Polar Opinion Dynamics in Social Networks

For decades, scientists have studied opinion formation in social networks, where information travels via word of mouth. The particularly interesting case is when polar opinions-Democrats versus Republicans or iOS versus Android-compete in the network. The central problem is to design and analyze a model that captures how polar opinions evolve in the real world. In this paper, we propose a general nonlinear model of polar opinion dynamics, rooted in several theories of sociology and social psychology. The model's key distinguishing trait is that unlike in the existing linear models, such as DeGroot and Friedkin-Johnsen models, an individual's susceptibility to persuasion is a function of his or her current opinion. For example, a person holding a neutral opinion may be rather malleable, while “extremists” may be strongly committed to their current beliefs. We also study three specializations of our general model, whose susceptibility functions correspond to different sociopsychological theories. We provide a comprehensive theoretical analysis of our nonlinear models' behavior using several tools from nonsmooth analysis of dynamical systems. To study convergence, we use nonsmooth max-min Lyapunov functions together with the generalized Invariance Principle. For our general model, we derive a general sufficient condition for the convergence to consensus. For the specialized models, we provide a full theoretical analysis of their convergence-whether to consensus or disagreement. Our results are rather general and easily apply to the analysis of other nonlinear models defined over directed networks, with Lyapunov functions constructed out of convex components.