space syntax 8242 1

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PAPERREF#8242

Proceedings:EighthInternationalSpaceSyntaxSymposium

EditedbyM.Greene,J.ReyesandA.Castro.SantiagodeChile:PUC,2012.

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ORDER,STRUCTUREANDDISORDERINSPACESYNTAXAND

LINKOGRAPHY:intelligibility,entropy,andcomplexity

measures

AUTHOR: TamerELKHOULY

BartlettSchoolofGraduateStudies,UCLFacultyoftheBuiltEnvironment,UniversityCollegeLondon

UCL,UnitedKingdom

email:[email protected]

AlanPENN

BartlettSchoolofGraduateStudies,UCLFacultyoftheBuiltEnvironment,UniversityCollegeLondon

UCL,UnitedKingdom

email:[email protected]

KEYWORDS: Linkography,Order,Disorder,Stringsofinformation,Intelligibility,Complexity,Entropy

THEME: MethodologicalDevelopmentandModeling

AbstractTherehasbeengreat interest in theuseof linkographies todescribe theevents that takeplace indesign

processeswiththeaimofunderstandingwhencreativitytakesplaceandtheconditionsunderwhichcreative

moments emerge in the design. Linkography is a directed graph network and because of this it gives

resemblancetothetypesoflargecomplexgraphsthatareused inthespacesyntaxcommunitytodescribe

urban systems. In thispaper,we investigate the applications of certainmeasures that comefrom space

syntaxanalyses

of

urban

graphs

to

look

at

linkography

systems.

One

hypothesis

is

that

complexity

is

created

indifferentscalesinthegraphsystemfromthelocalsubgraphtothewholesystem.Themethodofanalysis

illustratestheunderlyingstateofanysystem. Integration,complexityandentropyvaluesaremeasuredat

each individual node in the system to arrive at a better understanding on the rules that frame the

relationshipsbetweenthepartsandthewhole.

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SantiagodeChile:PUC,2012.

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INTRODUCTION

A linkographyisarepresentationoftheseriesofeventsthatcanbeobservedtooccurandcanbeusedto

helpanalyseprocessesofcreativityduringadesignsession.Thedifferencebetweenlinkographyandspatial

system is that linkography has a time factor. A linkograph is constructed from nodes that represent each

segment in the design process (according to time) and is based on parsing the dependency relationships

betweenthosenodes.

Becauseitisarepresentationthattracestheassociationsofeverysingleutterance,thedesignprocesscan

be looked at in terms of a linkographic pattern that displays the structure of the design reasoning. The

venuesofdense interrelations(clustersofthedesignutterances)areovertlyhighlightedonthegraphand

canbefurtherinterpretedthroughtheemergingartefactsalongtheprocess.

Thelinkographysystem ishypothesisedtodeliveravariationofcomplexitydegreesondifferentoccasions.

Theaimistouncoverthesignificanteventsthatmightbeassociatedwithcreative insightsandinspectthe

artefacts that are formulated at such events. Linkography and urban systems deal with multilevel

complexities,theoverallgoaloftheproposedanalyticalmethod istorevealtherelationshipbetweenthe

parts(subnetworks)thatconstitutethesystemandthewhole.

The relationship between the subsystems or the partial assemblies is inspected looked at from two

perspectives, information theory and entropy theory, to see whether a conflict occurs between

uncoordinated suborders despite being orderly structured (Arnheim, 1971; Laing, 1965) or whether an

order systemunderliesanentiredisorderstate (Planck,1969)anentitythat isdependenton arandom

dispersionoflimitedsuborders(Arnheim,1971;Kuntz,1968).

Acomputationalmodelisproposedthatcoversthedependencyrelationshipsoccurringbetweennodes,all

of which appear to have a sophisticated group of relations. The algorithm used is inspired by the Tcode

stringmeasuredevelopedbyTitchener(1998a;1998b;1998c;2004).

1.APOINTOFDEPARTURE

A gridiron urban system is perceived as a highly organised structure if it delivers different chances to

navigate fromoneplaceto another. It ishighly intelligible in this circumstance,butto some extent it can

becomeconfusing.Inaverysymmetricalandidenticalsystem,theexplorerhasequalchancestomovefrom

one point in the system to another and might get lost. Since intelligibility is the correlation between

connectivity and integration, hence the same correlation value is constituted for any element in this

particularsystem.

