Opinion formation models help us understand how macro-level behaviors like consensus or polarization emerge from micro-level interactions. Deffuant-Weisbuch (DW) model explores the impact of homophily, trusting similar beliefs more. This is modeled by allowing interactions to occur only between individuals whose opinions differ by less than a “confidence bound” parameter. Depending on the value of the confidence bound, the model can lead to consensus, polarization, or opinion fragmentation. Mean-field equations have been developed recently to approximate the dynamics of the DW model. These equations can describe both fully-mixed populations and the case where individuals interact only along the edges of a network.

In this work, we carry out a mathematical analysis of the mean-field equation of DW model to investigate the role of the confidence bound and boundaries on these important observables of the model. We study the extension of the Deffuant-Weisbuch model where interactions occur within a configuration-type network composed of several degree classes. 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 major 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.