Noise leads to quasi-consensus of Hegselmann-Krause opinion dynamics

This study aims to provide a theoretical analysis for investigating the consensus behavior of opinion dynamics in noisy environments. We present how random noises significantly help “synchronize” opinions in the Hegselmann-Krause (HK) model. We prove that when noise strength is below the critical value, the opinions of the noisy HK dynamics tend to achieve “consensus” (referred to as quasi-consensus due to noise) in finite time. Fragmentation of HK dynamics tends to eventually vanish when persistent noise is present. Meanwhile, we verify that the opinions almost surely divide infinite times when noise strength exceeds the critical value.