Hierarchical Fuzzy Opinion Networks: Top–Down for Social Organizations and Bottom–Up for Election

A fuzzy opinion is a Gaussian fuzzy set with the center representing the opinion and the standard deviation representing the uncertainty about the opinion, and a fuzzy opinion network is a connection of a number of fuzzy opinions in a structured way. In this article, we propose a top-down hierarchical fuzzy opinion network to model how the opinion of a top leader is penetrated into the members in social organizations, and a bottom-up fuzzy opinion network to model how the opinions of a large number of agents are agglomerated layer-by-layer into a consensus or a few opinions in the social processes, such as an election. For the top-down hierarchical fuzzy opinion network, we prove that the opinions of all the agents converge to the leader's opinion but the uncertainties of the agents in different groups are generally converge to different values. We demonstrate that the speed of convergence is greatly improved by organizing the agents in a hierarchical structure of small groups. For the bottom-up hierarchical fuzzy opinion network, we simulate how a wide spectrum of opinions are negotiating and summarizing with each other, in a layer-by-layer fashion in some typical situations.