Schelling's Segregation Model: Parameters, scaling, and aggregation

28 Pages·2009·2.51 MB·English

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DEMOGRAPHICRESEARCH
VOLUME21,ARTICLE12,PAGES341-366
PUBLISHED15SEPTEMBER2009
http://www.demographic-research.org/Volumes/Vol21/12/
DOI: 10.4054/DemRes.2009.21.12
ResearchArticle
Schelling’sSegregationModel:
Parameters,scaling,andaggregation
AbhinavSingh
DmitriVainchtein
HowardWeiss
(cid:176)c 2009AbhinavSinghetal.
Thisopen-accessworkispublishedunderthetermsoftheCreative
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TableofContents
1 Introduction 342
1.1 DescriptionoftheModel 343
2 Schelling’ssegregationisasmallcityphenomenon 344
3 Simulations 346
4 Analysis 351
4.1 MeasuresofAggregation 351
4.2 GlobalaggregationdependenceontheneighborcomfortthresholdT 353
4.2.1 T=3:sparseclusters 353
4.2.2 T=4:compactclustersandmesoscaleaggregation 355
4.2.3 T=5:finalstateswithmanyunhappyagents 356
4.3 Numberofstepsintheevolution 357
5 FinalstateswithN=50,N=100andN=200 358
6 Concludingremarksandacknowledgements 362
References 364
DemographicResearch:Volume21,Article12
ResearchArticle
Schelling’sSegregationModel: Parameters,scaling,andaggregation
AbhinavSingh1
DmitriVainchtein2
HowardWeiss3
Abstract
ThomasSchellingproposedasimplespatialmodeltoillustratehow,evenwithrelatively
mild assumptions on each individual’s nearest neighbor preferences, an integrated city
wouldlikelyunraveltoasegregatedcity, evenifallindividualspreferintegration. This
agentbasedlatticemodelhasbecomequiteinfluentialamongstsocialscientists,demogra-
phers,andeconomists.Aggregationrelatestoindividualscomingtogethertoformgroups
and Schelling equated global aggregation with segregation. Many authors assumed that
thesegregationwhichSchellingobservedinsimulationsonverysmallcitiespersistsfor
larger,realisticsizedcities. Wedescribehowdifferentmeasurescanbeusedtoquantify
the segregation and unlock its dependence on city size, disparate neighbor comfortabil-
itythreshold,andpopulationdensity. Wedevelophighlyefficientsimulationalgorithms
andquantifyaggregationinlargecitiesbasedonthousandsoftrials. Weidentifydistinct
scalesofglobalaggregation. Inparticular,weshowthatforthevaluesofdisparateneigh-
borcomfortabilitythresholdusedbySchelling,thestrikingglobalaggregationSchelling
observed is strictly a small city phenomenon. We also discover several scaling laws for
theaggregationmeasures.AlongthewayweprovethatintheSchellingmodel,inthepro-
cessofevolution,thetotalperimeteroftheinterfacebetweenthedifferentagentsalways
decreases,whichprovidesausefulanalyticaltooltostudytheevolution.
1SchoolofPhysicsandCenterforNonlinearScience,GeorgiaTech,USA,E-mail:[email protected].
2DepartmentofMechanicalEngineering,TempleUniversity,USA;SpaceResearchInstitute,Moscow,Russia,
E-mail:[email protected].
3School of Mathematics and Center for Nonlinear Science, Georgia Tech, USA, E-mail:
[email protected].
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1. Introduction
Inthe1970s,theeminenteconomicmodelerThomasSchellingproposedasimplespace-
time population model to illustrate how, even with relatively minimal assumptions con-
cerning every individual’s nearest neighbor preferences, an integrated city would likely
unravel to a segregated city, even if all individuals prefer integration (Schelling 1969;
Schelling 1971a; Schelling 1971b; Schelling 2006). His agent-based lattice model has
becomequiteinfluentialamongstsocialscientists, demographers, andeconomists. Cur-
rently, thereisaspiriteddiscussiononthevalidityofSchelling-typemodelstodescribe
actualsegregation,withargumentsbothfor(e.g.,Young1998;Fossett2006),andagainst
(e.g., Massey 1990; Laurie and Jaggi 2003), and a few authors have used and extended
theSchellingmodeltoaddressactualpopulationdata(Clark1991;BruchandMare2006;
Benensonetal.2006;Sander,Schreiber,andDoherty2000;ClarkandFossett2008).The
fewexamplesofquantitativeanalysesofsuchmodelsare(PollicottandWeiss2001;Fos-
sett 2006; Gerhold et al. 2008). Recently, Zhang (2004) proved analytically that, for
certain wedge-like utility functions and with additional random noise, the equilibrium
statespossessahighdegreeofsegregation.
