Opinion dynamics analysis for stubborn individuals in cooperation–competition networks based on path-dependence framework

The opinion evolution problem is studied in this paper for stubborn individuals in cooperation–competition networks, where the individuals’ opinion dynamics is described by the Friedkin–Johnsen model and the competitive relationship between individuals is characterized by negative weights. Then the lifting approach is successfully applied to the Friedkin–Johnsen model to deal with the influence of the negative weights while the path-dependence theory is used to the transition of two arbitrarily adjacent topics. Starting from the weight topology, the relationship between the weight topology and the augmented topology is established, and some sufficient conditions about achieving neutrality, bipartite consensus and convergence for the individuals’ opinions in a sequence of topics are obtained. Interestingly, a necessary and sufficient condition is given to ensure bipartite consensus for the individuals’ opinions under the premise that the weight topology is structurally unbalanced. Furthermore, the Friedkin–Johnsen model with dynamic stubbornness is also considered, and the concept of common topic subsequence is introduced. It is proved that all elements of the topic transfer matrix are positive for the common topic subsequence, and neutrality and bipartite consensus for the individuals’ opinions can be achieved for different types of weight topologies, respectively. Finally, numerical examples are given to support the correctness of the theoretical results.