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Trendsofgrowthanddifferentiationinstreetnetworks
Abstract.
ResearchintheareaofSpacesyntaxtendstobecentredonstaticnetworkrepresentationsofthe
built environment and its embedded social logic. Lacking the element of time, this synchronic
representationcannotcapturethedynamicsofgrowthandchangeinurbansystems.Inthispaper,
wearguethattheabstractvaluesofspace-timeasadualdimensionplayakeyroleasgeneratorsof
citysystems.Hence,wemapurbangrowthandseekexplanatorydescriptionsforthedrivingforces
thatcharacterisegrowthanddifferentiationinstreetnetworks.Intwocasestudies;Manhattanand
Barcelona, synchronic statesof thegrowing systemsare analysed.The states are separatedbya
short radius of time. The analysis leads to regularities thatmay help define the local and global
processesthatcharacteriseurbangrowthmarkedbyalternatingperiodsofexpansionandpruning
instreetnetworks.
Introduction
Instigatedbythecalltounderstandcitiesasmattersoforganisedcomplexity(Jacobs,1961),many
theorisationsweremadeonthedescriptionofsuchorganisationandhowthatorganisationcameto
be(Krafta,1999;Portugalietal,2011).Theconsequencesofaproclaimedself-organisedbehaviour
can-perhaps-betracedinthespatialsignatureofgrowth(Al_Sayed,2013),andtheinvarianttrends
thatcharacteriseit.InlinewithJacobs’call(1961),complexitytheoriesofcitiesregardedurban
systemsasemergentproductsofcomplexadaptiveprocessesthatinvolvenonlinearinteractions
betweendifferentsetsofvariables.Thedefinitionofcomplexadaptivebehaviourinurbansystems
wasverymuchdependantoninitialconditionsandtheactorsthatareputondisplayinsimulation
models(Allenet.al.,1977;AllenandSanglier,1979;Portugalietal,2011).Untilrecently(Masucciet
al,2013),therehasbeenlittleeffortdedicatedtooutlinetheverytrendsthaturbansystems
convergetointheirgrowthpatterns,wherethemajorityofstudiesweremorefocusedon
comparingcitiesacrossdifferentgeographies(CarvalhoandPenn,2004;Bettencourtetal,2007).
Recentlytherehasbeenawideinterestintheresearchcommunitytoempiricallydefinethelocal
mechanismsthatgeneratethepatternsweobserveincities(Al_Sayedetal,2009;2010;2012;
Stranoetal,2012;Barthelemyetal,2013;Serraetal,2016).
Withthescopeofidentifyingthelocalandglobaltrendsthatcitiesdisplayintheirgrowth
behaviour,anempiricalinvestigationisheldinthispaperfocusingontwocasestudies;Barcelona
andManhattan.Historicaldataonstreetnetworksismappedandanalysedusingspacesyntax
(HillierandHanson,1984).Thevaluesofstreetnetworkconfigurationsareusedtobuildstatistical
2
modelsofgrowth.Themappingrevealspositiveandreinforcingfeedbackmechanismsthatoperate
throughalternatingperiodsofexpansionandpruningintheurbangrid.
Streetnetworkdata
Forthepurposeofourinvestigation,streetnetworkmapsweremanuallytracedandextractedasa
vector layer on top of a set of historicalmaps (see appendix 1) that record the history of urban
development ofManhattan and Barcelona. Themaps ofManhattanwere observed in the years;
1642,1661,1695,1728,1755,1767,1789,1797,1808,1817,1836,1842,1850,1880,1920,and2008.
ThesequentialdatesofthemapsofBarcelonaare;1260,1290,1698,1714,1806,1855,1891,1901,
1920, 1943, 1970 and 2008. Street network maps were drawn in such a way as to reduce the
complexityofthestreetlayouttothefewestandlongestlines.Axial1linesweredrawnmanually.At
eachstageofgrowth,anaxiallinewasdrawntomatchtheexactcoordinatesoftheaxiallineinthe
formerstage.Thisrulewastotakethepreferencewhenconsideringthefewestandlongestlinesof
sight.
Tracinggrowthtrendsinhistoricaldata
Thenexttwosectionswillbedirectedtowardsunderstandingandanalysingdualhistorical
transformationsintheorganicanduniformgridsofManhattanandBarcelona.Whilewetakethe
synchronicrepresentationofcitiesasnetworksinterconnectingspatialelements,wemap
synchronicepisodesofgrowthtotracetheirdiachronicdevelopment(Griffiths,2009).The
assumptionisthat,theconfigurationsofspacearethemaininfluentialfactorsintheformationof
cities.Thesynchronicviewofspacemightbeidentifiedasaframeintime.Hence,aresultant
diachronicmodelwouldbeconstitutedofasequenceofsynchronicmodels.Weassumethe
presenceofinvariantfeaturesthatmarkthegeneraltrendsinadiachronicmodelofurbangrowth,
henceouranalysisisaimedatidentifyinganyregularityinthebehaviourofurbansystemsasthey
expandandchange.
Inordertobuildanexplanatorymodelofurbandevelopmentwechosetoanalysesequential
synchronicepisodes.InSpaceSyntaxterms,theseepisodesmightberepresentedbyspatial
networksthatcapturesstreetconfigurationsateachhistoricalstageofgrowth.Tocalculate
networkdistanceweusedthesegment2representationofSpaceSyntax(HillierandIida,2005).The
representationismanuallyconstructedasavectorlayerontopofhistoricalmapdata3.Considering
1Thetopologicaldescriptionofstreetnetworksmightbesimplydefinedasthefewestandlongestlinesofsight(axiallines)thatcoverallcontinuousspacesinanurbanregion.Eachaxiallinewillhaveacertainconnectivityvalue(degree);thatisthenumberofaxiallinesintersectingwithit.2Afiner-scaledescriptionofstreetsisthe(segmentlines);theuninterruptedstreetinterjunctionsthatlinktworoadintersections(Turner,2000).3seeappendix1
3
thetwocasesunderstudies,synchronicstatesofthegrowingsystemswereanalysed.Thestates
areseparatedbyashortradiusoftime.Foramoreaccuraterepresentation,thestreetswere
unlinkedweretherearemultilevelinterchangesintheroadinfrastructure4.PatternsofIntegration5
calculatedwithinlocal(500m)andglobal(R2000m)willbeexamined.Ofinterest,ishowvaluesof
integrationdistributeandchangeovertime.Furthertothat,statisticaldistributionswillbeoutlined
foreachstagetorevealwhethertheprobabilitydistributionsofgeometricnetworkproperties
changeasthesystemexpands.Thisisofgreatvalue,sincethedistributionofroadnetworks-
particularlyinwhatconcernsthenetworkgeometry-wasnotstudiedbefore(Waters,2006).
