An Analysis of One-Dimensional Schelling Segregation (1203.6346v2)

Abstract: We analyze the Schelling model of segregation in which a society of n individuals live in a ring. Each individual is one of two races and is only satisfied with his location so long as at least half his 2w nearest neighbors are of the same race as him. In the dynamics, randomly-chosen unhappy individuals successively swap locations. We consider the average size of monochromatic neighborhoods in the final stable state. Our analysis is the first rigorous analysis of the Schelling dynamics. We note that, in contrast to prior approximate analyses, the final state is nearly integrated: the average size of monochromatic neighborhoods is independent of n and polynomial in w.

Analysis of One-Dimensional Schelling Segregation

This paper provides a rigorous analysis of the one-dimensional Schelling model of segregation, which originally proposed by Thomas Schelling in 1969. This model is emblematic in demonstrating how local individual preferences could unwittingly lead to global patterns of segregation, despite individual preferences for integration.

Main Contributions

The research fills a gap in analytical understanding by examining a society of individuals arranged in a ring, with each person being one of two races. Satisfaction for an individual is contingent upon having at least half of their $2w$ nearest neighbors belonging to the same race. If unsatisfied, individuals will strategically swap locations until a stable state is achieved. This paper pioneers a rigorous analysis of these dynamics, proving that instead of ending in total segregation, the final configuration is nearly integrated, with the average size of monochromatic neighborhoods being independent of the total number of individuals $n$ and approximately polynomial in $w$.

Technical Approach and Findings

The paper's methodological framework involves:

1. Modeling Dynamics: Initially, individuals are randomly assigned one of two races on a ring topology. Their satisfaction is derived based on the racial configuration within a neighborhood defined by parameter $w$. Unhappy individuals swap positions in an attempt to become satisfied.
2. Differential Equation Framework: The model utilizes differential equations to approximate the dynamics of the system, offering a deterministic perspective on the average behavior over time. This is backed by Wormald's technique, allowing the calculation to be largely deterministic in the continuum limit.
3. Firewall Definition: The paper introduces the concept of a "firewall" which stabilizes certain configurations. An individual within a firewall (defined as a run of length $w+1$) will always be satisfied, ensuring they do not need to move.
4. Analytic Proofs and Results: Employing mathematical rigor, the paper demonstrates that the average run lengths in the stable configuration tend to be independent of $n$ and polynomial in $w$, suggesting a tendency towards integration rather than segregation. This contradicts previous conjectures and empirical observations that suggested inevitable segregation.

Implications and Future Directions

The implications of these findings are multifaceted. Theoretically, they reveal that the one-dimensional Schelling model does not necessitate segregation as earlier thought, providing a new understanding of the emergent dynamics in these systems. Practically, these results open inquiries into whether the integration observed in one-dimensional models might extend or transform in multi-dimensional ones, such as grid-expressed populations.

Future work could explore extending these insights into more complex topologies or incorporating stochastic elements into agent behavior, providing closer alignment with real-world urban dynamics. Understanding the transition from one-dimensional to two-dimensional configurations in segregation models remains another intriguing domain, given the complexity and variability of real-life social networks.

In conclusion, the paper significantly advances the field's understanding of Schelling's model, offering robust analytical insights that challenge and refine the narrative around segregation dynamics left by individual local preferences. Through rigorous analysis, a previously underexplored aspect of segregation dynamics hinges upon structural and contextual variables, revealing complexities that demand further exploration in diverse and dynamic sociological landscapes.