Symmetric preferences, asymmetric outcomes: Tipping dynamics in an open-city segregation model

TL;DR

This work presents a non-equilibrium, open-city segregation model where agent mobility is governed by a segregating chemical reaction network on a Moore-neighborhood lattice. Using mean-field theory, second-order moment closures, Gillespie simulations, and finite-size scaling, the authors identify a continuous tipping transition at a critical in-group rate and show that, while the transition exhibits Ising-like symmetry breaking, its full set of critical exponents does not match any known equilibrium universality class. The study highlights the role of vacancies and open-system dynamics in shaping segregation patterns and establishes a framework linking social segregation to non-equilibrium statistical physics. These results offer a data-friendly, stochastic alternative to utility-based models and point to rich directions for incorporating network structure and density fluctuations in social dynamics.

Abstract

Schelling's model of segregation demonstrates that even in the absence of social or governmental interventions, individuals with mild in-group preferences can self-organize into strongly segregated neighborhoods. Many variants of this celebrated model have been proposed by assuming agents tend to increase their satisfaction. Complementary to this traditional, utility-based approach, we model residential moves using satisfaction-independent reaction rates in a spatially extended chemical reaction network. The resulting model exhibits a counter-intuitive phenomenon: despite symmetric in-group preferences, the system undergoes a tipping transition at a critical preference level, beyond which one agent type dominates. We characterize this asymmetric phase transition in details using mean-field analysis, numerical simulations and finite size scaling methods. We find that while the transition shares key features with the Ising universality class, such as $\mathbb{Z}_2$ symmetry breaking and similar exponent ratios, the full set of critical exponents does not match any known universality class.

Figures (10)

• Figure 1: Overview of the reactions of our segregation model. Each lattice site can be in three possible states, corresponding to either an empty house or a resident of type $A$ or $B$. The reaction rules allow for random creation and annihilation of residents (moving in/out of the region) or pairwise reactions grouped into move-in (birth) and move-out (death) reactions. Both are subdivided further into those which promote like neighbors (in-group, or 'copy') and reactions that promote opposing neighbors (out-group, or 'split'). Some exemplary grid configurations are shown on a 2D lattice with $100\times 100$ sites. Increasing the in-group reaction rate $r_c$ gradually leads to a segregating transition, where regions of both red and blue types are equally large. Further increasing $r_c$ will lead to larger regions of one type dominating over smaller islands of the opposing type.
• Figure 2: The densities in the stable steady states for $T = \mu /\beta = 1$ and $r_s = 0.5$, as a function of $r_c$, obtained from the second order (full lines) and first order (dashed lines) mean-field approximations. We show the density of vacant sites (purple), of agents of type $A$ (red) and of agents of type $B$ (blue). The scatter data corresponds to averages of $10^4$ independent instances of the stochastic model, with grid-size $N = 50 \times 50$. The mean-field analysis shows a clear pitchfork bifurcation at some value of $r_c$ (here signaled with a black star), where the solution goes from the symmetric to the asymmetric phase. The mean field solutions agree with the stochastic model away from $r_c^*$ (here given by the grid-line $r_c \approx 3.8$). Note that there is also an asymmetric solution for which the densities of $A$ and $B$ are flipped with respect to the densities shown here.
• Figure 3: Schematic phase diagram of our model in the $(r_s, r_c)$-plane. The symmetric and asymmetric phases are separated by a tipping transition at $r_c^*$, while a cross-over transition from anti-segregating to segregating neighborhoods takes place when $r_c = r_s$. The inlay shows typical configurations for each phase, obtained from simulations of the model using Gillespie's algorithm.
• Figure 4: Snapshots of steady state solutions of the stochastic model obtained by varying $T = \mu/\beta$ and $r_c$ independently while fixing $r_s$ (here $r_s = 0.5$). Note that this plot is merely meant for schematic purposes, the axes are not to scale.
• Figure 5: The Binder cumulant for the magnetization characterizes the continuous phase transition to the asymmetric (ferromagnetic) phase where agents of one type become the dominant group in the region. The critical value, shown here for $r_s =0$, is determined at the crossing point for the Binder cumulants as $r_c^* = 0.7874(2)$.