Consensus in the two-state Axelrod model

TL;DR

This paper proves that in a one-dimensional two-state Axelrod model with any number of features, the system tends to consensus, confirming previous numerical conjectures about its clustering behavior.

Contribution

It provides a rigorous proof of the conjecture that the two-state Axelrod model clusters in one dimension for any number of features.

Findings

The model clusters when the number of features exceeds the number of states per feature.

The proof confirms the conjecture for the two-state case with arbitrary features.

The results enhance understanding of cultural consensus dynamics in social influence models.

Abstract

The Axelrod model is a spatial stochastic model for the dynamics of cultures which, similarly to the voter model, includes social influence, but differs from the latter by also accounting for another social factor called homophily, the tendency to interact more frequently with individuals who are more similar. Each individual is characterized by its opinions about a finite number of cultural features, each of which can assume the same finite number of states. Pairs of adjacent individuals interact at a rate equal to the fraction of features they have in common, thus modeling homophily, which results in the interacting pair having one more cultural feature in common, thus modeling social influence. It has been conjectured based on numerical simulations that the one-dimensional Axelrod model clusters when the number of features exceeds the number of states per feature. In this article, we prove this conjecture for the two-state model with an arbitrary number of features.