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A fuzzy set extension of Schelling’s spatial segregation model

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Abstract

This study explores a possible segregation mechanism assuming fuzzy group membership. We construct a fuzzy set extension of Schelling’s spatial segregation model. In the fuzzy Schelling model, each agent is assumed to have fuzzy membership in groups, which is typically assumed to represent the strength of the agent’s group identity. The model assumes that agents want to be with agents with the same or stronger (less fuzzy) group identity than themselves. Agents decide whether to stay or move depending on whether their neighborhood satisfies their desires. Analyzing a series of simulations reveals that: First, the fuzzy Schelling model can reproduce segregation at the macro level; here, segregation is formed by the accumulation of agents’ modest desires and actions. This is the most important property of the Schelling model. Second, agents’ behavior and situation differ depending on the fuzziness of their membership. Notably, agents with less fuzzy membership play an important role in the system’s equilibrium. Third, the tendency to reach equilibrium differs depending on the density of the space, required similarity level in the neighborhood, and initial distribution of membership values. Finally, we discuss the implications of the results.

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Data Availability

This study does not use any empirical data.

Notes

  1. For simplicity, we refer to this model as the Schelling model. However, we acknowledge that Sakoda made an important contribution to the development of this model [11, 18].

  2. As Fig. 11 in the Appendix shows, 200 steps are sufficient to reach equilibrium for most crisp Schelling model configurations. The long-term behavior of fuzzy Schelling models that do not reach equilibrium within 200 steps will be the subject of future research.

  3. For example, suppose that an agent with a membership value of 1 somehow reaches a neighborhood where four agents have membership value 1 and other four have a membership value 0. If the required similarity rate p is less than or equal to 0.5, then the agent can stay at the place as long as none of the other neighbors move elsewhere. In this situation, according to the definition of the measurement, the neighborhood similarity of the agent s(x) is 0.5, and the neighborhood fuzziness of the agent \(\phi (x)\) is 0, indicating low neighborhood similarity and less neighborhood fuzziness.

References

  1. Brewer, M. B., & Pierce, K. P. (2005). Social identity complexity and outgroup tolerance. Personality and Social Psychology Bulletin, 31(3), 428–437. https://doi.org/10.1177/0146167204271710

    Article  Google Scholar 

  2. Brown, R. (2000). Social identity theory: Past achievements, current problems and future challenges. European Journal of Social Psychology, 30(6), 745–778. https://doi.org/10.1002/1099-0992(200011/12)30:6<745::AID-EJSP24>3.0.CO;2-O

    Article  Google Scholar 

  3. Bruch, E. E., & Mare, R. D. (2006). Neighborhood choice and neighborhood change. American Journal of Sociology, 112(3), 667–709. https://doi.org/10.1086/507856

    Article  Google Scholar 

  4. Clark, W. A. V., & Fossett, M. (2008). Understanding the social context of the Schelling segregation model. Proceedings of the National Academy of Sciences, 105(11), 4109–4114. https://doi.org/10.1073/pnas.0708155105

    Article  Google Scholar 

  5. De Luca, A., & Termini, S. (1972). A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory. Information and Control, 20(4), 301–312. https://doi.org/10.1016/S0019-9958(72)90199-4

    Article  Google Scholar 

  6. Flache, A., & de Matos Fernandes, C. A. (2021). Agent-based computational models, In G. Manzo (Ed.) Research Handbook on Analytical Sociology, Research Handbooks in Sociology, (pp. 453–473). Edward Elgar Publishing. https://doi.org/10.4337/9781789906851.00033.

  7. Fossett, M. (2006). Ethnic preferences, social distance dynamics, and residential segregation: Theoretical explorations using simulation analysis. The Journal of Mathematical Sociology, 30(3–4), 185–273. https://doi.org/10.1080/00222500500544052

    Article  Google Scholar 

  8. Fossett, M. (2011). Generative models of segregation: Investigating model-generated patterns of residential segregation by ethnicity and socioeconomic status. The Journal of Mathematical Sociology, 35(1–3), 114–145. https://doi.org/10.1080/0022250X.2010.532367

    Article  Google Scholar 

  9. Grantham, E. O., & Giabbanelli, P. J. (2020). Creating perceptual uncertainty in agent-based models with social interactions. In 2020 Spring Simulation Conference (SpringSim) (pp. 1–12).

