Non-equilibrium phase transition and cultural drift in the continuous-trait Axelrod model

TL;DR

The paper remedies limitations of the discrete Axelrod model by introducing a continuous-trait version on a 2D lattice, with traits in $[0,1]$ and a continuous tolerance $d$ as the control parameter. Using finite-size scaling on the absorbing-state dynamics under perfect copying, it locates a critical point at $d_c \approx 0.0784$ and finds a continuous transition in the mean domain density with $\beta=1/3$ and $\nu=2$, while the largest-domain fraction $\rho$ shows a discontinuous jump at the same $d_c$, highlighting a hybrid transition. Under imperfect copying ($\sigma>0$), persistent noise generates two symmetry-related attractors at $d$ and $1-d$ but these attractors are fragile in the thermodynamic limit, preventing true consensus and leading to sustained fragmentation. The results imply that true monoculture is unlikely in large, continuously evolving societies and establish a link between discrete and continuous cultural-dynamics frameworks, with implications for understanding persistence of diversity under cultural drift.

Abstract

The standard Axelrod model of cultural dissemination, based on discrete cultural traits, exhibits a non-equilibrium phase transition but is inherently limited by its inability to continuously probe the critical behavior. We address this limitation by introducing a generalized Axelrod model utilizing continuous cultural traits confined to the interval $[0,1]$, and a similarity threshold, $d$, that serves as a continuous control parameter representing cultural tolerance. This framework allows for a robust analysis of the model's critical properties and its dynamics under cultural drift (copying noise). For the perfect copying scenario, we precisely locate the critical threshold $d_c$, which separates the disordered (fragmented) and ordered (polarized) phases. Through Finite-Size Scaling, we find that the mean domain density vanishes continuously at $d_c$ with the exponent $β= 1/3$. Simultaneously, the largest domain fraction displays a surprising discontinuous jump at $d_c$. We find that the finite size effects in the critical region are governed by the exponent $ν=2$ for both the continuous and discontinuous transitions. Under imperfect copying, persistent noise introduces a powerful selective pressure on the trait space, leading to the emergence of two symmetry-related attractors at the trait values $d$ and $1-d$. However, these noise-induced attractors prove fragile in the thermodynamic limit, becoming unstable at large lattice sizes, which directly accounts for the observed failure of the dynamics to freeze under sustained cultural drift. This suggests that in large, continuously evolving societies, true cultural convergence is highly unlikely, leading instead to sustained fragmentation and nonstationary dynamics where cultural domains never fully stabilize.

Figures (9)

• Figure 1: Mean density of domains $\mu$ (left panel) and coefficient of variation CV (right panel) as a function of the threshold $d$ for lattices of linear sizes $L=100, 200, 400$, and $800$ in the perfect copying case ($\sigma=0$). The solid curve in the left panel is a two-parameter fit function to the data for $L=800$, using the function $\mu = A(d_c-d)^\beta$. The best-fit parameters are $A= 1.643$ and $\beta = 0.335$, with $d_c =0.0784$. The dashed vertical line in the right panel indicates the location of the critical point, $d_c \approx 0.0784$, which is determined by the intersection of the CV data for the lattice sizes, $L=200$ and $L=400$. The error bars are smaller than the symbol sizes.
• Figure 2: (Left panel) Log-log plot of the mean density of domains $\mu$ at the critical threshold $d_c=0.0784$ as a function of the reciprocal of the linear lattice size in the perfect copying case ($\sigma=0$). The curve fitting the data is $\mu =BL^{-\beta/\nu}$ with $B= 0.35 \pm 0.01$ and $\beta/ \nu = 0.167 \pm 0.006$. (Right panel) Scaled order parameter against the scaled distance to the critical threshold for lattices of linear sizes $L=100, 200, 400$, and $800$. The parameters used are $\beta=1/3$, $\nu=2$, $d_c = 0.0784$, and $\sigma=0$. The error bars are smaller than the symbol sizes.
• Figure 3: (Left panel) Mean fraction of agents in the largest domain $\rho$ as a function of the threshold $d$ in the critical region for lattices of linear sizes $L=100, 200, 400$, and $800$ in the perfect copying case ($\sigma=0$). (Right panel) $\rho$ against the scaled distance to the critical threshold for lattices of linear sizes $L=400$ and $800$. The parameters used for the data collapse are the critical exponent $\nu=2$ and the critical point $d_c = 0.0784$. The dashed vertical lines indicate the location of the critical point $d_c$. The error bars are smaller than the symbol sizes.
• Figure 4: Histogram of the cultural traits $x$ in the absorbing configurations for the perfect copying case ($\sigma=0$). The results are shown for the ordered phase for the two traits $k=1$ and $k=2$ in a single simulation run (left panel) and averaged over $100$ runs (right panel). The parameters used are $L=400$ and $d=0.1$.
• Figure 5: Mean fraction of agents in the largest domain $\rho$ as a function of the threshold $d$. The left panel shows $\rho$ for a fixed lattice size ($L=50$) across different noise levels ($\sigma=0, 0.01$, and $0.02$). The right panel shows $\rho$ for a fixed noise level ($\sigma=0.01$) across different lattice sizes ($L=25, 50$, and $100$).