Probability of consensus in spatial opinion models with confidence threshold

Abstract

This paper gives lower bounds for the probability of consensus for two spatially explicit stochastic opinion models. Both processes are characterized by two finite connected graphs, that we call respectively the spatial graph and the opinion graph. The former represents the social network describing how individuals interact, while the latter represents the topological structure of the opinion space. The representation of the opinions as a graph induces a distance between opinions which we use to measure disagreements. Individuals can only interact with their neighbors on the spatial graph, and each interaction results in a local change of opinion only if the two interacting individuals do not disagree too much, which is quantified using a confidence threshold. In the first model, called the imitation process, an update results in both neighbors having the exact same opinion, whereas in the second model, called the attraction process, an update results in the neighbors' opinions getting one unit closer. For both models, we derive a lower bound for the probability of consensus that holds for any finite connected spatial graph. For the imitation process, the lower bound for the probability of consensus also holds for any finite connected opinion graph, whereas for the attraction process, the lower bound only holds for a certain class of opinion graphs that includes finite integer lattices, regular trees and star-like graphs.