Aggregation relates to individuals coming together to form groups or clusters, and
Schelling equated global aggregation with segregation. Many authors assume that the
striking global aggregation observed in simulations on very small ideal “cities" persists
forlarge, realisticsizecities. Arecentpaper(VinkovicandKirman2006)exhibitsfinal
statesforasmallnumberofmodelsimulationsofalargecity,andsomefinalstatesthat
do not exhibit significant global aggregation. However, quantification of this important
phenomenonislackingintheliterature,presumablydueinparttothehugecomputational
costsrequiredtorunsimulationsusingexistingalgorithms. Wedevelophighlyefficient
andfastalgorithmsthatallowustocarryoutmanysimulationsformanysetsofparame-
tersandtocomputemeaningfulstatisticsofthemeasuresofaggregation.
Theobjectiveofthispaperistoquantifytheaggregationandunlockitsdependence
on city size, disparate neighbor comfort threshold, and population density. One of the
measuresisthetotalperimeterofaconfiguration: thetotalnumberofcontactsbetween
theagentsofdifferentkind, adjustedtothepresenceofemptyspaces. Weprovethatas
the system evolves, the total perimeter decreases at every step. This provides a useful
analytical tool to study the evolution: it necessitates that the evolution of the Schelling
modelalwaysconvergestoalimitconfigurationafterafinitenumberoftimesteps. We
identifydistinctscalesofglobalaggregation,andinparticular,weshowthatforthevalues
of disparate neighbor comfort threshold used by Schelling, the striking global aggrega-
tion Schelling observed is strictly a small city phenomenon. We also discover several
remarkablescalinglawsfortheaggregationmeasures.
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1.1 DescriptionoftheModel
WeexpandSchelling’soriginalmodel4toathreeparameterfamilyofmodels. Thephase
space for these models is the N ×N square lattice with periodic boundary conditions
(opposite sides identified). We consider two distinct populations, that, in Schelling’s
words,referto“membershipinoneoftwohomogeneousgroups: menorwomen,blacks
and whites, French-speaking and English speaking, officers and enlisted men, students
and faculty, surfers and swimmers, the well dressed and the poorly dressed, and any
other dichotomy that is exhaustive and recognizable” (Schelling 2006). We denote by
B (black squares) and R (red squares, appear grey on black and white printing) these
two populations. See Fig. 1. Together these agents fill up some of the N2 sites, with
V remaining vacant sites (white squares). Each agent has eight nearest neighbors, cor-
responding to Moore, or Queen, neighborhood. Different neighborhoods were studied
indifferentpapers(see,e.g,Fossett2006; ClarkandFossett2008,wherethesizeofthe
neighborhood was referred to as ‘vision’). Fix a disparate neighbor comfort threshold
T ∈ {0,1,...,8}, and declare that a B or R is happy if T or more of its nearest eight
neighborsareB’sorR’s,respectively. Else,itisunhappy.
Figure1: A:AsimulationofSchelling’soriginalmodelwithN = 8;
B:OursimulationwithN = 100.
A B
4DifferentauthorsfrequentlyconsiderslightlydifferentversionsofSchelling’soriginalmodel,i.e.,different
waysofmovingboundaryagents.Allversionsseemtoexhibitthesamequalitativebehaviors,andthuswerefer
totheSchellingmodel.
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Demographically, theparameterN controlsthesizeofthecity, v = V/N2 controls
thepopulationdensityortheoccupancyratio(BusinessLocate2009),andT isan“agent
comfort index” that quantifies an agent’s tolerance to living amongst disparate nearest
neighbors.
In choosing the algorithm of evolution we followed the protocol introduced in the
originalSchellingpaper(Schelling1969)andlaterusedinPortugali,Benenson,andOmer
(1994) and Benenson et al. (2006). We begin the evolution by choosing an initial con-
figuration(describedinSect.3)andrandomlyselectinganunhappyBandavacantsite
surroundedbyatleastT nearestBneighbors. Providedthisispossible, interchangethe
unhappyBwiththevacantsite,sothatthisBbecomeshappy.Wethenrandomlyselectan
unhappyRandavacantsitehavingatleastT nearestneighborsoftypeR.Providedthis
ispossible,weinterchangetheunhappyRwiththevacantsite,sothatRbecomeshappy.
We repeat this iterative procedure, alternating between selecting an unhappy B and an
unhappyR,untilafinalstateisreached,wherenointerchangeispossiblethatincreases
happiness.Forsomefinalstates,some(andinsomecases,many)agentsmaybeunhappy,
buttherearenoallowableswitches.