1.SyntacticanalysisofManhattan’sgrowth
AfteracquiringhistoricalmapdataforManhattan6,itwaspossibletomapandanalysethegrowth
ofthestreetnetwork.Theaxialandsegmentintegrationvaluesofeachhistoricalstageofgrowth
werecomputedinDepthmap(Turner,2010),andthestatisticaldistributionsofsegmentintegration
mapswerevisualisedusingJMP(SAS/Statsoftware).Inthemapsandassociateddistributionplots,
thevaluesofangularsegmentIntegrationshowedanincreaseinlocalandglobalIntegrationasthe
urbansystemgrows(seeFigure1).LocalIntegrationvalueswerecalculatedwithin500metreradius
whichalmostequalsfiveminuteswalkingdistance.Thenumberofelementsretaininghighlocal
Integrationvaluesdroppedwhentheuniformgridwasimposed.Spatially,highervaluesappeared
toconcentrateindowntownManhattanandWashingtonHeightsareas.Duringthelaterphasesof
growth,clusterswithhighlocalIntegrationvaluesappearedtoconcentrateinareasthathavegrown
organicallyovertime.Onaglobalscaleandwithinaradiusthatcapturesametricdistanceof2000
metresfromeachsegmentelement,therewasamorerecognisableincreaseinIntegrationvaluesas
thesystemgrows(seeFigure1).ThehighestvaluesconcentratedintheLowerManhattanareaat
theinitialstagesofgrowth,spreadinginlaterstagestothemidtownarea.
Overall,thelocalandglobalIntegrationvaluesofthesegmentelementsprovedtofitwellintoa
heavytailedlognormaldistributioninmostphasesofgrowth(seeFigure1).Astheurbansystem
grew,therangeofglobalIntegrationvalueswasspreadingmorewidelythantherangeoflocal
Integrationvalues.Localstructurespreservedtheirrangeofvalueswithslightchangesonthemean
andmedianvalues.ThepeaksofthedistributionplotsforlocalIntegrationsharpenedinlaterphases
ofgrowth.ThiswaslessevidentwhenrenderingglobalIntegrationvalues.
Furthertopologicalstructuralpropertiesarelistedbelowthedistributionplotstoreflectonhow
changesontheshapeofthedistributionassociatedachangeonthetopologicalstructures.The4Anexampleforthatiswherebridgesandtunnelsconnecttothestreetnetwork.5Integrationisaspecialmeasureofangularclosenessinthesegmentnetwork.Itiscalculatedbyrelatingtotalangulardepthintheneighbourhoodtothenodecount(Turner,2000).Segmentintegrationwasnotnormalisedinthisparticularcase6seeappendix1
4
topologicalpropertiesweredefinedbyaxialIntelligibilityR2(R2,Rn)andSynergyR2(R2,R5).Along
withthestructuralproperties,thedistributionshapedefinedbythemean,medianandskewness
showedinterestingpatternswheretransitionalstatesinthestreetnetworkdistinguishedperiodsof
rapidchange.Withthesemeasures,twosharpchangeswererecognised;oneintheyears(1695,
1728)andtheotherintheyears(1850,1880).Bothperiodswitnessedcriticalchangesinthegrid
directionalityandstructure.Thesechangeshadanobservableeffectonthedistributionshapeas
wellasonIntelligibilityandSynergycorrelationcoefficients.Thegoodnessoffitmeasuredbya
Kolmogorov–Smirnovlimits(KSL)test7actedasanindicatortosharptransitionswheretheDvalue
suddenlydroppedtothehalf.Inthiscase,thedropmarkedanimprovementinthedegreetowhich
theempiricaldistributionfitsthereferencedistribution(thelognormaldistribution).Therewasno
particularrulethatassociatedtheriseanddropofIntelligibilityandSynergyononehandandthe
meanandmedianontheotherhand.Generally,thestructureofthegridwasweakenedwhen
exposedtoamassiveadditionofsegmentstreetelementswithinrelativelyshorterperiodsoftime.
AfterasuddendropinIntelligibility,thesystemappearedtoretrievethestructuralpropertiesithad
before.Inparallel,thedistributionshapesettledwithnosignificantchanges.Thesefindingsare
significantastheyindicatetoanautonomousprocessthattakesplaceinurbanstructures.Thisis
mostlyvisibleinthewaythestreetnetworkpreservedcertainpatternsofpersistenceandchange.
Whensuchpatternsweredisturbedbyanartificialintervention,thesystemreadapteditsstructure
toretrieveitspriordistributionpatterns.
Overall,theanalyticalattemptstounderstandthespatialstructureofManhattanunderlinedseveral
consistenciesthatwerenoticeableinthegrowthprocessoftheurbanregion.Onbothlocaland
globalmetricradii,segmentangularIntegrationvaluesincreasedasthesystemgrew.Thepaceof
changeontheglobalscalewasmorerapidthanthatonthelocalscale.Thelatterwaslessaffected
afteracertainstage,probablyduetothescaleofmeasurement.Whereevident,theincreasein
Integrationvaluesseemedtobedirectional.Itgenerallystartedfromcertaincentresanddeveloped
inspaceandtime.Anotherconsistencywasevidentinhowthestreetnetworkpresentedsharp
changesinIntelligibilityandSynergythatappearedtosynchronisewithchangesontheshapeof
statisticaldistributions.Thesharptransitionalstageswereoftenassociatedwithamassiveaddition
ofstreets,andwerecharacterizedbyachangeonthefitnesslevelofintegrationvaluestoa
lognormaldistribution.Thesetransitionalstatesofthesystemwerefollowedbyaperiodwherethe
urbansystemadaptedbacktoitspriorstructuralpatternstoimproveitsfitnesstoalognormal
distribution.