In reality, no system is perfectly set up as a 100% identical gridiron. Every city has some sense of

differentiation that adds to the structure and provides the capacity to grasp the relation between the

whole and the parts. The example of two forests, naturalspontaneous and farmedgrid, reflects two

differentstates. Inthefirstcase,treesarenotalignedandthedistribution ischaotic,while inthesecond,

treesarestrictlyplantedalongstraight linesandthearrangement issimilareverywhere inthenetwork.In

bothcases,systemsdisorientatetheexplorer fromapropernavigation.Yetthehighlyordered forestand

thedisorderedonearebothconsideredextremeexamplesintermsofintelligibility.

Whatdeservesattentionishowweconstructasystem;acityfor instance.Hanson(1989)haspointedthat

order might be misleading about its function and that it could be a manifestation of another underlying

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state.Hence,the importanceofdistinguishingthiskindofrelationship iscrucialtorevealtherealstateat

each stage of a multilevelcomplex system.What wemean is that something might occur onthe system

midwaybetweentotalchaosandtotalorder,acertainpointwhereitstartstobehavedifferentlyfromthe

precedingstate(s).

Thedemonstrationofthegridironorder,despitebeingasingularityinintelligibilityterms,isasunintelligible

as the total chaos state. Both systems deliver lack of intelligibility for the thinking subject. Order, in this

particular case, isjust the same as complete disorder in delivering a lack of intelligibility. However, if we

imposeadifferentiationonthegridironbyaddingsomediagonalsandroutes,thewholestructurehasnot

drastically changed but its intelligibility moves from one state to another (the system becomes more

intelligibleotherwise).

Infact,workingwithsystemsthathavemultilevelcomplexityondifferentscales iscommon inurbanand

linkography systems. One view is that there is a clear order and that the structure of the system can be

easilygraspedandunderstood.Theotherview isthatthere isnorule inthecomplexworldandthat it is

actuallyjust random. The paradox is that if it is truly random is there a simple way to describe it? Can a

complexworldbereducedtoasinglevalue?

Thispaperproposesahypothesisthatinmultilevelcomplexsystemshighorderlinesstendstobecomeless

complexoverall,andthereforeahighlylinkednodedeliversfewchoicesandprobabilities.Thealternativeto

inspectingthesystemisthereforetomeasuretheprobabilityforeachnodeandcomplexityateachlevel(at

every subnetwork) included within the system. In doing so, we propose the adoption of strings of

informationtocodeprobabilitiesateachpointandcomputetheinformationcontentfromit.Thepractical

aimsofusingthismethodaretwofold.

First,sinceallthe inspectedsubnetworkshavethesamesubgraphsizeeffect,themeasuresofstringsat

eachpointinthesystemarealreadyrelativisedandeligibleforcomparison.Thisisbecausetheinformation

isextractedforallthepossiblerelationsthatcouldbemadefromanypointinthesystemtotheothers(the

subgraphsizealwaysequalsn1).

Second,integrationvaluesarealsorelativisedtothesubgraphsize.Thus,integration,complexity,rateand

contentof informationarerelativisedparametersthatwe lookat inordertospecifytherelationbetween

thepartsconstitutingthewhole.

2.ENTROPYANDINFORMATION

Space syntax and design process are multiscaled complex contexts. The information content at different

scalesreflectsthecomplexityateachlevelinthesystem.Intheproposedmethod,thesystemcanbereadin

two ways. The first looks at the probability of choice at any item, point, or node, while the second

looksattherateofinformationmeasuredforasequenceofitems.

The methods correspond to entropy theory and information theory respectively. But while entropy is

concernedwithsetsofindividualitems,informationisconcernedwiththeindividualsequenceofthose

items. The entropy theory asserts that a set should be treated as a microstate; the microstates

constitutethecomplexionsoftheoverallprocess.1 Atthispoint,themainobjectof inquiry in information

1 Arnheim(1971)describedthemicrostateintheprincipleofentropytheoryas:theparticularcharacterofanymicrostatedoesnot

matter;itsstructuraluniqueness,orderlinessordisorderlinessdoesnotcount.Whatdoesmatteristhetotalityoftheseinnumerable

complexionsaddinguptoaglobalmacrostateofthewholeprocess.Itisnotconcernedwiththeprobabilityofsuccessioninaseriesof

itemsbutwiththeoveralldistributionofkindsofitemsinagivenarrangement.

