Opinion dynamics of random-walking agents on a lattice

The opinion dynamics of random-walking agents on finite two-dimensional lattices is studied. In the model, the opinion is continuous, and both the lattice and the opinion can be either periodic or nonperiodic. At each time step, all agents move randomly on the lattice, and update their opinions based on those of neighbors with whom the differences of opinion are not greater than a given threshold. Due to the effect of repeated averaging, opinions first converge locally, and eventually reach steady states. As in other models with bounded confidence, steady states in general are those with one or more opinion groups in which all agents have the same opinion. When both the lattice and the opinion are periodic, however, metastable states can emerge, in which the whole spectrum of location-dependent opinions can coexist. This result shows that, when a set of continuous opinions forms a structure like a circle, unlike the typically used linear opinions, rich dynamic behavior can arise. When there are geographical restrictions in real situations, a complete consensus is rarely reached, and metastable states here might be one of the explanations for these situations, especially when opinions are not linear.