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

2025-07-11

AI Summary

This paper investigates the finite-time quasi-synchronization time of the high-dimensional Hegselmann–Krause (HK) model under stochastic noise, focusing on the impact of spatial boundedness and dimensionality on quasi-synchronization capability. Using stochastic process analysis, multidimensional dynamical systems theory, and probabilistic convergence methods, the integrability of the quasi-synchronization time is rigorously characterized. The main contributions are threefold: (1) It is proven that, in any bounded domain of arbitrary dimension, the system achieves quasi-synchronization in finite time almost surely; (2) In unbounded domains, finite-time quasi-synchronization is guaranteed only in one and two dimensions, while it fails in higher dimensions; (3) A fine-grained classification of synchronization time integrability across spatial regimes is established for the first time: exponential integrability in bounded domains, polynomial integrability in low-dimensional unbounded domains, and non-integrability in high-dimensional unbounded domains. These results unify and deepen the understanding of convergence mechanisms in high-dimensional social consensus dynamics.

Technology Category

• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction
• Reasoning under Uncertainty: Stochastic Optimization
• Application Domains: Humanities & Computational Social Science

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Security and Privacy: Large-scale security measurements

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.

Problem

• Investigates quasi-synchronization time in high-dimensional HK dynamics
• Determines if bounded or unbounded space affects synchronization outcomes
• Proves integrability of random time for different dimensional cases

Innovation

• Investigates high-dimensional HK model quasi-synchronization time
• Proves bounded space ensures finite-time synchronization
• Shows unbounded space limits synchronization to low dimensions