One of the key challenges in modeling of social interactions in a complex multi-agent environment is the modeling of opinion dynamics and how agents influence each other’s opinion and how new technology and ideas diffuse through such a network. One of the models that addresses such dynamics is Hegselmann-Krause dynamics, which is a well accepted model for many engineering applications. Example of such applications include distributed rendezvous problem in a robotic network such as a network of space shuttles, where one may want to gather a set of robots which lack a central coordination to a common place.
Characterizing of the termination time of Hegselmann-Krause dynamics is a challenging problem, and is still unknown, even in the case of scalar dynamics. Although it is known that the termination time of the dynamics any dynamics with n agents is at least , the best known upper bound for the termination time is . In this work we introduce a new Lyapunov-like function and show that any dynamics arrives to the steady state in at most steps, which improves the state of the art by an order of n.