the known O(n³) results of the symmetric case. Our proof technique relies on a novel Lyapunov-like function, which measures the spread of the opinion profile. As a by-product, we fully characterize the switching pattern in the opinions of the agents.