Emergence of Polarization in a Sigmoidal Bounded-Confidence Model of Opinion Dynamics

We study a nonlinear bounded-confidence model (BCM) of continuous-time opinion dynamics on networks with both persuadable individuals and zealots. The model is parameterized by a nonnegative scalar ', which controls the steepness of a smooth influence function. This influence function encodes the relative weights that individuals place on the opinions of other individuals. When ' = 0, this influence function recovers Taylor's averaging model; when ' - co, the influence function converges to that of a modified Hegselmann-Krause (HK) BCM. Unlike the classical HK model, however, our sigmoidal bounded-confidence model (SBCM) is smooth for any finite '. We show that the set of steady states of our SBCM is qualitatively similar to that of the Taylor model when ' is small and that the set of steady states approaches a subset of the set of steady states of a modified HK model as ' - co. For certain special graph topologies, we give analytical descriptions of important features of the space of steady states. A notable result is a closed-form relationship between graph topology and the stability of polarized states in a simple special case that models echo chambers in social networks. Because the influence function of our BCM is smooth, we are able to study it with linear stability analysis, which is difficult to employ with the usual discontinuous influence functions in BCMs.