Abstract
Schelling’s models of segregation, first described in 1969 (Am Econ Rev 59:488–493, 1969) are among the best known models of self-organising behaviour. Their original purpose was to identify mechanisms of urban racial segregation. But his models form part of a family which arises in statistical mechanics, neural networks, social science, and beyond, where populations of agents interact on networks. Despite extensive study, unperturbed Schelling models have largely resisted rigorous analysis, prior results generally focusing on variants in which noise is introduced into the dynamics, the resulting system being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory (Young in Individual strategy and social structure: an evolutionary theory of institutions, Princeton University Press, Princeton, 1998). A series of recent papers (Brandt et al. in: Proceedings of the 44th annual ACM symposium on theory of computing (STOC 2012), 2012); Barmpalias et al. in: 55th annual IEEE symposium on foundations of computer science, Philadelphia, 2014, J Stat Phys 158:806–852, 2015), has seen the first rigorous analyses of 1-dimensional unperturbed Schelling models, in an asymptotic framework largely unknown in statistical mechanics. Here we provide the first such analysis of 2- and 3-dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from Brandt et al. in: Proceedings of the 44th annual ACM symposium on theory of computing (STOC 2012), 2012).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
N. Immorlica, R. Kleinberg, B. Lucier, M. Zadomighaddam, Exponential Segregation in a Two-Dimensional Schelling Model with Tolerant Individuals, preprint, arXiv:1511.02537.
The C++ code for these simulations is available at http://barmpalias.net/schelcode.shtml.
One should think of \(\mathtt {pn}\) as p-artial n-eighbourhood, \(\mathtt {uh}\) as u-nh-appy, and \(\mathtt {ruh}\) as r-ight-extended neighbourhood u-nh-appy.
Here it is to be understood that how large one must take w, and how large n must be compared to w, may depend on our particular choices of \(\tau \) and \(\gamma \) (as well as the given values \(\tau _{\alpha }\) and \(\tau _{\beta }\)).
Think of \(\mathtt {ju}\) as j-ust u-nhappy and think of \(\mathtt {rju}\) as r-ight extended neighbourhood j-ust u-nhappy.
One should think of \(\mathtt {ln}\) as (standing for) l-ower n-eighbourhood, and \(\mathtt {rn}\) as r-otated n-eighbourhood.
References
Amit, D.J., Gutfreund, H., Sompolinsky, H.: Spin-glass models of neural networks. Phys. Rev. A 32, 1007 (1985)
Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)
Barmpalias, G., Elwes, R., Lewis-Pye, A.: Digital morphogenesis via Schelling segregation. In: FOCS 2014 55th Annual IEEE Symposium on Foundations of Computer Science, Oct. 18–21, Philadelphia (2014)
Barmpalias, G., Elwes, R., Lewis-Pye, A.: Tipping points in 1-dimensional schelling models with switching agents. J. Stat. Phys. 158, 806–852 (2015)
Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2001)
Brandt, C., Immorlica, N., Kamath, G., Kleinberg, R.: An analysis of one-dimensional schelling segregation. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC 2012)
Byrka, K., Jdrzejewski, A., Sznajd-Weron, K., Weron, R.: Difficulty is critical: the importance of social factors in modeling diffusion of green products and practices. Renew. Sustain. Energy Rev. 62, 723–735 (2016)
Canning, A., Naef, J.-P.: Phase diagrams and the instability of the spin glass states for the diluted hopfield neural network model. J. Phys. I 2, 1791–1801 (1992)
Castillo, I.P., Skantzos, N.S.: The Little-Hopfield model on a sparse random graph. J. Phys. A 37, 9087–9099 (2004)
Dall’Asta, L., Castellano, C., Marsili, M.: Statistical physics of the Schelling model of segregation. J. Stat. Mech. 7, L07002 (2008)
Gauvin, L., Vannemenus, J., Nadal, J.-P.: Phase diagram of a Schelling segregation model. Eur. Phys. J. B 70, 293–304 (2009)
Henry, A.D., Prałat, P., Zhang, C.: Emergence of segregation in evolving social networks. Proc. Natl. Acad. Sci. 108(21), 8605–8610 (2011)
Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. 79(8), 2554–2558 (1982)
Kleinberg, J.: Cascading behavior in networks: algorithmic and economic issues. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Game Theory. Cambridge University Press, New York (2007)
Ódor, G.: Self-organising, two temperature Ising model describing human segregation. Int. J. Mod. Phys. C 3, 393–398 (2008)
Pancs, R., Vriend, N.: Schellings spatial proximity model of segregation revisited. J. Public Econ. 91(1–2), 1–24 (2007)
Pollicott, M., Weiss, H.: The dynamics of Schelling-type segregation models and a non-linear graph Laplacian variational problem. Adv. Appl. Math. 27, 17–40 (2001)
Schelling, T.: Models of segregation. Am. Econ. Rev. 59, 488–493 (1969)
Schelling, T.: Micromotives and Macrobehavior. Norton, New York (1978)
Schuman, H., Steeh, C., Bobo, L., Krysan, M.: Racial Attitudes in America: Trends and Interpretations, revised edn. Harvard University Press, Cambridge, MA (1997)
Stauffer, D., Solomon, S.: Ising, Schelling and self-organising segregation. Eur. Phys. J. B 57, 473–479 (2007)
Vinković, D., Kirman, A., Physical, A.: Analogue of the Schelling model. Proc. Natl. Acad. Sci. 51(103), 19261–19265 (2006)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)
Wormald, N.C.: Differential equations for random processes and random graphs. Ann. Appl. Probab. 5, 1217–1235 (1995)
Young, H.P.: Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, Princeton (1998)
Zhang, J.: A dynamic model of residential segregation. J. Math. Soc. 28(3), 147–170 (2004)
Zhang, J.: Residential segregation in an all-integrationist world. J. Econ. Behav. Organ. 54(4), 533–550 (2004)
Zhang, J.: Tipping and residential segregation: a unified Schelling model. J. Reg. Sci. 51, 167–193 (2011)
Acknowledgments
The authors thank Sandro Azaele for some helpful comments on an earlier draft of this paper. Barmpalias was supported by the 1000 Talents Program for Young Scholars from the Chinese Government, and the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative No. 2010Y2GB03. Additional support was received by the CAS and the Institute of Software of the CAS. Partial support was also received from a Marsden Grant of New Zealand and the China Basic Research Program (973) Grant No. 2014CB340302. Andrew Lewis-Pye (previously Andrew Lewis) was supported by a Royal Society University Research Fellowship.
Author information
Authors and Affiliations
Corresponding author
Additional information
Authors are listed alphabetically.
Rights and permissions
About this article
Cite this article
Barmpalias, G., Elwes, R. & Lewis-Pye, A. Unperturbed Schelling Segregation in Two or Three Dimensions. J Stat Phys 164, 1460–1487 (2016). https://doi.org/10.1007/s10955-016-1589-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-016-1589-6