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.

中文翻译：

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基于路径依赖框架的合作竞争网络中顽固个体的意见动态分析


本文研究了合作竞争网络中顽固个体的观点演化问题，其中个体的观点动态由Friedkin-Johnsen模型描述，个体之间的竞争关系用负权重来表征。然后将提升方法成功应用于Friedkin-Johnsen模型以处理负权重的影响，同时将路径依赖理论应用于两个任意相邻主题的转换。从权重拓扑出发，建立了权重拓扑和增广拓扑之间的关系，得到了主题序列中个体意见实现中立、双方共识和收敛的一些充分条件。有趣的是，在权重拓扑结构不平衡的前提下，给出了保证双方意见达成共识的充要条件。此外，还考虑了具有动态顽固性的Friedkin-Johnsen模型，并引入了公共主题子序列的概念。证明了主题转移矩阵的所有元素对于公共主题子序列都是正的，对于不同类型的权重拓扑，可以分别实现个体意见的中立性和双方共识。最后给出数值算例来支持理论结果的正确性。