The consensus in the two-feature two-state one-dimensional Axelrod model revisited

TL;DR

This paper revisits the one-dimensional Axelrod model with two features and states, clarifying the discrepancy between simulation results and analytical predictions regarding cultural diversity and consensus.

Contribution

It demonstrates that the apparent coexistence of multicultural states in simulations is due to the order of limits taken, reconciling simulation and analytical results.

Findings

Simulations show multicultural configurations due to limit order.

Analytical results confirm convergence to monocultural states.

Order of limits affects observed cultural diversity.

Abstract

The Axelrod model for the dissemination of culture exhibits a rich spatial distribution of cultural domains, which depends on the values of the two model parameters: $F$, the number of cultural features and $q$, the common number of states each feature can assume. In the one-dimensional model with $F=q=2$, which is closely related to the constrained voter model, Monte Carlo simulations indicate the existence of multicultural absorbing configurations in which at least one macroscopic domain coexist with a multitude of microscopic ones in the thermodynamic limit. However, rigorous analytical results for the infinite system starting from the configuration where all cultures are equally likely show convergence to only monocultural or consensus configurations. Here we show that this disagreement is due simply to the order that the time-asymptotic limit and the thermodynamic limit are taken in the simulations. In addition, we show how the consensus-only result can be derived using Monte Carlo simulations of finite chains.