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SCHELLING is an independent Swiss family company that offers diverse packaging, print and display solutions. It manufactures packaging and displays made of corrugated cardboard and solid board, as well as leaflets, labels, and other printed products. It was founded in 1876 and is based in Lenzburg, Switzerland.

Headquarters Location

Industriestrasse 11

Rupperswil, CH-5102,





Mobility constraints in segregation models

Jul 26, 2023

Abstract Since the development of the original Schelling model of urban segregation, several enhancements have been proposed, but none have considered the impact of mobility constraints on model dynamics. Recent studies have shown that human mobility follows specific patterns, such as a preference for short distances and dense locations. This paper proposes a segregation model incorporating mobility constraints to make agents select their location based on distance and location relevance. Our findings indicate that the mobility-constrained model produces lower segregation levels but takes longer to converge than the original Schelling model. We identified a few persistently unhappy agents from the minority group who cause this prolonged convergence time and lower segregation level as they move around the grid centre. Our study presents a more realistic representation of how agents move in urban areas and provides a novel and insightful approach to analyzing the impact of mobility constraints on segregation models. We highlight the significance of incorporating mobility constraints when policymakers design interventions to address urban segregation. Introduction Understanding urban segregation, which refers to the spatial separation and concentration of different social groups within an urban area, is of paramount importance given its impact on various social, economic, and cultural facets of our society 1 , 2 , 3 , 4 . For example, high levels of segregation lead to limited access to quality education, healthcare, and employment opportunities for marginalized communities, exacerbating socioeconomic disparities and hindering social mobility 5 , 6 , 7 . Additionally, concentrated poverty resulting from segregation strain public resources, contribute to higher crime rates, and foster social isolation, further impeding community development and cohesion 8 , 9 . Particularly crucial is the mathematical modelling of the mechanisms underlying segregation dynamics, as it provides a robust tool for conducting insightful what-if analyses, understanding the intricacies of social inequities, facilitating integration, and promoting social cohesion 2 , 10 . Thus, it is unsurprising that modelling urban segregation has attracted the attention and efforts of scientists from different disciplines 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 . In 1971, economist Thomas Schelling proposed an agent-based model to explain how individual actions could result in global phenomena, focusing on urban segregation 20 , 21 , 22 , 23 . He observed that segregation dynamics emerge due to homophily among social groups across various demographic factors such as ethnicity, language, income, and class affiliation 23 . To illustrate this idea, Schelling used a simple spatial proximity model that divided the population into two groups based on a homophily threshold. Agents of two colours were placed randomly on a two-dimensional grid, and each agent preferred to live next to people in their group. If an agent is unhappy with their current location, they will move to the nearest square that satisfies them. Schelling found that segregation emerges above a homophily threshold of 1/3, and other factors affecting segregation include the ratio of individuals, the homophily threshold, and individual demands. Numerous variants and enhancements of the Schelling model have been proposed so far, modifying agents’ behaviour 11 , 12 , 13 , environmental configuration 24 , 25 , 26 , 27 , 28 , 29 , considering geographical regions 14 , 15 , 16 , including real-world segregation data along with strategies to validate simulated behaviour with observations 17 , 18 , 19 , implementing agent behaviours based on psychological and sociological theories 30 , 31 , 32 , 33 , and allowing for sensitivity analysis to quantify outcome dependency on various parameters and initial conditions 10 , 34 , 35 , 36 . Other works show how even milder preferences or integration policies can eventually lead to unexpected segregation scenarios 37 , 38 , and how the introduction of venues can have an impact on segregation dynamics 39 . Despite these advancements, all proposed models assume that unhappy agents move randomly on the grid without any preference for nearby or far away locations. However, recent empirical studies have shown that human movement, far from being random, follows specific statistical patterns across various spatial scales, including daily movements and migrations 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 . These individual mobility patterns are characterized by a preference for short distances and relevant places over longer distances and sparse ones 40 , 41 , 42 , 43 , 44 , 45 , 48 , 54 , 55 , 56 , 57 . Despite considerable interest in modelling and predicting human mobility 43 , 52 , it remains unclear how mobility patterns relate to segregation patterns. Recent empirical studies suggest a link between experienced income segregation and an individual’s tendency to explore new places and visitors from different income groups 58 , but this has not been systematically studied in an agent-based, Schelling-like simulation framework. At the same time, although a few simulation studies show how restricting relocation options 25 and considering collective factors 33 , 59 affect segregation dynamics, these findings have not been related to mobility constraints. Thus, we still lack a comprehensive understanding of how mobility constraints impact segregation dynamics. This study fills this gap by designing a segregation model that considers mobility constraints and exploring how they influence segregation dynamics. Drawing on the gravity law of human mobility 48 , 55 , 60 , 61 , 62 , 63 , our model allows unhappy agents to select the next location to move based on distance and location relevance. Our findings reveal that mobility-constrained models exhibit lower levels of segregation than the Schelling model, albeit with a longer convergence time, and that agents that end in the periphery are more segregated than those in the grid centre. We attribute these phenomena to a small group of persistently unhappy agents from the minority group who gravitate towards the grid centre due to the preference for nearby and relevant locations imposed by mobility constraints. Mobility-constrained segregation models In our model, agents may be of two types, moving on a bi-dimensional grid of size \(m = N \times N\). As in the original Schelling model 20 , 21 , 22 , 23 , an agent is happy when it is surrounded by a number of agents of the same type above a predetermined homophily threshold. An unhappy agent located at a cell A moves to a new cell B based on a probability function, p(B), which depends on two factors: the distance d(A, B) between A and B, and the relevance r(B) of destination B. This probability captures the gravity law of human mobility 43 , 47 , 48 , 52 , 55 , 60 , 61 , 62 , 63 , 64 , positing that people tend to travel to nearby and relevant locations, a concept that has been supported by extensive research in fields ranging from transport planning 65 and spatial economics 62 , 66 , 67 to epidemic spreading 57 , 68 , 69 , 70 , 71 . The distance between points A and B, represented by coordinates \((x_A, y_A)\) and \((x_B, y_B)\), is computed as their Euclidean distance on the grid, \(d(A, B)=\sqrt{(x_A-x_B)^2 + (y_A-y_B)^2}\). Mathematically, we define the probability of an agent moving to cell B, given its current cell A, as a product of two power-law functions: $$\begin{aligned} p(B) \propto r(B)^\alpha d(A, B)^\beta \end{aligned}$$ (1) where parameter \(\alpha > 0\) models the tendency to move preferably to relevant places, while \(\beta\) captures the tendency to prefer (\(\beta > 0\)) or avoid (\(\beta <0\)) large displacements. These two parameters govern the influence of distance and relevance on the simulation outcomes, encapsulating the essential factors that shape the dynamics of the model. We assume a core-periphery structure to model the distribution of relevance across the grid cells 72 and use a radial distribution where the relevance value of each cell decreases with its distance from the grid centre C: $$\begin{aligned} r(A) \propto \frac{1}{d(A,C)^\kappa } \end{aligned}$$ (2) with \(\kappa =2\). The results obtained with a uniformly random spatial distribution of relevance can be found in Supplementary Note 1 . Note that since all agents share the information about cell relevance, \(\alpha = -x\) means being repelled by a cell to the same extent that \(\alpha = x\) means being attracted to it. The case where \(\alpha = 0\) and \(\beta = 0\) corresponds to the original Schelling model. The model simulation ends when all agents are happy. From the gravity segregation model, we derive two other families of models: the distance models (\(\alpha =0, \beta \ne 0\)), which only imposes constraints on distance, and the relevance models (\(\alpha \ne 0, \beta = 0\)), which only considers relevance. See Table 1 for algorithmic details about the gravity, relevance, and distance models. Table 1 Schema of the distance, relevance, and gravity models.

SCHELLING Frequently Asked Questions (FAQ)

  • When was SCHELLING founded?

    SCHELLING was founded in 1876.

  • Where is SCHELLING's headquarters?

    SCHELLING's headquarters is located at Industriestrasse 11, Rupperswil.

  • Who are SCHELLING's competitors?

    Competitors of SCHELLING include Solvias and 4 more.


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