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

TL;DR

This paper explores how the quasi-synchronization time of high-dimensional Hegselmann-Krause dynamics varies with space boundedness and dimension, revealing finite-time convergence in bounded spaces and limitations in unbounded spaces.

Contribution

It extends the understanding of HK dynamics by analyzing the quasi-synchronization time in high-dimensional spaces, highlighting the impact of space boundedness and dimension.

Findings

Quasi-synchronization occurs almost surely in bounded spaces within finite time.

In unbounded spaces, quasi-synchronization is only achievable in low dimensions (1D and 2D).

Different integrability properties of the random synchronization time are established.

Abstract

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.