Opinion dynamics in social networks with stubborn agents: An issue-based perspective

Classic models on opinion dynamics usually focus on a group of agents forming their opinions interactively over a single issue. Yet generally agreement cannot be achieved over a single issue when agents are not completely open to interpersonal influence. In this paper, opinion formation in social networks with stubborn agents is considered over issue sequences. The social network with stubborn agents is described by the Friedkin-Johnsen (F-J) model where agents are stubborn to their initial opinions. Firstly, we propose a sufficient and necessary condition in terms of network topology for convergence of the F-J model over a single issue. Secondly, opinion formation of the F-J model is investigated over issue sequences. Our analysis establishes connections between the interpersonal influence network and the network describing the relationship of agents' initial opinions for successive issues. Taking advantage of these connections, we derive the sufficient and necessary condition for the F-J model to achieve opinion consensus and form clusters over issue sequences, respectively. Finally, we consider a more general scenario where each agent has bounded confidence in forming its initial opinion. By analyzing the evolution of agents' ultimate opinions for each issue over issue sequences, we prove that the connectivity of the state-dependent network is preserved in this setting. Then the conditions for agents to achieve opinion consensus over issue sequences are established. Simulation examples are provided to illustrate the effectiveness of our theoretical results.