A Smooth Bounded Confidence Model Maintaining Clustering Phenomenon

In this paper, we propose a stochastic bounded confidence model. At every time slot, each individual receives an external social signal which is the average of opinions of its neighbors. Then agents compare the signals with their own personal biases which are defined as the initial values that ones hold. With positive probability, the agents either accept the opinion or persuade themselves with the help of personal prejudices. The probability of acceptance is reversely proportional to the opinion discrepancy between the signal and the bias. The model is modified as a continuous opinions discrete actions (CODA) model and thus is a Markov chain taking values on a finite state space. It is verified that the chain is aperiodic and finally converges in distribution to some invariant measure. The classification of states shows that the influences of distant opinions will boost consensus while the presence of personal biases promote clustering. The model also combines DeGroot model with Friedkin-Johnson model as well, by using a bounded confidence framework.