  10. Hazan, A., & Randon-Furling, J. (2013). A Schelling model with switching agents: Decreasing segregation via random allocation and social mobility. The European Physical Journal B, 86(10), 421. https://doi.org/10.1140/epjb/e2013-31142-1

    Article  Google Scholar 

  11. Hegselmann, R. (2017). Thomas C. Schelling and James M. Sakoda: The intellectual, technical, and social history of a model. Journal of Artificial Societies and Social Simulation, 20(3), 15. https://doi.org/10.18564/jasss.3511

    Article  Google Scholar 

  12. Ishida, A. (2016). Imagined conditions of the “Japanese’’: Social category analysis of internet survey data. Japanese Sociological Review, 67(2), 182–200. https://doi.org/10.4057/jsr.67.182

    Article  Google Scholar 

  13. Jones, N., Marks, R., Ramirez, R. & Ríos-Vargas, M. (2021). Improved race and ethnicity measures reveal U.S. population is much more multiracial. [Online] https://www.census.gov/library/stories/2021/08/improved-race-ethnicity-measures-reveal-united-states-population-much-more-multiracial.html. Accessed 1 Aug 2023.

  14. Leszczensky, L., & Pink, S. (2019). What drives ethnic homophily? A relational approach on how ethnic identification moderates preferences for same-ethnic friends. American Sociological Review, 84(3), 394–419. https://doi.org/10.1177/0003122419846849

    Article  Google Scholar 

  15. Levy, A., Saguy, T., van Zomeren, M., & Halperin, E. (2017). Ingroups, outgroups, and the gateway groups between: The potential of dual identities to improve intergroup relations. Journal of Experimental Social Psychology, 70, 260–271. https://doi.org/10.1016/j.jesp.2016.09.011

    Article  Google Scholar 

  16. Liu, Z., Li, X., Khojandi, A., & Lazarova-Molnar, S. (2019). On the extension of Schelling’s segregation model. In 2019 Winter Simulation Conference (WSC) (pp. 285–296).

  17. Perlmann, J., & Waters, M. C. (Eds.). (2002). The new race question: How the census counts multiracial individuals. Russell Sage Foundation.

    Google Scholar 

  18. Sakoda, J. M. (1971). The checkerboard model of social interaction. The Journal of Mathematical Sociology, 1(1), 119–132. https://doi.org/10.1080/0022250X.1971.9989791

    Article  Google Scholar 

  19. Schelling, T. C. (1971). Dynamic models of segregation. The Journal of Mathematical Sociology, 1(2), 143–186. https://doi.org/10.1080/0022250X.1971.9989794

    Article  Google Scholar 

  20. Schelling, T. C. (1978). Micromotives and macrobehavior. W. W. Norton & Company.

    Google Scholar 

  21. Silver, D., Byrne, U., & Adler, P. (2021). Venues and segregation: A revised Schelling model. PLoS One, 16(1), 1–35. https://doi.org/10.1371/journal.pone.0242611

    Article  Google Scholar 

  22. Smithson, M., & Verkuilen, J. (2006). Fuzzy set theory: Applications in the social sciences. Sage Publications.

    Book  Google Scholar 

  23. Urselmans, L., & Phelps, S. (2018). A Schelling model with adaptive tolerance. PLoS One, 13(3), 1–24. https://doi.org/10.1371/journal.pone.0193950

    Article  Google Scholar 

  24. Verkuyten, M. (2018). The social psychology of ethnic identity (2nd ed.). Routledge.

    Book  Google Scholar 

  25. Vinković, D., & Kirman, A. (2006). A physical analogue of the Schelling model. Proceedings of the National Academy of Sciences, 103(51), 19261–19265. https://doi.org/10.1073/pnas.0609371103