7KSLtestmeasureshereadistancebetweentheempiricaldistributionfunctionofthesegmentIntegrationvaluesandthecumulativedistributionfunctionofthelognormaldistribution
5
Figure1ComparingangularIntegrationmaps(radius500andradius2000meters),theirdistribution
plots,andtheaxialIntelligibilityR2(R2,Rn)andSynergyR2(R2,R5)foreachstageofgrowthin
Manhattan.
2.SyntacticanalysisofBarcelona’sgrowth
Inthissection,wepresentasimilarsetofanalyticalinvestigationstothatappliedonManhattan’s
case.Thescopeistoexaminewhetheradifferentcasestudywouldpresentdifferenttrendsof
persistenceandchangeinthestatisticaldistributionofaccessibilityvalues.SimilartoManhattan’s
1642 à Angular integration maps (radius 500 meters)
_______________________________________________________________________________________
1642 à Angular integration maps (radius 2000 meters)
Dis
trib
utio
n R
500m
Dis
trib
utio
n R
2000
m
year
16422
16611
1695 1728 1755 1767 1789 1797 1808 1817 1836 1842 1850 1880 1920 2008
Inte
l
0.77 0.84 0.87 0.69 0.68 0.56 0.58 0.61 0.63 0.72 0.73 0.74 0.70 0.50 0.62 0.66
Syn 0.77 0.84 0.89 0.69 0.70 0.88 0.87 0.86 0.86 0.84 0.85 0.85 0.79 058 0.71 0.71
Severity of transition from one synchronic state to another given structural and distribution indicators
Higher angular integration values
__ Log normal distribution curve
__ Mean/median values
0
100
200
300
0200400600800100012001400
-----
6
case,thechoiceofgrowthphasestobeexaminedwaslimitedtothedataavailable(seeappendix1).
Axialmapsweredrawnmanuallyfollowingtheprocedureexplainedinearliersections.Theaxial
representationwasanalysedtoobtainaxialIntelligibilityandSynergy.Amorerefinedsegmental
representationwasalsousedtoobtainangularsegmentIntegration.Statisticaldistributionswere
plottedtodisplayhowaggregatepropertiesoftheurbansystemchangeintime.
ThesegmentalangularmeasuresappearedtoplotanalogouspatternsinBarcelonatothose
exploredinManhattan(seeFigure2).Similaritiesappearedwhereorganicordeformedlocal
structurespreservedthehighestvalues.Overtheperiodofgrowth,theoldcitycontinuedto
preservethehighestvaluesofangularsegmentIntegration(R500m).Equally,emergentsuburban
towncentresintheurbanfringemaintainedsimilarvaluesoflocalIntegration.Similarto
Manhattan’scase,segmentintegrationmapsradius(2000m)renderedhowhighlyintegrated
centresexpandandshiftleavinglowervaluesontheperipheries.InBarcelona,higherIntegration
valuesshiftedfromtheoldcitytocentreattheheartoftheuniformgrid.Thelinesrepresenting
majorroadsinthestreetnetworkappearedtobeparticularlyintegratedconnectingallpartsofthe
urbangridwiththelocaloldcitycentreandthesuburbantowncentres.Asthespatialsystemgrew
anddeformed,thedistinctfeaturesofthetwodifferentgridpatterns(organicanduniform)faded
andaconnectivestructurearosetracingtheruralfreewaysthatprecededtheuniformgrid.
ThepatternsofaggregatedistributionswereconsistentwithourobservationsinManhattan.In
moststages,thedistributionplotsforangularsegmentIntegrationradius500andradius2000
appearedtofitintoalognormalcurve(seeFigure2).Theperiodfrom1698to1855presenteda
multimodaldistributionpatternthathardlyfitsintheformerlognormalcurve.Thisisdueto
distinctdifferencesindensity,angularityandsegmentlinelengthbetweenfullyurbanised
agglomerationsaroundtheoldcitycentreandtheruralfreewaynetworkontheperipheries.This
phenomenonmayhelpdelineatecompacturbanformstatistically.Distincttransitionalperiodsthat
characterisetransformationsinthestreetnetworkmightberecognisedwhencomparingthe
shapesofstatisticaldistributions.Intwocasesoutofthree,wefoundthedistributionoflocal
Integrationturningfromarelativelyunimodaltomultimodal.ThecaseislessclearinManhattan.
Withthatcomesalsoachangeonthedistributionmomentsincludingthemedianandmean.
IntelligibilityandSynergycoefficientscameastoconfirmthesetransitionalperiodsrevealing
changesonthelevelofthestructuralunitybetweenthepartsandthewhole.Again,similarto
Manhattan’scase;thechangesonthestructurewereassociatedwithamassadditionofsegment
elementstotheurbannetworks.
Onthewhole,theanalysesexposedseveralparticularitiesthathadtodowiththeinclusionofless
urbanisedareasintheanalysis.Wheresuchdifferenceswereclear,theyhadasignificantimpacton
7
theshapeofstatisticaldistributioninBarcelona’scase.Theimpactofstubsonthesegment
representationandtheeffectofhighwayinfrastructurewerefoundtodisrupttheoverall
distributionpatterns.Thisfindingmighthelpoutlinethecharacteristicsofcompacturbanform;
definedbyhowaggregateangularconfigurationsofstreetstructuresfitintolognormal
distributions.
Theeffectofanomaliesontherepresentationisnotlikelytodismissthemoreprevailing
consistencies,particularlyinhowlocalcentresconservehighlocalIntegrationvaluesandinhow
globalIntegrationspreadstocentralareasasthesystemgrows.Consistenciesarealsoevidentin
howtransitionalperiodsarecharacterisedbyanassociationwithmassadditionofsegmentstreet
elements.Suchassociationsoftensynchronisedwithchangesonthefitnessofthelognormal
distributionfunction;wheretheDvalue–heredescribingthegoodnessoffit-droppedtohalforelse
doubled.Thisallneedstobeconsideredgivendifferentlocalradii.Herearadiusof2000mprovedto
beastrongerindicatorofthetransitionsinthegrowingstreetnetwork.Themoretheangulardepth
fitsalognormaldistribution,thelesslikelythesystemwouldcallforchanges.Onthecontrary,the
moreerraticaretheIntegrationvaluesaroundthelognormaldistributioncurvethemorelikelyitis
forthesystemtochange.Inthisway,thesystem’sinclinationtochangeisstronglyrelatedtohow
gooditfitsalognormaldistributionfunction.
