  Mean-field equations have been developed recently to approximate the dynamics
of the Deffuant model of opinion formation. These equations can describe both
fully-mixed populations and the case where individuals interact only along
edges of a network. In each case, interactions only occur between individuals
whose opinions differ by less than a given parameter, called the confidence
bound. The size of the confidence bound parameter is known to strongly affect
both the dynamics and the number and location of opinion clusters. In this work
we carry out a mathematical analysis of the mean-field equations to investigate
the role of the confidence bound and boundaries on these important observables
of the model. We consider the limit in which the confidence bound interval is
small, and identify the key mechanisms driving opinion evolution. We show that
linear stability analysis can predict the number and location of opinion
clusters. Comparison with numerical simulations of the model illustrates that
the early-time dynamics and the final cluster locations can be accurately
approximated for networks composed of two degree classes, as well as for the
case of a fully-mixed population.
