The behavior of one-dimensional Hegselmann-Krause (HK) dynamics driven by noise has been extensively studied. Previous research has indicated that within no matter the bounded or the unbounded space of one dimension, the HK dynamics attain quasi-synchronization (synchronization in noisy case) in finite time. However, it remains unclear whether this phenomenon holds in high-dimensional space. This paper investigates the random time for quasi-synchronization of multi-dimensional HK model and reveals that the boundedness and dimensions of the space determine different outcomes. To be specific, if the space is bounded, quasi-synchronization can be attained almost surely for all dimensions within a finite time, whereas in unbounded space, quasi-synchronization can only be achieved in low-dimensional cases (one and two). Furthermore, different integrability of the random time of various cases is proved.

Publication:

AUTOMATICA

http://dx.doi.org/10.1016/j.automatica.2026.113168

Author:

Wei Su

School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China

E-mail addresses: su.wei@bjtu.edu.cn

Meiru Jiang

School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China

E-mail addresses: 23121692@bjtu.edu.cn

Yongguang Yu

School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China

E-mail addresses: ygyu@bjtu.edu.cn

Ge Chen

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, & University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China

Corresponding author.

E-mail addresses:chenge@amss.ac.cn