Rebels Lead to the Doctrine of the Mean: A Heterogeneous DeGroot Model

Zhigang Cao, Fengliang Jiao, Xinglong Qu, Wen-Xu Wang, Mingmin Yang, Xiaoguang Yang, Boyu Zhang

Abstract: The DeGroot model is one of the most classical models in the field of opinion dynamics. The standard DeGroot model assumes that agents are homogeneous and update their opinions in the direction of a weighted average of their neighbors’ opinions. One natural question is whether a second type of agents could significantly change the main properties of the model. The authors address this question by introducing rebels, who update their opinions toward the opposite of their neighbors’ weighted average. The authors find that the existence of rebels remarkably affects the opinion dynamics. Under certain mild conditions, the existence of a few rebels will lead the group opinion to the golden mean, regardless of the initial opinions of the agents and the structure of the learning network. This result is completely different from that of the standard DeGroot model, where the final consensus opinion is determined by both the initial opinions and the learning topology. The study then provides new insights into understanding how heterogeneous individuals in a group reach consensus and why the golden mean is so common in human society.