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 scalar γ, which controls the steepness of a
smooth influence function. This influence function encodes the relative weights
that nodes place on the opinions of other nodes. When γ=0, this
influence function recovers Taylor's averaging model; when γ→∞, 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 γ. We
show that the set of steady states of our SBCM is qualitatively similar to that
of the Taylor model when γ is small and that the set of steady states
approaches a subset of the set of steady states of a modified HK model as
γ→∞. For several 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 the stability of a
polarized state and the graph topology in a simple model of 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.Comment: 29 pages, 7 figure