Thewayinwhichtheurbansystemsettledbackintoalognormaldistributionafterthetransitional
periodsisagainsuggestiveofanautonomousbehaviourthatcitiesexhibitintheirgrowthpatterns.
Suchhypothesisneedstobesupportedbyamoreextensiveinvestigationintothelocal
mechanismsthatgoverngrowth.Beforegoingforahigherresolution,itisimperativetothinkabout
thetimedimensionalityandhowitcouplesspatialtransitions.InManhattan,wenotedatime
factor,orwhatwemighttermasaradiusoftimethatassociatestheradiusofspace,inthatlocal
Integrationmaintainsaslowerpaceofchangewhencomparedtoglobalradii.Tofurtherexposethis
dualityweneedtoplotspatialchangesagainstthetimeaxis.
8
Figure2ComparingangularIntegrationmaps(radius500andradius2000meters),theirdistribution
plots,andtheaxialIntelligibilityR2(R2,Rn)andSynergyR2(R2,R5)foreachstageofgrowthin
Barcelona.
GenericTrendsofGrowthinstreetnetworks
MeasuringontheoverallgrowthpatternsofthespatialstructureinManhattanandBarcelona,
growthcouldbeoutlinedbyexponentialtrendsthatplottheincreaseinthenumberofstreet
segments(seeFigure3).Overlaidontopofthatexponentialtrendisatrendthatmarksthechanges
1290 à
Angular integration maps (radius 500 meters)
___________________ ________________________________________________________
1290 à
Angular integration maps (radius 2000 meters)
Dis
trib
utio
n R
500m
Dis
trib
utio
n R
2000
m
year
1260 1290 1698 1714 1806 1855 1891 1901 1920 1943 1970 2008
Inte
l
0.90 0.62 0.57 0.58 0.58 0.52 0.68 0.67 0.67 0.62 0.47 0.40
Syn 0.90 0.71 0.64 0.65 0.64 0.64 0.71 0.72 0.71 0.69 0.63 0.59
Severity of transition from one synchronic state to another given structural and distribution indicators
Higher angular integration values
__ Log normal distribution curve
__ Mean/median values
-----
9
onthefractaldimensionoftheurbansystemasitgrows.Therangeisrelativetotheimagesizeof
thetwostreetstructures.Thefractaldimensioniscalculatedusingthebox-countingalgorithm,by
iterativelyoverlayingboxeswithdifferentsizesandcalculatingtheblackpixelswithin(seeappendix
2).Theoutputofthisprocedureisaregressionline.Theslopeofthisregressionlineisthefractal
dimensionofthestreetstructure.
Theexponentialgrowthparametersrecordeddifferentvaluesforthetwourbanregions.This
differencemightbereasonedbythesystem’ssensitivedependenceoninitialconditions;a
phenomenonthatcharacterisescomplexsystems.Thecircumstancesthatlimitgrowthatcertain
stagesmightalsoplayaroleindrivingtheexponentialtrend.Physicalboundarylimitation–for
example-presentsaconstraintduringthelaststagesofgrowthdivertingtheexponentialtrendin
Manhattantoalineartrendthatsettlesatacertainglobalmaxima.Atthispoint,wewereableto
distinguishapointofinflectionthatmarkedachangeontheconcavityofthecurve.InBarcelona,
theadditionofnewelementscontinuedtofollowanexponentialtrendprovidedthatthestructure
hasnotreacheditsmaximumgeographicalboundaries.Ontheexponentialmodels,wemarked
transitionalstatesthatwerederivedfromcriticalchangesontheprobabilitydistributionsofthe
streetstructurestoexposeanycorrespondenceswiththegenerictrends.Wenotedaremarkable
correspondencebetweenthesetransitionsandasuddenriseorfallinthenumberofelements,
alongwiththegrid’sfractaldimension.Suchtransitionswerealsotraceableinthechangesonthe
overallsizeoftheurbansystemandinthepatternsitrendered.
Whilebothgrowthtrendsfollowedanonlinearexponentialmodel,theydidthatwithdifferent
rates.Manhattan’sexponentialgrowthcoveredaperiodof240yearsstartingfromyear0=1640,
whereasBarcelona’sgrowthspannedoveraperiodof750yearsstartingfromyear0=1260.The
numberofelements(NoE)variablewasmodelledasanonlinearfunctionofyear.Tobuildthe
exponentialmodel,apredictionformulawasset.Thepredictionformulaincludedparametric
estimates(X0,X1).X0isthepredictionofthenumberofelementsatyear1240inBarcelonaandyear
1698inManhattan.Itshouldbeneartheactualvalueofthenumberofstreetsegmentsinthe
networkrepresentationofthehistoricaldata.Theformulacontainedtheparameters’initialvalues.