    Article  Google Scholar 

  26. Woo, Y. (2022). Homogenous Japan? An empirical examination on public perceptions of citizenship. Social Science Japan Journal, 25(2), 209–228. https://doi.org/10.1093/ssjj/jyac001

    Article  Google Scholar 

  27. Xie, Y., & Zhou, X. (2012). Modeling individual-level heterogeneity in racial residential segregation. Proceedings of the National Academy of Sciences, 109(29), 11646–11651. https://doi.org/10.1073/pnas.1202218109

    Article  Google Scholar 

  28. Young, H. P. (1998). Individual strategy and social structure: An evolutionary theory of institutions. Princeton University Press.

    Book  Google Scholar 

  29. Zadeh, L. A. (1965). Fuzzy sets. Information Control, 8, 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

    Article  Google Scholar 

  30. Zhang, J. (2004). A dynamic model of residential segregation. The Journal of Mathematical Sociology, 28(3), 147–170. https://doi.org/10.1080/00222500490480202

    Article  Google Scholar 

  31. Zhang, J. (2004). Residential segregation in an all-integrationist world. Journal of Economic Behavior & Organization, 54(4), 533–550. https://doi.org/10.1016/j.jebo.2003.03.005

    Article  Google Scholar 

  32. Zhang, J. (2011). Tipping and residential segregation: A unified Schelling model. Journal of Regional Science, 51(1), 167–193. https://doi.org/10.1111/j.1467-9787.2010.00671.x

    Article  Google Scholar 

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Acknowledgements

I am grateful to the two anonymous reviewers for valuable suggestions that helped improve this article. This study was supported by JSPS KAKENHI Grant Number 19K02065, 23K01808.

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  1. School of Sociology, Kwansei Gakuin University, 1-155 Uegahara Ichiban-cho, Nishinomiya, 662-8501, Hyogo, Japan

    Atsushi Ishida

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Correspondence to Atsushi Ishida.

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Supplementary information

The animation of the simulation of Fig. 3 and the Python code for the simulation is available at the author’s GitHub repository (https://github.com/aishidajt9/fuzzySchelling).

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Appendix A

Appendix A

See Figures 11, 12, 13, 14, 15, 16, 17 and 18.

Fig. 11
figure 11

Change in the mean and standard deviation of the number of steps to reach equilibrium with different densities d and required similarity rates p in the crisp Schelling model (100 replications)

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Fig. 12
figure 12

Change in the mean and standard deviation of the neighborhood similarity at equilibrium with different densities d and required similarity rates p in the crisp Schelling model (100 replications)

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Fig. 13
figure 13

Change in the mean and standard deviation of percent unhappy for each entropy value of the simulation with uniform distribution (100 replications, \(\alpha =1, \beta =1, d = 0.7\), and \(p = 0.3\))

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Fig. 14
figure 14

Change in the mean and standard deviation of average neighborhood similarity for each entropy value of the simulation with uniform distribution (100 replications, \(\alpha =1, \beta =1, d = 0.7\), and \(p = 0.3\))

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Fig. 15
figure 15

Change in the mean and standard deviation of average neighborhood fuzziness for each entropy value of the simulation with uniform distribution (100 replications, \(\alpha =1, \beta =1, d = 0.7\), and \(p = 0.3\))

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Fig. 16
figure 16

Change in the mean and standard deviation of the three measures for the uniform distribution (\(\alpha =1\) and \(\beta =1\)) with different densities d and required similarity rates p (100 replications)

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Fig. 17
figure 17

Change in the mean and standard deviation of the three measures for the unimodal distribution (\(\alpha =2\) and \(\beta =2\)) with different densities d and required similarity rates p (100 replications)

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Fig. 18
figure 18

Change in the mean and standard deviation of the three measures for the bimodal distribution (\(\alpha =0.5\) and \(\beta =0.5\)) with different densities d and required similarity rates p (100 replications)

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Ishida, A. A fuzzy set extension of Schelling’s spatial segregation model. J Comput Soc Sc (2023). https://doi.org/10.1007/s42001-023-00234-7

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