Asynchronous Expressed and Private Opinion Dynamics on Influence Networks

In this article, we consider an asynchronous discrete-time opinion dynamics model for a social influence network, which consists of a population of interacting individuals, each holding private and expressed opinions. At each time instant, a single individual activates and updates both his or her expressed and private opinions simultaneously. While an individual's new private opinion is shaped by his or her private opinion, the expressed opinions of his or her neighbors, and attachment to an existing prejudice, his or her expressed opinion is instead a tradeoff between his or her true private opinion and a "public opinion," representing the current norm or standard opinion from the perspective of the individual. We show that the expressed and private opinions of all individuals converge to the same value exponentially fast when no individual holds an existing prejudice, with the lower bound on the convergence speed based on a general assumption on the network topology. If there are individuals who hold an existing prejudice, then the final opinions of every individual lie between the minimum and maximum values of those constant prejudices. Our results are also extended to consider the case where two or more individuals activate and update their opinions at each time instant. Simulations are provided to illustrate the results.