Opinion separation in leader–follower coopetitive social networks.

Opinion dynamics on social networks describe the time evolution of the opinions within the leader–follower coopetitive (cooperative-competitive) interactions. When does opinion formation result in opinion separation with the single leader? This paper investigates a continuous-time model of such opinion separation problems with signed networks. Our model combines the positive/negative weighting rule of Degroot to describe the impact of their neighbors and the reflected leader–follower mechanism of Jadbabaie to describe trust/distrust relationships from followers to the leader. When the leader is stationary, we establish a necessary and sufficient condition, which ensure all opinions finally converge to bipartite consensus as long as the network topology is structurally balanced. When the leader is dynamic, a sufficient condition is proposed to ensure the opinions of followers move with the opinion of leader or the opposite value in the same velocity. Subsequently, we introduce Laplacian schemes, the framework of signed networks and matrix analysis to develop the stability of networked systems. Numerical simulation examples are given to illustrate the performance of the proposed leader–follower coopetitive systems. [ABSTRACT FROM AUTHOR]