We propose and study a simple model for the evolution of political opinion through a population. The model includes a nonlinear term that causes individuals with more extreme views to be less receptive to external influence. Such a term was suggested in 1981 by Cobb in the context of a scalar-valued diffusion equation, and recent empirical studies support this modeling assumption. Here, we use the same philosophy in a network-based model. This allows us to incorporate the pattern of pairwise social interactions present in the population. We show that the model can admit two distinct stable steady states. This bi-stability property is seen to support polarization and can also make the long-term behavior of the system extremely sensitive to the initial conditions and to the precise connectivity structure. Computational results are given to illustrate these effects.