  This paper analyzes a nonlinear opinion dynamics model which generalizes the
DeGroot model by introducing a bias parameter for each individual. The original
DeGroot model is recovered when the bias parameter is equal to zero. The
magnitude of this parameter reflects an individual's degree of bias when
assimilating new opinions, and depending on the magnitude, an individual is
said to have weak, intermediate, and strong bias. The opinions of the
individuals lie between 0 and 1. It is shown that for strongly connected
networks, the equilibria with all elements equal identically to the extreme
value 0 or 1 is locally exponentially stable, while the equilibrium with all
elements equal to the neutral consensus value of 1/2 is unstable. Regions of
attraction for the extreme consensus equilibria are given. 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
unstable for all strongly connected networks if individuals all have weak bias,
becomes locally exponentially stable for complete and two-island networks if
individuals all have strong bias, and its stability heavily depends on the
network topology when individuals have intermediate bias. Analysis on star
graphs and simulations show that additional equilibria may exist where
individuals form clusters.