TheX1growthrateparameterwasgiventheinitialvalueof0.006forBarcelona,and0.013for
Manhattan,whichwouldmatchtheestimateoftheslopethatisderivedfromfittingthenaturallog
ofNoEtoyearwithastraightline.Theinitialvaluesdidfitreasonablywellwiththefinalparameter
estimatesoftheexponentialmodel.Thepredictionformulawasdefinedasfollows;
ƒ(x)=X0*Exp(X1*(year–year0)) (1)
wheretheparametersinitialvaluesare;
X1=0.013,X0=159forManhattan
10
X1=0.006,X0=90forBarcelona
Thenonlinearfittingprocesswastailoredsothattheconfidencelimits8retainedavaluefor
Alpha=0.05andtheconvergencecriterionwassetto0.00001.Usingthismethod,solutionswere
foundforaminimisedSumSquareErrors(SSE)value.Thefitnesstoanexponentialtrendappeared
tobedifferentforthetwourbanregions.Themeansquarederror(MSE)inManhattan’sgrowth
trendwasabout1978047.3whileinBarcelona’strendMSEapproximated7270391.8.TheMSEcan
besignificantlyreducedforManhattan,ifwedidnotconsiderthelaststageofgrowth(year=2005)
inthecalculation.Ifweweretomissthatyearfromtheanalysis,wewouldhavereachedouttoan
MSEvalueof531727.42;fourtimeslessthantheMSEvalueforthefullperiodofgrowth.Thistest
highlightedtheroleofphysicalboundariesinshapingthegrowthprocess.Itisimportantto
mentionherethatsuchconstraintisonlyoneofmany.Forthisreason,itmightbeimpossibleto
findanaveragedsolutionthatfitsbothurbanscenarios.Thisisnotonlymadedifficultbythe
sensitivedependenceoninitialconditionsbutisalsoverymuchdependantonthecircumstancesand
conditionsthatdirectgrowthateachtimeintervalandforeachlocalisedadditionofstreetsegment
tothenetwork.Giventhedifficultyinfindingdatathatmapstheprogressofurbandevelopmentat
thisspatiotemporalscale,itismaybereasonabletopursueanotherformofmorphogeneticmapping
thatonlyreflectsonhowstructureschangein-betweendifferenttimeframesfollowingthe
methodsdevelopedin(AlSayedetal,2012).Suchanapproachwouldhelpinferringtherulesand
forcesthatoperateonthelocallevelthatultimatelybuildintothegrowthpatternsoutlinedinthis
section.Beforeoptingforaninvestigationintothelocaltransformationsofurbanstructures,we
needtocastsomelightonthelocalisedprocessesthatarethoughttoconstitutetheprincipal
feedbackdynamicsinurbansystems.
8Theiterationsforconfidencelimitsdonotfindtheprofile-likelihoodconfidenceintervalssuccessfully.
11
Figure3AnonlinearexponentialgrowthmodelforManhattanandBarcelonamarkingtransitional
statescorrespondingtothoseoutlinedinthestatisticaldistributionsofintegrationvalues(figure1
&2).
Localmechanismsofpreferentialattachmentandpruninginstreetnetworks
Theprevioussectionelucidatedthegeneraltrendsofgrowththatoutlinedtheyearlyincreaseinthe
numberofstreetelementsinManhattanandBarcelona.Lookingattheexponentialmodels,itis
imperativetoregardthefactorsthatmighthaveaffectedanddistortedthegrowthpatterns.The
increaseinthenumberofstreetelementsdidnotgowithoutresistance.Atdifferentratesandon
differentscales,streetsemergedordisappearedlocallyduetoartificialandnaturalcauses.Artificial
interventionscouldbeaneffectofintentionalplanningactionsthatalterthestreetlayout.Natural
causescouldbetheoreticallydefinedasatendencyintheurbansystemtofollowanautonomous
processofself-organisationinwhichtheurbansystemadaptsfollowingartificialinterventionsto
preservethestructuralunitybetweenthewholeandtheparts.Itisimplicittoourknowledge,how
suchprocessesbuildfromthelocaltotheglobaltorenderthehierarchicalorganisationthatmakes
Manhattan
Linear and nonlinear fit solutions
Barcelona
Deformity R2 = 0.80 Natural logarithm fit Fractal D R2 = 0.53 Linear fit
Deformity R2 = 0.95 Linear fit Fractal D R2 = 0.83 Linear fit
Nonlinear fit solution
SSE DFE MSE RMSE 27692661.637 14 1978047.3 1406.4307 Parameter
Estimate
ApproxStdErr
b1 0.007164405 0.00106851 b0 733.0202662 237.867098 b1 0.013 0 b0 159 0
SSE DFE MSE RMSE 72703917.749 10 7270391.8 2696.3664 Parameter
Estimate
ApproxStdErr
b1 0.0105310111 0.00088276 b0 10.25766467 6.91634685 b1 0.006 0 b0 90 0
year year
—— Deformity trend —— Fractal Dimension trend
▅▅▅ Transitional states in the growth trend
Manhattan
Linear and nonlinear fit solutions
Barcelona
Deformity R2 = 0.80 Natural logarithm fit Fractal D R2 = 0.53 Linear fit
Deformity R2 = 0.95 Linear fit Fractal D R2 = 0.83 Linear fit
Nonlinear fit solution
SSE DFE MSE RMSE 27692661.637 14 1978047.3 1406.4307 Parameter
Estimate
ApproxStdErr
b1 0.007164405 0.00106851 b0 733.0202662 237.867098 b1 0.013 0 b0 159 0
SSE DFE MSE RMSE 72703917.749 10 7270391.8 2696.3664 Parameter
Estimate
ApproxStdErr
b1 0.0105310111 0.00088276 b0 10.25766467 6.91634685 b1 0.006 0 b0 90 0
year year
—— Deformity trend —— Fractal Dimension trend
▅▅▅ Transitional states in the growth trend
Manhattan
Linear and nonlinear fit solutions
Barcelona
Deformity R2 = 0.80 Natural logarithm fit Fractal D R2 = 0.53 Linear fit
Deformity R2 = 0.95 Linear fit Fractal D R2 = 0.83 Linear fit
Nonlinear fit solution
SSE DFE MSE RMSE 27692661.637 14 1978047.3 1406.4307 Parameter
Estimate
ApproxStdErr
b1 0.007164405 0.00106851 b0 733.0202662 237.867098 b1 0.013 0 b0 159 0
SSE DFE MSE RMSE 72703917.749 10 7270391.8 2696.3664 Parameter
Estimate
ApproxStdErr
b1 0.0105310111 0.00088276 b0 10.25766467 6.91634685 b1 0.006 0 b0 90 0
year year
—— Deformity trend —— Fractal Dimension trend
▅▅▅ Transitional states in the growth trend
Manhattan
Linear and nonlinear fit solutions
Barcelona
Deformity R2 = 0.80 Natural logarithm fit Fractal D R2 = 0.53 Linear fit
Deformity R2 = 0.95 Linear fit Fractal D R2 = 0.83 Linear fit
Nonlinear fit solution
SSE DFE MSE RMSE 27692661.637 14 1978047.3 1406.4307 Parameter
Estimate
ApproxStdErr
b1 0.007164405 0.00106851 b0 733.0202662 237.867098 b1 0.013 0 b0 159 0
SSE DFE MSE RMSE 72703917.749 10 7270391.8 2696.3664 Parameter
Estimate
ApproxStdErr
b1 0.0105310111 0.00088276 b0 10.25766467 6.91634685 b1 0.006 0 b0 90 0
year year
—— Deformity trend —— Fractal Dimension trend
▅▅▅ Transitional states in the growth trend
12
urbansystems.Yet,itmightbepossibletosetforthsomeassumptionsonthemechanismsinvolved
bylookingattheelementaryprocessesofadditionanddeletion,mergenceandsubdivisionthat
leadtogrowthanddifferentiationincities.
