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Noise effects in Schelling metapopulation model with underlying star topology

  • Regular Article – Statistical and Nonlinear Physics
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Abstract

The Schelling model, one of the most famous agent-based models in social physics, describes the formation of social segregated groups of agents based on their preferences. It has received significant attention, and various versions of the model have been extensively developed and studied as well. In this paper, we examine the impact of noise on Schelling’s metapopulation segregation model, specifically focusing on an underlying star topological structure. Our findings demonstrate that the fascinating effects caused by the star topology become fragile when the model is subjected to noise. We conducted theoretical analyses and numerical simulations to investigate the stationary state of the systems, their evolutionary process, and the distribution of the agents. Our results indicate that the anomaly in optimizing collective utility by egoists diminishes at a non-vanishing level of noise. On the other hand, altruists regain their ability to optimize collective utility. As the noise level increases, the resulting randomness becomes dominant in the movement of agents, leading to a reduced distinction between the two types of agents and eventually become random walkers.

Graphical abstract

The intriguing effects of Schelling’s metapopulation segregation model with underlying star topology become fragile when noise parameter is introduced.

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Data Availability Statement

The source code that produced the data and supports the findings of the current study is openly available at the following URL: https://github.com/schelling-physics-shnu/Schelling-su-zhang.

References

  1. M. Perc, The Social Physics Collective. Sci. Rep. 9, 16549 (2019)

    Article  ADS  Google Scholar 

  2. M. Jusup et al., Social Mechanics. Phys. Rep. 948, 1–148 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  3. R.K. Pathria, P.D. Beale, Statistical Mechanics, 3rd edn. (Elsevier, Academic Press, 2011)

    Google Scholar 

  4. R. Mantegna, and H. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, (1999)

  5. D. Sornette, Why Stock Markets Crash: Critical Events in Complex Financial Systems. Princeton University Press, (2017)

  6. J.-P. Bouchaud, and M. Potters, Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management, 2nd ed. Cambridge University Press, (2003)

  7. C. Castellano, S. Fortunato, V. Loreto, Statistical physics of social dynamics. Rev. Modern Phys. 81, 591–646 (2009)

    Article  ADS  Google Scholar 

  8. M. Barthelemy, The statistical physics of cities. Nat. Rev. Phys. 1, 406–415 (2019)

    Article  Google Scholar 

  9. D. Chowdhury, L. Santen, A. Schadschneider, Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329, 199–329 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  10. D. Helbing, Traffic and related self-driven many-particle systems. Rev. Modern Phys. 73, 1067–1141 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  11. T. Nagatani, The physics of traffic jams. Rep. Progress Phys. 65, 1331–1386 (2002)

    Article  ADS  Google Scholar 

  12. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Epidemic processes in complex networks. Rev. Modern Phys. 87, 925–979 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  13. Z. Wang et al., Statistical physics of vaccination. Phys. Rep. 66, 1–113 (2016)

    ADS  MathSciNet  Google Scholar 

  14. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  15. S. Fortunato, D. Hric, Community detection in networks: a user guide. Phys. Rep. 659, 1–44 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  16. M. Barthélemy, Spatial networks. Phys. Rep. 499, 1–101 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  17. S. Boccaletti et al., The structure and dynamics of multilayer networks. Phys. Rep. 544, 1–122 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  18. G.F. de Arruda, F.A. Rodrigues, Y. Moreno, Fundamentals of spreading processes in single and multilayer complex networks. Phys. Rep. 756, 1–59 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  19. M. Perc et al., Statistical physics of human cooperation. Phys. Rep. 687, 1–51 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  20. T.C. Schelling, Dynamic models of segregation. J. Math. Sociol. 1, 143–186 (1971)

    Article  Google Scholar 

  21. T.C. Schelling, Micromotives and Macrobehavior (Norton, New York, 1978)

    Google Scholar 

  22. A.J. Laurie, N.K. Jaggi, Role of “vision’’ in neighbourhood racial segregation: a variant of the Schelling segregation model. Urban Stud. 40, 2687–2704 (2003)

    Article  Google Scholar 

  23. W. Clark, M. Fossett, Understanding the social context of the Schelling segregation model. Proc. Natl. Acad. Sci. U.S.A. 105, 4109–4114 (2008)

    Article  ADS  Google Scholar 

  24. M. Fossett, D.R. Dietrich, Effects of city size, shape, and form, and neighborhood size and shape in agent-based models of residential segregation: Are Schelling-style preference effects robust? Environ. Plann. B 36, 149–169 (2009)

    Article  Google Scholar 

  25. G. Fagiolo, M. Valente, N.J. Vriend, Segregation in networks. J. Econ. Behav. Organization 64, 316–336 (2007)

    Article  Google Scholar 

  26. R. Pancs, N.J. Vriend, Schelling’s spatial proximity model of segregation revisited. J. Publ. Econ. 91, 1–24 (2007)

    Article  Google Scholar 

  27. S. Grauwin, F. Goffette-Nagot, P. Jensen, Dynamic models of residential segregation: an analytical solution. J. Publ. Econ. 96, 124–141 (2012)

    Article  Google Scholar 

  28. M. Pollicott, H. Weiss, The dynamics of Schelling-type segregation models and a nonlinear graph laplacian variational problem. Adv. Appl. Math. 27, 17–40 (2001)

    Article  MathSciNet  Google Scholar 

  29. S. Gerhold, L. Glebsky, C. Schneider, H. Weiss, B. Zimmermann, Computing the complexity for Schelling segregation models. Commun. Nonlinear Sci. Numer. Simul. 13, 2236–2245 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  30. R. Durrett, Y. Zhang, Exact solution for a metapopulation version of Schelling’s model. Proc. Natl. Acad. Sci. U.S.A. 111, 14036–14041 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  31. D. Vinkovic, A. Kirman, A physical analogue of the Schelling model. Proc. Natl. Acad. Sci. U.S.A. 103, 19261–19265 (2006)

