Fixation results for the two-feature Axelrod model with a variable number of opinions

Abstract

The Axelrod model is a spatial stochastic model for the dynamics of cultures that includes two key social mechanisms: homophily and social influence, respectively defined as the tendency of individuals to interact more frequently with individuals who are more similar and the tendency of individuals to become more similar when they interact. The original model assumes that individuals are located on the vertex set of an interaction network and are characterized by their culture, a vector of opinions about $F$ cultural features, each of which offering the same number $q$ of alternatives. Pairs of neighbors interact at a rate proportional to the number of cultural features for which they agree, which results in one more agreement between the two neighbors. In this article, we study a more general and more realistic version of the standard Axelrod model that allows for a variable number of opinions across cultural features, say $q_i$ possible alternatives for the $i$th cultural feature. Our main result shows that the one-dimensional system with two cultural features fixates when $q_1 + q_2 \geq 6$.