Stability Analysis of a Nonlinear Opinion Dynamics Model for Biased Assimilation

Abstract

This paper studies the stability of equilibria of a nonlinear opinion dynamics model proposed in [1] for biased assimilation, which generalizes the DeGroot model by introducing a bias parameter. When the bias parameter is zero, the model reduces to the original DeGroot model. A positive value of this parameter reflects the degree of how biased an agent is. The opinions of the agents lie between 0 and 1. When the bias parameter is positive, it is shown that the equilibria with all elements equal identically to the extreme value 0 or 1 is locally stable, while the equilibrium with all elements equal to the intermediate consensus value $\frac{1}{2}$ is unstable. For the equilibrium consisting of both extreme values 0 and 1, which corresponds to opinion polarization according to the model, it is shown that the equilibrium is locally stable if the bias parameter is greater than one for two-island networks, becomes unstable if the bias parameter is less than one, and its stability heavily depends on the network topology when the parameter equals one, in which case the limiting behavior of the model is established for certain initial conditions. It is also shown that for a small negative bias parameter, with which the agents can be regarded as anti-biased, the equilibrium with all elements equal to $\frac{1}{2}$ is locally stable.