Thetwobasicprocessesthatgoverntrendsofgrowthorshrinkingincitiesarebasedonapositive
feedbackloopthatresultsfromtheadditionofnewelementsandareinforcingfeedbackloopthat
resultsfrompruningcertainelements(AlSayedetal,2012).Inthisway,urbangrowthisaproduct
oftheelementaryprocessesofaddition,pruning,mergenceandsubdivision.Thelatterprocesses
branchfromthemainloops(Figure).
Figure4Positiveandreinforcingfeedbackloopsinthespatialnetworksofstreets.
Itissuggestedthatinperiodsofexpansion,apositivefeedbackmechanismoperatesandtakesthe
form of exponential addition of elements. New street structures are preferentially added where
there is a potential increase in the street network accessibility. At each stage of growth the
connectivity (degree) of street lines follows power law distributions, a phenomenon that is also
observedinscale-freenetworks.Theobservedhistoricalgrowthbehaviourcomesastoconfirmthis
finding. In a process that resembles preferential attachment in information networks; the
emergenceofnewpatchesofgridstructures in18thcenturyBarcelona followshighvalues in the
connectivityofstraightstreetlines.
Preferential attachment was previously suggested to be a possible model for growth in street
networks taking the topological Space Syntax representation (Volchenkov andBlanchard, 2008).
Thissuggestionwasmadeafterobservingthatcontrolvalues(degreeofchoice)fallintopowerlaw,
an observation that was also compatible with Porta et. al.’s findings (2006). The model was
presented in the formof cautious suggestion since no evidencewas explicitly presented forwhy
suchmodel –famous in information networks-would fit the case of physically constrained urban
systems.Herewefoundhistoricaltracesforsuchprocessesinboththegeometricandtopological
representationsofBarcelona’surbangrowth, lessclearly inManhattan.Theobservedphenomena
we reported indicated thataprocessofpreferentialattachmentdidnotexactly follow themodel
initially proposed in (Barabási-Albert, 1999). In the famousBarabási-Albertmodel description for
13
preferentialattachment,nodesthatarehighlyconnected(highdegree)willhavemorelikelihoodto
attach tomorenodes. Inotherwords;nodes thatare richwithconnectionswillbemore likely to
increase their connections. The presence of physical constraints in urban networks, might have
contributedtowhatourobservationshere,that isaformofpreferentialattachmentwherenodes
thatarehighlyconnecteddevelopedawholeneighbourhoodclusterintheirlocalities(AlSayedet
al,2012).
This Barabási-Albert model will probably need to be adjusted to fit with the nature of street
networks.Consideringthetopologicalaxialrepresentationofstreetsandhowitdevelopedintime
(Figure 5), we might note some patterns arising from the use of simple network measures like
connectivity (degree). These observations were reconstructed from historical map data to track
howthesystemchangedwiththeadditionofnewelements inBarcelona.Theemergenceofnew
gridstructuresseemedtocoincidewithhighvaluesofconnectivity.Ateachstageinthesystem,the
connectivityvaluesexhibitedPowerlawdistributions(Figure6).Theseobservationswouldstandas
an empirical validation for the applicability of the Barabasi-Albert model on urban networks.
However,it isimportanttoemphasiseherethatspatialnetworksincitiesarenotreallyscale-free;
since theyhavegeometric properties that are particularly crucial for them tooperate.Moreover,
there isacaponhowmanyconnectionsanaxial linemayconnecttogiven itsrelative lengthand
theintensityofthegridstructurethatneighboursit.Someadditionalconstraints-possiblyaxialline
length-maybeappliedtodistinguishthoseelementsinvacantland.
14
Figure5AreconstructionofBarcelona’sgrowthprocessconsideringthehistoricalmapsdated(at
thebottom).TheConnectivityvalues(atthetop)renderedagainthephenomenonofpreferential
attachmentinhowhighvaluesprecedetheemergenceofnewgridpatches.
Figure 6 Log-Log plot of Connectivity (degree) values for different developmental stages of
Barcelona’saxialstructurefittingwithPowerlawdistributions.
Another observed regularity in street network growth is the disappearance ofweakly connected
street structures (Pruning). Once the urban system reached its maximum boundary, another
process of reinforcing feedback took place. This mechanism was mostly evident in Manhattan,
where the filling of the central park rendered this exact area as poorly connected indicating a
process of pruning of weak local structures (Figure 7). The process of pruning was less clear in
Barcelona,wherethefillingofthecitadelareashowsaslightincreaseinlocalintegrationvaluesin
Highigher connectivity values
1806 1855 1891
15
theareaadjacenttotheoldcitycentre.Themechanismofpruningtookplaceatacertainstageof
growthandcontributedtothedemarcationofastructuredeformingthehomogeneityofthegrid.It
isimportanttoconsiderherethehistoricalcircumstancesthatledtotheformationofavoidwithin
a densely occupied urban structure in both cases. Some are explained in (Al Sayed, 2014).
Ultimately, the abstract physical form of the urban grid is a materialisation of many less easily
delineated social and economic conditions that historically shape urban regions (Griffiths, 2009;
Vaughanetal,2013).
Manhattan2008
IntegrationR500m-Range(60,100)
Manhattan–CentralParkareafilledwithgrid
IntegrationR500m-Range(60,100)
Barcelona2008
IntegrationR500m-Range(60,200)
Barcelona–citadelareafilledwithgrid
IntegrationR500m-Range(60,200)
Highervaluesofaccessibility
Figure7Apruningprocessofelementswithlowintegrationvalues(radius500m)inManhattanand
Barcelona.ThemodelsontherightareproducedbyfillingtheCentralParkvoidinManhattanand
theCitadelareainBarcelonawithagridthathasananalogousstructuretoitssurrounding.
