Expressed and Private Opinion Dynamics on Influence Networks with Asynchronous Updating

In this paper, an asynchronous discrete-time opinion dynamics model on a social influence network is considered. At each time instant, a single individual activates and updates two state variables simultaneously. The individual's new private opinion is a weighted average of her current private opinion, the expressed opinions of her neighbors, and a constant prejudice. Meanwhile, the individual's new expressed opinion is equal to her current private opinion, altered due to a pressure to conform to the public opinion as perceived by the individual, being the average expressed opinion among her neighbors. We analyze the system for social networks which are rooted, and show that if no individual holds a prejudice, then a mild assumption on the activation sequence of the individuals guarantees convergence. In particular, the expressed and private opinions of all individuals converge to the same value exponentially fast, with two lower bounds on convergence speeds based on two different assumptions on the network topology. Simulations are provided to illustrate the result, and provide support to the conjecture that the system dynamics may converge even if individuals hold an existing prejudice.