  Recently the theory of noise-induced synchronization of Hegselmann-Krause
(HK) dynamics has been well developed. As a typical opinion dynamics of bounded
confidence, the HK model obeys a synchronous updating rule, i.e., \emph{all}
agents check and update their opinions at each time point. However, whether
asynchronous bounded confidence models, including the famous Deffuant-Weisbuch
(DW) model, can be synchronized by noise have not been theoretically proved. In
this paper, we propose a generalized bounded confidence model which possesses a
stochastic asynchronous rule. The model takes the DW model and the HK model as
special cases and can significantly generalize the bounded confidence models to
practical application. We discover that the asynchronous model possesses a
different noise-based synchronization behavior compared to the synchronous HK
model. Generally, the HK dynamics can achieve quasi-synchronization
\emph{almost surely} under the drive of noise. For the asynchronous dynamics,
we prove that the model can achieve quasi-synchronization \emph{in mean}, which
is a new type of quasi-synchronization weaker than the "almost surely" sense.
The results unify the theory of noise-induced synchronization of bounded
confidence opinion dynamics and hence proves the noise-induced synchronization
of DW model theoretically for the first time. Moreover, the results provide a
theoretical foundation for developing noise-based control strategy of more
complex social opinion systems with stochastic asynchronous rules.
