[Paper Review] Urban and Scientific Segregation: The Schelling-Ising Model

Dietrich Stauffer, Christian Schulze|ArXiv.org|Oct 27, 2007

Opinion Dynamics and Social Influence4 references3 citations

TL;DR

This paper proposes a two-temperature Ising-like model to explain urban segregation as an emergent phenomenon from individual preferences, not external forces. By introducing noise and dynamic tolerance (temperature) adjustments, the model shows that small, non-zero noise enables the formation of large, stable ghettos—resolving the failure of the original Schelling model to produce infinite clusters.

ABSTRACT

Urban segregation of different communities, like blacks and whites in the USA, has been simulated by Ising-like models since Schelling 1971. This research was accompanied by a scientific segregation, with sociologists and physicists ignoring each other until 2000. We review recent progress and also present some new two-temperature multi-cultural simulations.

Motivation & Objective

• To resolve the scholarly segregation between sociology and physics in studying urban segregation.
• To investigate whether the original Schelling model can produce large-scale ghettos without external forces.
• To explore how noise and dynamic tolerance (temperature) influence the emergence of segregation in multi-cultural urban simulations.
• To demonstrate that small, non-zero noise is essential for achieving macroscopic segregation in Ising-like models.
• To bridge the gap between sociological models of residential segregation and statistical physics approaches.

Proposed method

• Adapts the Schelling-Ising model using a two-temperature framework: one temperature $T_1$ for individual tolerance and another $T_2$ for external noise.
• Applies Glauber dynamics to update spins probabilistically based on energy change and both temperatures $T_1$ and $T_2$.
• Introduces site-dependent $T_1(i)$ that increases when all four neighbors are of the same group and decreases when none are, simulating adaptive tolerance.
• Applies a global forgetting rate (0.3% per step) to prevent $T_1(i)$ from growing indefinitely.
• Uses a $Q=5$ Potts-like model to represent five cultural groups, allowing for richer segregation dynamics than binary models.
• Simulates the system on a 2D lattice with random sequential updates and measures correlation $C$ and average $T_1$ over time.

Experimental results

Research questions

• RQ1 Can the original Schelling model produce large, stable ghettos without external forces or randomness?
• RQ2 How does the inclusion of noise at temperature $T_2$ affect the strength and continuity of segregation in the model?
• RQ3 What role does dynamic tolerance ($T_1(i)$) play in enabling or suppressing the growth of segregated clusters?
• RQ4 Does the system exhibit a phase transition to large-scale segregation as noise increases?
• RQ5 How do surface versus bulk noise affect the final degree of segregation in the system?

Key findings

• The original Schelling model fails to produce large ghettos due to blocking at interfaces, even with large lattices.
• Small, non-zero noise ($T_2$) is essential to break blocking and enable the formation of large, stable segregated domains.
• Noise at $T_2 > 5$ strongly reduces segregation, and effects saturate beyond this level, showing a threshold-like behavior.
• Bulk noise reduces segregation more effectively than surface noise, as confirmed by normalization in simulations.
• For $Q=5$, no sharp phase transition is observed with increasing $T_1$ or $T_2$, but segregation strength decreases continuously with noise.
• Dynamic tolerance $T_1(i)$, adjusted based on neighborhood homogeneity and subject to global forgetting, enables adaptive learning and prevents infinite tolerance buildup.