    Article  ADS  Google Scholar 

  32. L. Dall’Asta, C. Castellano, M. Marsili, Statistical physics of the Schelling model of segregation. J. Stat. Mech. 2008, L07002 (2008)

    Article  Google Scholar 

  33. D. Stauffer, S. Solomon, Ising, Schelling and self-organising segregation. Euro. Phys. J. B 57, 473–479 (2007)

    Article  ADS  Google Scholar 

  34. L. Gauvin, J. Vannimenus, J.-P. Nadal, Phase diagram of a Schelling segregation model. Euro. Phys. J. B 70, 293–304 (2009)

    Article  ADS  Google Scholar 

  35. S. Grauwin, E. Bertin, R. Lemoy, P. Jensen, Competition between collective and individual dynamics. Proc. Natl. Acad. Sci. U.S.A. 106, 20622–20626 (2009)

    Article  ADS  Google Scholar 

  36. L. Gauvin, J.-P. Nadal, J. Vannimenus, Schelling segregation in an open city: A kinetically constrained Blume-Emery-Griffiths spin-1 system. Phys. Rev. E 81, 066120 (2010)

    Article  ADS  Google Scholar 

  37. N.G. Domic, E. Goles, S. Rica, Dynamics and complexity of the Schelling segregation model. Phys. Rev. E 83, 056111 (2011)

    Article  ADS  Google Scholar 

  38. P. Jensen, T. Matreux, J. Cambe, H. Larralde, E. Bertin, Giant catalytic effect of altruists in Schelling’s segregation model. Phys. Rev. Lett. 120, 208301 (2018)

    Article  ADS  Google Scholar 

  39. G. Su, Q. Xiong, Y. Zhang, Intriguing effects of underlying star topology in Schelling’s model with blocks. Phys. Rev. E 102, 012317 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  40. G. Su, Y. Zhang, Significant suppression of segregation in Schelling’s metapopulation model with star-type underlying topology. Euro. Phys. J. B 96, 91 (2023)

    Article  ADS  Google Scholar 

  41. S.P. Anderson, A. De Palma, and J (The MIT Press, F. Thisse. Discrete Choice Theory of Product Differentiation, 1992)

  42. L.E. Blume, The statistical mechanics of strategic interaction. Games Econ. Behav. 5, 387–424 (1993)

    Article  MathSciNet  Google Scholar 

  43. G. Szabó, and Csaba Tőke, Evolutionary prisoner’s dilemma game on a square lattice. Phys. Rev. E, 58:69–73, (1998)

  44. M. Perc, Coherence resonance in a spatial prisoner’s dilemma game. New J. Phys. 8, 22 (2006)

    Article  ADS  Google Scholar 

  45. A. Traulsen, J.M. Pacheco, M.A. Nowak, Pairwise comparison and selection temperature in evolutionary game dynamics. J. Theor. Biol. 246, 522–529 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  46. A. Szolnoki, M. Perc, and G. Szabó1, Topology-independent impact of noise on cooperation in spatial public goods games. Phys. Rev. E 80:056109, (2009)

  47. A. Traulsen, D. Semmann, R.D. Sommerfeld, H.-J. Krambeck, M. Milinski, Human strategy updating in evolutionary games. Proc. Natl Acad. Sci. USA 107, 2962–2966 (2010)

    Article  ADS  Google Scholar 

  48. F. Fu, D.I. Rosenbloom, L. Wang, M.A. Nowak, Imitation dynamics of vaccination behaviour on social networks. Proc. R. Soc. B 278, 42–49 (2011)

    Article  Google Scholar 

  49. A. Szolnoki, M. Perc, Conformity enhances network reciprocity in evolutionary social dilemmas. J. R. Soc. Interface 12, 20141299 (2015)

    Article  Google Scholar 

  50. M. Tanaka, J. Tanimoto, Is subsidizing vaccination with hub agent priority policy really meaningful to suppress disease spreading? J. Theoretical Biol. 486, 110059 (2020)

  51. J. Huang, J. Wang, C. Xia, Role of vaccine efficacy in the vaccination behavior under myopic update rule on complex networks. Chaos, Solitons Fractals 130, 109425 (2020)

    Article  Google Scholar 

  52. H. Zhang et al., Exploring cooperative evolution with tunable payoff’s loners using reinforcement learning. Chaos, Solitons & Fractals 178, 114358 (2024)

    Article  MathSciNet  Google Scholar 

  53. F. Battiston et al., The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098 (2021)

    Article  Google Scholar 

  54. W. Wang et al., Epidemic spreading on higher-order networks. Phys. Rep. 1056, 1–70 (2024)

    Article  MathSciNet  Google Scholar 

  55. Y. Zhang, M. Lucas, F. Battiston, Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes. Nat. Commun. 14, 1605 (2023)

    Article  ADS  Google Scholar 

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Acknowledgements

The authors acknowledge the financial support of the National Natural Science Foundation of China via Grant No. 11505115.

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Authors and Affiliations

  1. Department of Physics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Yihan Liu, Guifeng Su & Yi Zhang

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  1. Yihan Liu
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  2. Guifeng Su
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All the authors contributed equally to the analytical and numerical calculations contained in the present manuscript.

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Correspondence to Guifeng Su or Yi Zhang.

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Liu, Y., Su, G. & Zhang, Y. Noise effects in Schelling metapopulation model with underlying star topology. Eur. Phys. J. B 97, 31 (2024). https://doi.org/10.1140/epjb/s10051-024-00667-7

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