From the DeGroot Model to the DeGroot-Non-Consensus Model: The Jump States and the Frozen Fragment States

Abstract

Non-consensus phenomena are widely observed in human society, but more attention is paid to consensus phenomena. One famous consensus model is the DeGroot model, and there are a series of outstanding works derived from it. By introducing the cognition bias, resulting in over-confidence and under-confidence in the DeGroot model, we propose a non-consensus model, namely the DeGroot-Non-Consensus model. It bridges consensus phenomena and non-consensus phenomena. While different in meaning, the new opinion model can reproduce the DeGroot model’s behaviors and supply a series of interesting non-consensus states. We find frozen fragment states for the over-confident population and time-dependent states for strong interaction strength. In frozen fragment states, the population is polarized into opinion clusters formed by extremists. In time-dependent states, agents jump between two opinions that only differ in the sign, which provides a possible explanation for the swing in opinions in elections and the fluctuations in open questions in the absence of external information. All of these states are summarized in the phase diagrams of the self-confidence and the interaction strength plane. Moreover, the transition scenarios along different parameter paths are studied. Meanwhile, the influence of the nodes’ degree is illustrated in the phase diagrams and the relationship is given. The finite size effect is found in the not quite over-confident population. An interesting phenomenon for small population sizes is that neutral populations with large opinion variance are robust to the fluctuations induced by a finite population size.