  The Hegselmann-Krause (HK) model is a wellknown opinion dynamics, attracting
a significant amount of interest from a number of fields. However, the
heterogeneous HK model is difficult to analyze - even the most basic property
of convergence is still open to prove. For the first time, this paper takes
into consideration heterogeneous HK models with environment or communication
noise. Under environment noise, it has been revealed that the heterogeneous HK
model with or without global information has a phase transition for the upper
limit of the maximum opinion difference, and has a critical noise amplitude
depending on the minimal confidence threshold for quasi-synchronization. In
addition, the convergence time to the quasi-synchronization is bounded by a
negative exponential distribution. The heterogeneous HK model with global
information and communication noise is also analyzed. Finally, for the basic HK
model with communication noise, we show that the heterogeneous case exhibits a
different behavior regarding quasi-synchronization from the homogenous case.
Interestingly, raising the confidence thresholds of constituent agents may
break quasi-synchronization. Our results reveal that the heterogeneity of
individuals is harmful to synchronization, which may be the reason why the
synchronization of opinions is hard to reach in reality, even within that of a
small group.
