Expressed and Private Opinions Coevolution for Stubborn Individuals with Cooperation-Competition Interactions

In this paper, the opinion coevolution problem is investigated for multiple stubborn individuals within a cooperation-competition network. A new Expressed-Private-Opinion (EPO) model is first proposed, which involves the evolution of two opinions, namely, expressed opinion and private opinion. In order to overcome the adverse effects of competitive factors among individuals, the lifting approach is applied to this EPO model, resulting in an augmented dynamic system. Then, based on the network topology, the convergence of this system is analyzed in two different cases. Further, the relative error bound between the final expressed opinion and private opinion is established, which shows that the upper bound is determined by the individual's resilience coefficient. In addition, the EPO model for multiple DeGroot individuals is investigated, and it is proved that all expressed opinions and private opinions tend to zero regardless of the network topology of the augmented dynamic system. Finally, four numerical examples are presented to demonstrate the accuracy of the theoretical results. © 2025 IEEE.