Trends of growth and differentiation in street networks

1

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

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500m

Dis

trib

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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

-----

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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

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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

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2000

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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.