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theoryistoinvestigatetheprobabilityofoccurrencebyestablishingthenumberofpossiblesequences.The

sequence of items is not covered in entropy theory but is necessary for information theory. Table 1

illustratesthedifferencesthatdistinguishthetwoperspectives.

Entropytheory Informationtheory

Structure Itemsconstitutethemaincharacteristicsofthe

structure

Structuremeansnothingisbetterthanthosecertainsequencesofitemsthatcanbeexpectedtooccur

Central

points

Concernedwithsetsofindividualitems Isabouttheoveralldistributionofkindsofitemsina

givenarrangement

Focusedontheindividualsequenceofitems Isaboutsequencesandarrangementsofitem.

Themoreremotethearrangementofsetsisfromarandomdistribution,theloweritsentropy,andthe

higheritsorderrepresentation

Thelesspredictablethesequence,themoreinformationthesequencewillyield,andthemoreremoteits

representationfromorder

Example

Arandomiseddistributionwillbecalledbytheentropytheoristhighlyprobableandthereforeofloworder

becauseinnumerabledistributionsofthiskindcanoccur

Ahighlyrandomisedsequencewillbesaidtocarrymuchinformationbytheinformationtheoristbecause

informationinthissenseisconcernedwiththeprobabilityofthisparticularsequence

Application

Forexample,KanandGeros(2005a;2007;2008)estimationmethodtoacquireentropyfrom

linkography

Forexample,Turners(2007)adoptionofShannonsformulatoestimateentropyforurbansystemswith

Depthmap

Forexample,Titcheneretal.s(2005)computationofstringsofinformation

Forexample,Brettels(2006)adoptionofTitcheners(2004)tcodemeasurestoestimateentropyfor

navigationroutes

Table1:Thedifferencesthatdistinguishentropytheoryandinformationtheory

An observer would findthat the most highly orderedsystem provides maximum informationcontent and

thusisoppositetoprobabilisticentropysincethepredictionisveryhigh.Iftotaldisorderprovidesmaximum

informationaswell,thenmaximumorderisconveyedbymaximumdisorder(Arnheim,1971).Howeverthe

distinction can be made through a parameter that measures the underlying system of any order. Since

information isacrudemeasurethatconfirmsaclear increase inregularityoverall,extremeregularityand

apparentsimilarityarelikelytodeliveraverylowprobabilityvalue.

Entropygrowswiththeprobabilityofastateofaffairswhile informationdoestheoppositeand increases

withtheimprobability.Thelesslikelyaneventistohappen,themoreinformationitsoccurrencerepresents.

Theleastpredictablesequenceofeventswillcarrythemaximuminformation.Hence,thispaperfocuseson

howentropycouldbeestimated formultilevelsystems inawaythatviewstherelationshipbetweenthe

nature of complexion between the partial assemblies that are made at each point and the whole. The

proposed method therefore adopts entropy and complexity as independent measures to assess complex

systemssuchas linkography. However, it should benotedthat the structure state of any systemneeds a

variationofcharacteristicsinordertoconstructanintelligiblesystem.2 Thenextsectionreviewsmethodsto

estimateentropyandintroducesthecomputationalmethodofstringsofinformation.

2 Referringbacktotheexampleofthegridironsystem,allelementsdeliverthesamecorrelationvaluebetweenconnectivityand

integration,howeveranyimposeddifferentiationonthegridironcausevariationsonintelligibility,andthensystemchangesfroma

statetoanother.

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3.ENTROPYOFSPATIALSYSTEMSANDLINKOGRAPHY

The estimation of entropy for spatial systems is based on the frequency distribution of the point depths

(Turner, 2007). The point depth entropy of a location, si, is expressed by utilising Shannons formula of

uncertaintyasshownintheequation:

dma x

PointDepthEntropyforspatialsystem si = Pd log2Pd (I)d=1

WheredmaxisthemaximumdepthfromvertexviandPdisthefrequencyofpointdepthdfromthevertex

Estimatingpointdepthentropyinthiswayshowshoworderlyaspatialsystemisstructuredfromacertain

location.Themethodisafunctionalequationbasedonmeandepth.InDepthmap,theinformationfroma

point is calculated with respect to the expected frequency of locations at each depth. Turner (2007)

explained that the expected frequency is based on the probability of events occurring depending on a

single variable,themeandepthofthejgraph.The benefitofcalculatingentropyor information fromapoint in space syntax pertains to how easy it is to traverse to a certain depth within the system. Low

disorderiseasy;highdisorderishard.

