Noise Leads to Quasi-Consensus of Hegselmann-Krause Opinion Dynamics

This paper aims at providing the theoretical analysis to investigate consensus behavior of opinion dynamics in noisy environments. In fact, the well-known opinion dynamics, the Hegselmann-Krause (HK) model, demonstrates various agreement or disagreement opinion behaviors. Here we strictly prove an interesting conclusion that the random noise can almost surely "synchronize" the opinions in finite time for the HK model. To be specific, when all the opinions are affected by noises, we prove that the noisy opinions merge as a quasi-consensus in finite time and moreover, we give a critical noise strength to achieve the quasi-consensus. While only part of opinions are noise-infected, we prove that all the opinions will be "synchronized" even when the noise strength is very weak. In other words, the fragmentation phenomenon of the HK dynamics eventually vanishes in noisy environment.