On S1 as an alternative continuous opinion space in a three-party regime

In this work, we propose a discrete system to model the dynamics of individual opinions when the agents of a population have three equally likely choices. The social network consists of a finite number of agents with pairwise interactions at discrete times, and the opinion space is identified as a triangle in the plane. After a suitable homotopic transformation, one may convert the opinion space into the classical circle S1 of the Cartesian plane. The opinion of each agent is updated following a general nonlinear law which considers individual parameters of the members. We establish conditions that guarantee the existence of attracting points (or strong consensus), and infer the existence of attracting intervals (identified here as weak consensus). Moreover, we notice that the conditions that lead to global consensuses are independent of the weight matrix and the number of agents in the network. The simulations obtained in this work confirm the validity of the analytical results.