Forthesakeofcompleteness,wecarriedoutsimulationsusingotheragentselection
protocols, including random selection schemes. We observed no significant differences
in the final states using the other selection schemes. This supports the claim in Young
(1998)thatthe finedetails ofthe evolutionhavenegligibleinfluence on thestructure of
thefinalstates.
2. Schelling’ssegregationisasmallcityphenomenon
SchellingconsideredthecasescitysizeN = 8,neighborcomfortthresholdT = 3,and
vacancyratiov = 33%. ForT = 3or4, andv = 0, a“checkerboard"configurationof
B’s and R’s (imagine placing B’s on the red squares and R’s on the black squares of an
actualcheckerboard)isafinalstate,sinceallagentshavefourlikenearestneighbors.
To generate his initial configurations, Schelling begins with a checkerboard config-
urationwithoutperiodicboundaryconditionsandrandomlyremovesapproximatelyone
thirdoftheB’sandR’s,keepingequalnumbersofbothagents(Schelling2006). Werefer
totheresultasadeletedcheckerboardconfiguration. Removingtheseagentsmakessome
oftheremainingagentsunhappyanddrivestheevolution. Severalauthorshaveobserved
that removing such a large percentage of agents is unnatural, but it is crucial to attain
aggregationinSchelling’smodel. Removingfeweragentsresultsinafinalconfiguration
closetotheinitialconfiguration. Finally,Schellingmodifiesthedeletedcheckerboardby
randomlyaddingatotalof5B’sand5R’sinvacantspaces. Suchaninitialconfiguration
isassumedtobeaproxyforanearlyintegratedcity.
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The final state of a typical run of Schelling’s original model system is presented in
Fig. 1A. Schelling performed many simulations by hand using an actual checkerboard,
andobservedthatthefinalstatespresentedasignificantdegreeofglobalaggregation. He
equatedtheglobalaggregationwithsegregationofacity.
In this paper, we investigate whether the global aggregation that Schelling observed
for very small lattices persists for larger lattices. In Fig. 1B, we present a characteristic
finalstateforoursimulationswithcitysizeN =100. ComparingFigs.1Aand 1B,one
canseeastrikingqualitativedifferencebetweenthetwofinalstates. Whilethereissome
localaggregationinthefinalstatewithN =100,thereisnoglobalaggregation. Visually
inspecting this and other final states, one immediately sees that the global aggregation
observedbySchellingisasmalllatticephenomenon.
Toquantifythedifferenceinaggregationbetweenasmallcity(N = 8)andalarger
city(N =100)weusedacombinationoftwoaggregationmeasures: thenumberofclus-
tersinthefinalstateofthemodelandthenormalizedaveragesizeofindividualclusters
(see Sect. 4.1 for a detailed description of these and several other measures of aggrega-
tion). Wedeterminethenormalizedaveragesizeofaclusterbydividingtheaveragesize
ofaclusterbythetotalnumberofagentsinthecity. Thelatterdeterminestheproportion
ofacitycoveredbyanindividualclusterandprovidesawaytocompareaggregationbe-
tweencitiesofdifferentsizes. Figure2showsthemeanvaluesofthesetwoaggregation
measuresinthefinalstatesofcitiesoftwosizes(N =8and100). Wecomputethemean
valuesofthetwomeasuresbasedon100trialsforeachchoiceofthevacancyratiovand
neighbor comfort threshold T = 3. We observe that the normalized average size of a
clusterinthelargecityissmallerthanoneinasmallcity. Thisimpliesthatanindividual
clusterinalargecitycoversasmallerproportionofthecityascomparedtoaclusterin
asmaller city. Most finalstates ofthe smallcity aresegregatedintotwoclusters forall
choicesofthevacancyratiowhilethenumberofclustersinthelargecityincreasesfrom
22 for a city with 24% empty locations to 55 for a city with 33% empty locations. As
we move from a small city to a large one, the relative size of a cluster in the final state
decreases and the number of clusters increases. This shows that the large scale global
aggregationobservedbySchellingisstrictlyasmallcityphenomenonanddoesnotoccur
forlargercities.
In Sections 3 through 5 we quantify the global aggregation using several new mea-
sures, and we analyze the structure of the final states for different values of neighbor
comfortthresholdT,vacancyratiov,andcitysizeN.
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Figure2: Aggregationmeasurestodistinguishbetweenasmallcity(N = 8)
andalargecity(N = 100)forconstantneighborcomfortthresh-
oldT = 3anddifferentvaluesofvacancyratiov.
A:Normalizedaveragesizeofanindividualcluster.
B:Numberofclustersinthefinalstateofacity.
