Bifurcation analysis of Friedkin–Johnsen and Hegselmann–Krause models with a nonlinear interaction potential

Opinion dynamics of a group of individuals is the change in the members’ opinions through mutual interaction with each other. The related literature contains works in which the dynamics is modeled as a continuous system, of which behavioral patterns are analyzed in regard to the parameters contained in the system. These models are constructed by the assumption that the individuals are interdependent. Besides, the decisions of the individuals are only affected by two forces: self-bias force and group influence force. In this work, a nonlinear dynamical system which models the evolution of the decision of a group under the existence of a leader is considered. The two well-known opinion evolutions Friedkin–Johnsen and Hegselmann–Krause models are incorporated with a nonlinear exponentially decaying interaction potential to analyze the dynamics. The model considered in this work, which reflects simultaneously the effect of a leader and a nonlinear potential in the group dynamics, is analyzed the first time in the literature. Through this mechanism, the aim of the study is to understand the transitions between the final stable situations of the system which can be agreement, majority rule or disagreement. Bifurcation analysis of the system is performed to obtain stability results on the system. It is shown that one of these transition mechanisms is an imperfect pitchfork bifurcation. The study successfully produces boundary curves of the different regions in the parameter space that separates the stable states. The results of the work well present the distinctions between these final situations analytically, in case of N=3 agents plus a leader. The group mechanism considered is promising in the sense of having possible further modeling opportunities. For a large N number of people, clusters of opinions can be studied numerically, for analyzing how a leader can affect a big group of agents. The same can be tried for a big community with several leaders, which can represent the dynamics of electoral processes. The basic construction in this work will be a starting point for such further analyses.