Noise-induced synchronization of Hegselmann-Krause dynamics in full space

TL;DR

This paper investigates how noise can induce synchronization in the Hegselmann-Krause model when the system's state space is unbounded, extending existing theory beyond bounded cases.

Contribution

It develops a new theoretical framework for noise-induced synchronization in unbounded state spaces using topological properties and stopping time theory.

Findings

Established the conditions for noise-induced synchronization in unbounded spaces.

Provided a rigorous mathematical interpretation of randomness-driven synchronization.

Extended the theoretical understanding of self-organizing systems beyond bounded state assumptions.

Abstract

The Hegselmann-Krause (HK) model is a typical self-organizing system with local rule dynamics. In spite of its widespread use and numerous extensions, the underlying theory of its synchronization induced by noise still needs to be developed. In its original formulation, as a model first proposed to address opinion dynamics, its state-space was assumed to be bounded, and the theoretical analysis of noise-induced synchronization for this particular situation has been well established. However, when system states are allowed to exist in an unbounded space, mathematical difficulties arise whose theoretical analysis becomes non-trivial and is as such still lacking. In this paper, we completely resolve this problem by exploring the topological properties of HK dynamics and by employing the theory of independent stopping time. The associated result in full statespace provides a solid interpretation of the randomness-induced synchronization of self-organizing systems