Log Choice Rn Range (4, 6)
Integration Rn Range (3000, 5000)
Integration R500m Range (60, 100)
Cur
rent
Man
hatta
n
A d
ense
r uni
form
grid
Higher values of accessibility
Log Choice Rn Range (4, 6)
Integration Rn Range (3000, 5000)
Integration R500m Range (60, 100)
Bro
adw
ay o
mitt
ed
Cen
tral P
ark
area
fille
d w
ith g
rid
Higher values of accessibility
16
Conclusion
Thispaperpresentsempiricalmodelsofstreetnetworkgrowththatareexplanatoryofsomeofthe
mechanismsofgrowthandchangeinManhattanandBarcelona.Lookingatthegenerictrends
elucidatedinthispaper,thereseemstobewhatsuggeststhatspatialsystemsareinclinedtofollow
asimpleexponentialmodelintheirgrowththatpertaintotheinvariantgeometrictransformations
inthenetworkstructure.Theexponentialtrendwasalsomarkedinpopulationsize(Bretagnolleet.
al.,2002;Pumainet.al.,2006).Themappingmethodwasinstrumentalinlookingforpatternsof
growthanddifferentiationintheurbanregionsunderstudy.Ingeneral,thesepatternsseemtobe
drawnbycertainforcesgoverninghowthephysicalprocessesofaddition,subdivision,mergence
anddisappearanceoperateonthelevelofsegmentelementoraxialline.Theadditionand
subdivisionareproductsofaprocessofpreferentialattachmentthatissimilartothemechanism
thatgoverngrowthininformationnetworks,inthatnodes(hereaxiallines)thatarehighly
connectedaremorelikelytoattachtonewnodesinsubsequentstagesofgrowth.Themergence
anddisappearanceofstreetsareproductsofaprocessofpruning,whereweaklocalstructures
disappearinlaterstagesofgrowthtoreinforcestructuraldifferentiationwithintheurbangrid.The
physicalprocessesappeartobeassociatedwithchangesonangularIntegrationvaluesinthestreet
network.ThelocalandglobalIntegrationvaluesseemtoincreasetoacertainleveltosettleat
certainaveragevaluesoncethesystemhasgrownbeyondtheradiiofmeasurement.While
integratedcentresarepreserved,changesseemtotravelincertaindirections.Wheresuch
processesareinterruptedbyamassiveadditionofsegmentstreetelements,theyseemtorevert
backtotheirinitialrates.Theseinterruptionsmightberecognisedasperiodsoftransitions.The
transitionsmightalsobeconcurrenttoinnovationcycleswhereinformation,economicand
industrialdevelopmentstakeplaceonalargescaletransformingthelandscapeofacity(Pumain
andMoriconi,1997;Pumainet.al.,2006).Theyappeartohaveadestabilisationeffectonthe
structuralunityofthesystemandtheaggregatedistributionsofIntegrationvaluesthat
consequentlyreduceditsfitnessleveltoalognormaldistribution.
Theambitiontofindamodelthatunlockstheprocessofgrowthshouldnotbetakenwithout
raisingconcernsoverthespecificinitialconditionsthatwouldinevitablydeterminethecourseof
growthinsomecases.Thegeographicandtopologicalaffordancesofbothregionshadclearimpact
ongrowthbehaviour.InBarcelona,theinclusionofperi-urbanareasinthedistributionplots
distortedtheaggregatedistributionpatternsthatwouldotherwisesettleintoalognormal
distribution.InManhattan,theeffectofgeographywasvisibleinallsetsofanalyses.Thephysical
boundaryconstraintwasmarkedasaninflectionpointontheexponentialtrendthatasystem
followsasnewsegmentstreetelementsareadded.Whatischaracterisedbyachangeonconcavity
17
intheexponentialcurveisalsomarkedontheaveragedIntegrationtrends,andtheprobability
distributionsofvaluesofchangeonangularIntegration.Thesystemdivertsatthatpointfrom
growthtodifferentiation.Thedifferentiationprocessisprobablyaugmentedbythesystem’s
tendencytoreacttotheimpositionoftheuniformgrid,whereweaklyintegratedlocalstructures
areprunedtogiverisetoaheterogeneousstructure.
Thefindingsathandscomeastosupportthehypothesisthaturbangrowthisnotentirelyarandom
process,andthatthereseemstobesomeinherentorganisationthatgovernsspatialstructures.We
mightevengofurtherwiththattoclaimthatstreetnetworksexhibitsomeformofautonomy,in
thewaytheyadaptbacktotheiroriginalpatternsofgrowthaftercertainperiodsoftransition.
Throughtheactofsomeimplicitself-organisationmechanism,thetwourbanregionsobservedin
thispaperappearedtoretrievethecharacteristicsoftheirmacrostatefollowinglarge-scale
planninginterventions.Toexposethemechanismsthatcontributetothisbehaviour,thereisaneed
togofurtherwithouranalyticalinvestigationtoobservehowelementschangelocallyandhowthey
settleintocertainhierarchies.
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Appendix1
HistoricalstreetmapsofManhattanandBarcelona
NewYorkCity1642[Mapoverlaysstreetsasof1880]FromReportontheSocialStatisticsofCities,
CompiledbyGeorgeE.Waring,Jr.,UnitedStates.CensusOffice,PartI,1886.
NewYorkCity1661[Mapoverlaysstreetsasof1880]FromReportontheSocialStatisticsofCities,
CompiledbyGeorgeE.Waring,Jr.,UnitedStates.CensusOffice,PartI,1886.
NewYorkCity1695FromManualoftheCorporationoftheCityofNewYorkfor1852byD.T.
Valentine,1852
NewYorkCity1728[Mapoverlaysstreetsasof1880]FromReportontheSocialStatisticsofCities,
CompiledbyGeorgeE.Waring,Jr.,UnitedStates.CensusOffice,PartI,1886.
