Effect of long-range interactions on the phase transition of Axelrod's model

Abstract

Axelrod's model with $F=2$ cultural features, where each feature can assume $k$ states drawn from a Poisson distribution of parameter $q$, exhibits a continuous nonequilibrium phase transition in the square lattice. Here we use extensive Monte Carlo simulations and finite size scaling to study the critical behavior of the order parameter $\rho$, which is the fraction of sites that belong to the largest domain of an absorbing configuration averaged over many runs. We find that it vanishes as $\rho \sim {\left(q_c^0-q\right)}^{\beta }$ with $\beta \approx 0.25$ at the critical point $q_c^0\approx 3.10$ and that the exponent that measures the width of the critical region is ${\nu }^0\approx 2.1$. In addition, we find that introduction of long-range links by rewiring the nearest-neighbors links of the square lattice with probability $p$ turns the transition discontinuous, with the critical point $q_c^p$ increasing from $3.1$ to $27.17$, approximately, as $p$ increases from $0$ to $1$. The sharpness of the threshold, as measured by the exponent ${\nu }^p\approx 1$ for $p>0$, increases with the square root of the number of nodes of the resulting small-world network.

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• Erdös-Rényi phase transition in the Axelrod model on complete graphs
• Non-equilibrium phase transition and cultural drift in the continuous-trait Axelrod model