3. Simulations
Wearethefirstauthorstoquantifyaggregationinalargecitybasedonalargenumberof
simulations. We develop highly efficient algorithms to simulate the model and quantify
theaggregation. Wecurrentlyneedapproximatelyoneminutetorunasinglesimulation
for a city of size N = 100 and we ran more then nine thousand simulations for this
manuscript. We achieve this boost in speed by coming up with innovative ways to de-
terminethehappinessofeachagentandtofindasuitablelocationforanunhappyagent.
We exploit the ability of modern software packages such as NumPy (Oliphant 2006) to
efficientlymanipulatematrices;wecannowcomputepropertiesofallagentsinacitysi-
multaneouslythusescapingtheslowprocessofdealingwitheachagentindividually. The
speedatwhichwefindanunhappyagentinthecityandasuitablelocationforitdeter-
minesthespeedofthesimulation. Naivelylookingateachagentinthecitytodetermine
ifitisunhappyandtestingeachemptylocationtodetermineifitisasuitablelocationfor
an unhappyagent, makesthe simulation much slower. A much more efficient approach
istoconstructmatriceshighlightingtheunhappyagentsinthecityandsuitablelocations
forunhappyagents. Theunhappinessofanagentandthesuitabilityofalocationareboth
based on the number of similar neighbors in the neighborhood of a given location. As
anexampleoftheefficiencygainedbymatrixmethods,weoutlinethestepstodetermine
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DemographicResearch:Volume21,Article12
the number of R agents surrounding each agent in the city simultaneously. In our ‘city
matrix’, an R agent is represented by 1, a B agent by -1 and an empty location by 0.
Therefore,theproblemofdeterminingthenumberofsurroundingRagentsisreducedto
addingupthe1’sintheneighborhoodofeachagentandignoringthe-1’s. Weignorethe
-1’sbysimplyfindingtheabsolutevalueofeachelementinthecitymatrix;thisconverts
the -1’s into 1’s but leaves the 1’s and 0’s unchanged. We call this modified matrix the
‘absolutevaluematrix’. Whenweaddthecitymatrixandtheabsolutevaluematrix,all
the -1’s are gone and the sum of all the elements gives the number of R agents in the
8-point neighborhood. Similar methods can be used to speed up the process of finding
suitablelocationsforunhappyagentsandcomputingaggregationmeasures.
WestudythedynamicsforlargelatticesandpresentourresultsmostlyforcitysizeN =
100. Figures 2-7 are all based on N = 100. In the last section, we discuss the cases
N =50andN =200,andshowthatN greaterthan100doesnotleadtoqualitativelyor
quantitativelydifferentstatesandphenomena. Werestrictourdiscussiontocitieshaving
anequalnumberofB’sandR’s. Wewillreporttheresultsonthedynamicswithdifferent
proportionsofB’sandR’sinaseparatemanuscript(Singh,Vainchtein,andWeiss2009).
WeconsiderneighborcomfortthresholdT =3,4,5andvacancyratiovbetween2%
and 33%. The system does not evolve very much for other values of T: for T = 1,2
almostalloftheagentsaresatisfiedinmostoftheinitialconfigurations,whileforT ≥6
therearealmostnolegalswitches. Valuesofvlargerthan33%correspondtounrealistic
environments. For each pair of parameters T and v, we perform 100 simulations. This
number of simulations was chosen to ensure a 95% confidence interval for parameter
estimation. TheCentralLimitTheoremprovidesconfidenceintervalsforthemeanvalues
oftheaggregationmeasures.
We choose the initial configuration by starting with a checkerboard with periodic
boundaryconditions. Demographically,acheckerboardconfigurationisamaximallyin-
tegratedconfiguration. Wethenrandomlyremovehalftheintendedvacantlocationsv/2
ofbothB’sandR’s(thuskeepingequalnumbersofbothagents). Werandomlypermute
agentsintwo3×3blocks. Alternatively, wecouldchooseacompletelyrandominitial
configuration. Ingeneral, exceptforsmallvaluesofv, thefinalstatesarequantitatively
similartotheonesobtainedusingtheSchelling-likeinitialconditions.
In Figs. 3-5, we present characteristic final states for different values of T and v.
Visually,theaggregationinthefinalstateswithfixedvaresubstantiallydifferentthanin
thefinalstateswithfixedT.
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Singhetal.:Schelling’sSegregationModel:Parameters,scaling,andaggregation
Figure3: CharacteristicfinalstatesforneighborcomfortthresholdT = 3
fordifferentvacancyratiosv:
A:v = 2%,B:v = 6%,C:v = 10%,D:v = 15%,
E:v = 20%,F:v = 24%,G:v = 28%,H:v = 33%.
A B C
D E F
G H
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