Limiting behavior for a general class of voter models with confidence threshold

This article is concerned with a general class of stochastic spatial models for the dynamics of opinions. Like in the one-dimensional voter model, individuals are located on the integers and update their opinion at a constant rate based on the opinion of their neighbors. However, unlike in the voter model, the set of opinions is represented by the set of vertices of a finite connected graph that we call the opinion graph: when an individual interacts with a neighbor, she imitates this neighbor if and only if the distance between their opinions, defined as the graph distance induced by the opinion graph, does not exceed a certain confidence threshold. Our first result shows that, when the confidence threshold is at least equal to the radius of the opinion graph, the process fluctuates and clusters. We also establish a general sufficient condition for fixation of the process based on the structure of the opinion graph, which we then significantly improve for opinion graphs which are distance-regular. Our general results are used to understand the dynamics of the system for various examples of opinion graphs: paths and stars, which are not distance-regular, and cycles, hypercubes and the five Platonic solids, which are distance-regular.