A novel analytical formulation of the Axelrod model

TL;DR

This paper introduces a new analytical mean field approach to the Axelrod model for cultural dissemination, providing insights into phase transitions and system behavior on complete graphs.

Contribution

It develops novel master equations for F=2 and F=3, offering a detailed analytical framework that aligns with simulations and explains state transitions.

Findings

Analytical master equations match simulation results.

Transition driven by initial conditions and system size.

Scaling relation accounts for missing links during dynamics.

Abstract

The Axelrod model of cultural dissemination has been widely studied in the field of statistical mechanics. The traditional version of this agent-based model is to assign a cultural vector of $F$ components to each agent, where each component can take one of $Q$ cultural trait. In this work, we introduce a novel set of mean field master equations to describe the model for $F=2$ and $F=3$ in complete graphs where all indirect interactions are explicitly calculated. We find that the transition between different macroscopic states is driven by initial conditions (set by parameter $Q$) and the size of the system $N$, who measures the balance between linear and cubic terms in master equations. We also find that this analytical approach fully agrees with simulations where the system does not break up during the dynamics and a scaling relation related to missing links reestablishes the agreement when this happens.