Social Power Evolution of Multiple DeGroot Individuals With Centralized Media

In this article, the social power evolution problem is investigated for a social network with a centralized media and multiple DeGroot individuals, where the centralized media impacts the evolution of individuals’ opinions through the centralization parameter at the broadcast moment, while the network topology among the individuals is influenced by the relative interaction matrix. Then, the convergence of the corresponding opinion dynamics on the time scale is derived by discussing three distinct initial social powers. By integrating the reflected appraisal mechanism, a social power evolution model with the centralized media is established, which is essentially a nonlinear mapping. Based on the Jacobian matrix of this nonlinear mapping, it is proved that both the social powers of the centralized media and DeGroot individuals can converge provided that the centralization parameter exceeds a certain threshold; furthermore, a lower bound for the centralized media’s final social power is estimated, which helps to demonstrate that the centralized media possesses the greatest social power within the whole network. Additionally, concerning individuals’ final social powers, all individuals are first divided into three categories, and a sufficient condition is presented to ensure that the balanced individual has the greatest social power except for the centralized media. Finally, the obtained results are illustrated by a numerical example.