Properties of Quasi-synchronization Time of High-dimensional Hegselmann-Krause Dynamics

The Hegselmann-Krause (HK) model was first introduced in the field of opinion dynamics to describe the opinion evolution of individuals who interact with others and whose opinions are influenced by those of the people around them [1]. In the HK model, the individuals update their opinions over time by taking the average of the opinions of all their neighbors whose opinions are close enough to their own. This closeness is determined by a bounded confidence threshold, such that agents influence each other's opinion only if their opinions lay within the confidence threshold. Though initially proposed in the context of opinion dynamics, the HK model captures a fundamental self-organizing mechanism in complex systems. Beyond its original application, it has also been adopted as a basic game learning algorithm [2, 3] and has found widespread use in diverse fields, including demand response programs in smart grids [4] and hybrid energy storage management [5]. Among the many properties, one of the interesting features of the model is that it can be synchronized by random noise. This phenomenon, also known as "noise-induced order" in self-organizing systems, was first found in some simulation studies [6-8]. Then the analysis of the phenomenon was considered based on some noisy HK-type models.