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

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.

Figure 1

Mean fraction of sites in the largest domain $\rho$ as function of the Poisson parameter $q$ for the square lattice ($p=0$), small-world networks with $p=0.1$, and random networks ($p=1$). The different symbols represent different number of sites $N=L^2$, viz. $L=100$ ($\diamond$), $L=200$ ($\times$), and $L=400$ ($\square$). The error bars are smaller than the symbol sizes.

Figure 2

Mean fraction of sites in the largest domain $\rho$ in the critical region for the square lattice ($p=0$) with linear size $L=200$ ($\times$), $L=400$ ($\square$), $L=600$ ($▵$), $L=800$ ($▿$), and $L=1000$ ($◯$). The dotted vertical line indicates the location of $q_c^0\approx 3.1$ and the solid curve is the two-parameters fitting function $\rho =\mathcal{A}{\left(q_c^0-q\right)}^{\beta }$, with $\mathcal{A}=0.40\pm 0.01$ and $\beta =0.25\pm 0.02$. The error bars are smaller than the symbol sizes.

Figure 3

Log-log plot of $\rho$ against $q_c^0-q$ with $q_c^0=3.1$ for the square lattice ($p=0$) with $L=200$ ($\times$), $L=600$ ($▵$) and $L=1000$ ($◯$). The solid line is the fitting function $\rho =\mathcal{A}{\left(q_c^0-q\right)}^{\beta }$ with $\mathcal{A}=0.40\pm 0.01$, and $\beta =0.25\pm 0.02$.

Figure 4

Log-log plot of $\rho$ against the reciprocal of the linear lattice size at the critical point $q_c^0\approx 3.1$. The curve fitting the data is $\rho =\mathcal{B}{L}^{\beta /{\nu }^0}$ with $\mathcal{B}=0.371\pm 0.005$ and $\beta /{\nu }^0=0.117\pm 0.002$. The error bars are smaller than the symbol sizes.

Figure 5

Scaled order parameter against the scaled distance to the critical point for the square lattice ($p=0$) with linear size $L=200$ ($\times$), $L=400$ ($\square$), $L=600$ ($▵$), $L=800$ ($▿$), and $L=1000$ ($◯$). The error bars are smaller than the symbol sizes. The parameters are $q_c^0=3.10,\beta /{\nu }^0=0.117$, and ${\nu }^0=2.1$.

Figure 6

Order parameter $\rho$ against the reciprocal of $L$ for random networks and $q=27.0$ ($⧫$), 27.1 ($▾$), 27.15 ($◯$), 27.2 ($▴$), and 27.5 ($\blacksquare$). The critical point is $q_c^1=27.175\pm 0.125$. The error bars are smaller than the symbol sizes and the lines are guides to the eye.

Figure 7

Order parameter against the scaled distance to the critical point for random networks ($p=1$) with $L=600$ ($▵$), $L=800$ ($▿$), and $L=1000$ ($◯$). The error bars are smaller than the symbol sizes. The parameters are $q_c^1=27.175$, and ${\nu }^1=1.0$.

Figure 8

Order parameter against the scaled distance to the critical point for small-world networks with $p=0.1$ and $L=600$ ($▵$), $L=800$ ($▿$), and $L=1000$ ($◯$). The error bars are smaller than the symbol sizes. The parameters are $q_c^{0.1}=14.40$ and ${\nu }^{0.1}=1.0$.