Revisiting the non-equilibrium phase transitions of the continuous-trait Axelrod model

Abstract

We investigate the non-equilibrium phase transitions of the continuous-trait Axelrod model, an agent-based framework where individual culture is represented by a vector of $F$ continuous features confined to the interval $(0,1)$. Local interactions are governed by a metric similarity threshold $d$, which acts as a continuous control parameter of social tolerance. The dynamics inevitably freeze into one of two absorbing configuration classes: an ordered, homogeneous monocultural state at high tolerance, or a highly fragmented, disordered state at low tolerance. While previous studies characterized the transition as hybrid based on the continuous behavior of the domain density $\mu$ alongside a discontinuous jump in the largest domain fraction $\rho$, we show that this apparent continuity is an artifact of severe finite-size masking effects. By shifting the methodological focus to the scaling of the median $\tilde{\mu}$ and analyzing the full probability distributions $P(\mu)$, we unveil a clear bimodal structure with disjoint maxima across independent simulation runs. Our results reveal that for $F=2$, the system undergoes a genuinely hybrid transition in the contemporary sense, featuring a tiny but finite latent jump ($\mu_c \approx 0.089$) at the critical threshold $d_c \approx 0.0784$ while scaling toward it from below via a non-analytical power law with a mean-field exponent $\beta \approx 1/2$. Conversely, for $F=3$, the higher trait-space dimensionality suppresses local fluctuations, yielding a traditional, non-hybrid first-order transition. We apply this framework to the alternative discrete Poisson variant of the model, successfully confirming its known continuous transition for $F=2$ and discontinuous, non-hybrid transition for $F=3$, thereby establishing a unified characterization of phase transitions in Axelrod-like systems.