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We study a nonlinear bounded-confidence model (BCM) of continuous-timeopinion dynamics on networks with both persuadable individuals and zealots. Themodel is parameterized by a scalar $\gamma$, which controls the steepness of asmooth influence function. This influence function encodes the relative weightsthat nodes place on the opinions of other nodes. When $\gamma = 0$, thisinfluence function recovers Taylor's averaging model; when $\gamma \rightarrow\infty$, the influence function converges to that of a modifiedHegselmann--Krause (HK) BCM. Unlike the classical HK model, however, oursigmoidal bounded-confidence model (SBCM) is smooth for any finite $\gamma$. Weshow that the set of steady states of our SBCM is qualitatively similar to thatof the Taylor model when $\gamma$ is small and that the set of steady statesapproaches a subset of the set of steady states of a modified HK model as$\gamma \rightarrow \infty$. For several special graph topologies, we giveanalytical descriptions of important features of the space of steady states. Anotable result is a closed-form relationship between the stability of apolarized state and the graph topology in a simple model of echo chambers insocial networks. Because the influence function of our BCM is smooth, we areable to study it with linear stability analysis, which is difficult to employwith the usual discontinuous influence functions in BCMs.

Emergence of polarization in a sigmoidal bounded-confidence model of opinion dynamics