NewYorkCity1755[Mapoverlaysstreetsasof1880]FromReportontheSocialStatisticsofCities,
CompiledbyGeorgeE.Waring,Jr.,UnitedStates.CensusOffice,PartI,1886.
NewYorkCity1767FromAHistoryoftheAmericanPeople,WoodrowWilson,HarperandBrothers
Publishers,NewYorkandLondon,(c)1902,VolII
20
NewYorkCity1775AplanofthecityofNew-York&itsenvirons:toGreenwich,ontheNorthor
HudsonsRiver,andtoCrownPoint,ontheEastorSoundRiver,showingtheseveralstreets,public
buildings,docks,fort&battery,withthetrueform&courseofthecommandinggrounds,withand
withoutthetown:FromJohnMontresor,engineer;P.Andrews,sculp
NewYorkCity1782[Mapoverlaysstreetsasof1880]FromReportontheSocialStatisticsofCities,
CompiledbyGeorgeE.Waring,Jr.,UnitedStates.CensusOffice,PartI,1886.
NewYorkCity1808FromD.Longworth'smapof1808forD.T.Valentine'sManualfor1852,byG.
Hayward,lithr.
NewYorkCity1817FromThos.H.Poppleton,citysurveyor;P.Mavericksc.Newark.
TopographicalmapofthecityandcountyofNew-York,andtheadjacentcountry1836:Publishedby
J.H.Colton&Co.,No.4SpruceSt.,1836(New-York:Engraved&printedbyS.Stiles&Co.)
NewYorkCity1842"New-York"FromTanner,H.S.TheAmericanTraveller;orGuideThroughthe
UnitedStates.EighthEdition.NewYork,1842.
NewYorkCity1850FromMitchellSr.,S.A.,ANewUniversalAtlasContainingMapsofthevarious
Empires,Kingdoms,StatesandRepublicsOfTheWorld.
NewYorkCity1880[Mapoverlaysstreetsasof1880]"TheOriginalTopographyofManhattan
IslandfromtheBatteryto155thStreet"FromReportontheSocialStatisticsofCities,Compiledby
GeorgeE.Waring,Jr.,UnitedStates.CensusOffice,PartI,1886.
NewYorkCity1891FromBromley,G.W.andBromley,W.S.,AtlasofthecityofNewYork/
Manhattanislandfromactualsurveysandofficialplans.Philadelphia:G.W.Bromley&Co.,1891.
[web:rumsey]
NewYorkCity(LowerManhattan)1920"ChiefPointsofInterestinLowerManhattan"From
AutomobileBlueBook1920
NewYorkCity(UpperManhattan)1920"ChiefPointsofInterestinUpperManhattan"From
AutomobileBlueBook1920
Barcelona1200-1300FromBensch,S.P.(1995)BarcelonaandItsRulers,1096-1291.Cambridge\
NewYork\Melbourne:CambridgeUniversityPress.pp.27
Barcelona1260FromPlantadelaciutatromanaambl’hipòtesidesituaciódelselements
distribuïdorsd’aigua,FromsegonsC.MiróiH.Orengo
Barcelona1494-1495FromHieronymusMünzeraccount
Barcelona1698Frenchmapfrom1698ofthecityofBarcelonawithindicationsforasiege.Title:
PlanduSiègedelavilledeBarcelonne:AveclaCartedelacôtedelaMerdepuisleCapdeCervera
jusqu'auxenvironsdeLlobregat.
21
Barcelona1706PlandelaVilledeBarceloneetChateaudeMontIuy(PlanoftheCityofBarcelona
andCastleofMontjuic)FromAnnaBeeck
Barcelona1711FromNicholasDeFer,publishedin1711inParis.
Barcelona1745FromI.BASIRE1745.preparedforMrTindal’scontinuationofMrRapin’sHistoryof
England.
Barcelona1751FromBodenehr'sCuriosesStaatsundKriegsTheatrum.
Barcelona1806ThewalledcityofBarcelonaandtheCitadel1806.PlanoftheCityandPortof
Barcelona.From:"Voyagedel'Espagne"byAlexandredeLaborde.-Paris,1806-1820arroworiented
withthenorthtothenortheastofthemap.
Barcelona1851FromGHeck
Barcelona1860publishedbyJ&CWalkerfortheBritishAdmiralty.
Barcelona1862FromCoronel,Teniente-CoroneldeIngenierosD.FranciscoCoello.AidedbyD.
PascualMadozauthorofthestatisticsandhistoricalnotesMadrid1862.
Barcelona1890PlanodeBarcelonaysusAlrededoresen1890.FromD.J.M.Serra.byGerona;S.
bytheMediterraneanSea;S.W.byTarragona;andW.andN.W.byLrida.
Barcelona1914FromPlànolgeneraldeBarcelona.LafebretifoideaBarcelona:gràficdel’epidèmia
del’any1914
Barcelona1929FromWagner&Debes,Leipzig,
Barcelona1943From[http://www.lib.utexas.edu/maps/ams/spain_city_plans/txu-pclmaps-oclc-
6478271-barcelona.jpg]
Barcelona1970FromtheDavidWilliamscollection
BarcelonaFieldStudiesCentre.Source:DeBarcinoaBarcelona1992BarcelonaContemporary
CultureCentre[online]Availablefrom<http://geographyfieldwork.com/Poll2.htm>[Dateaccessed:
14October 2012]
Appendix2
BoxCountingmethod
ThefractalDdimensionmightbeobtainedusingthefollowingmethod.Asquaremeshofvarious
sizessislaidoveranimage(containingtheobjectthatwewanttocomputeitsfractaldimension).
ThenumberofmeshboxesN(s)thatcontainpartoftheimagearecounted.Thefractal(box)
dimensionDisgivenbytheslopeofthelinearportionofalog(N(s))vslog(1/s)graph.
log 𝑁 𝑠 = 𝐷 log(1𝑠) (2)
22
Sincethereisnopreferredoriginfortheboxeswithrespecttothepixelsintheimage,multiple
measuresN(s)canbecomputedfordifferentmeshorigins.ThegraphedvalueofN(s)isusuallythe
averageofN(s)fromdifferentmeshorigins.
Therangeofboxsizesusedhereare;2,3,4,6,8,12,16,32,64.