In linkography,withreferralto(Geroetal.;2011,KanandGero,2005b;2005c;2007;2008;2009a;2011b;

andKanetal.;2006;2007),Shannonstheoryofinformation(1949)isadoptedtoinspecttheoccurrenceof

dependency relationships between utterances.3 This gives two possible choices to code the system:

linkedandunlinked(oronandoff).Theformulausedis:

ShannonEntropyforLinkography H = (plinked.log2plinked)+(punlinked.log2punlinked (II)

Kan and Geros method looks at the overall distribution of sets (items of relations) regardless of the

sequenceofoccurrenceofelementsconstitutingthelinkographyaccordingtotime.TheexampleinFigure

1emphasisesthatthedifferencesbetweentwo linkographicpatternsarenotconsidered intheestimation

processofentropy.Thisisowingtothesummationstepprocessedoverthewholenetworkforeachof

thetwoprobabilities,linkedandunlinked,regardlessofthepositionofitemsinthesystemthatshould

precedetheestimation.4

3 Goldschmidt(1992)definedadesignmoveorstepinthefollowingterms:amoveisanactofreasoningthatpresentsacoherent

propositionpertainingtoanentitythatisbeingdesigned.Goldschmidt(1995)alsostated:astep,anact,oranoperation,which

transformsthedesignsituationrelativetothestateinwhichitwaspriortothatmove.Seealso:Goldschmidt(1990).

4 Rememberthatlinkographyisadirectedgraphaccordingtotimefactor.

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Bothgraphshavethesameentropyvaluedespitethecleardifferenceofarrangementsineachsystem.This

isbecausetheequationisbasedonsummingthevaluesofeachprobabilitywithoutconsideringtheposition

of each in the existing pattern. The next section provides a synopsis on intelligibility in space syntax. It

illustrates a brief from a previous study (Brettel 2006) that combined string measures with integration

values on spatial networks with distinctive configurations, investigating the connectivity between nodes

throughnavigationinvarioussamples.

4.THECOMPUTATIONOFSTRINGSOFINFORMATION

AninclinationtowardsthehypothesisisdeliveredthroughoutBrettels(2006)study,whichinvestigatedhow

order, structure, and disorder of street layouts are perceived when navigating through an urban

environment.Shestatedthatanorderedenvironmenttendstobemoreintelligiblewhenbrokenupbyan

irregularityoccasionally.Inourstudy,weask:Underwhichcircumstancesdoesthesystemchangefromone

statetoanother?ButmorespecificallyonBrettel,weask:Arehighlyintelligiblespatialsystemspredictable

tonavigatethrough?andDoessimpletraversethroughurbanfabricdeliverlesscomplexstructure?5

The string of information measures to deal with event structure was introduced in Brettels study6 in

ordertocomputebarcodesofeventsequencesextractedfromnavigationroutes,inadditiontosyntactical

analysis.Thestringmeasureswereexpectedtorelatetotheperceivedorderalongaroute.Theentropyof

5 Inotherwords,ifthemechanismofaccessfromonepointtoanotherissimple,doesthesynthesisformofitsroutedeliverlow

complexity?

6 Aneventisdefinedasasegmentoftimeatagivenlocationthatisperceivedbyanobservertohaveabeginningandanend

(TverskyandZacks,2001).

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each routes string was interpreted as the probability of the uncertainty that a route provides for the

traveller,andwasexpectedtorelatetotheperceivedstructurealongroutes.7 Whenaroutehasveryfew

turns, the probability of choices is too low (for example, gridiron patterns such as New York and San

Francisco).However,entropydelivershighvalues(relatively)withcomplexpatternswhentherouteconsists

of some turns and deviations within it (for example, composite fabrics such as London and Rome).

Moreover,theisovistfieldsowedthedifferentiationofvisualcatchmentareasbetweentheanalysedcities

not only according to the delineation in the route but also because of picking up structurally different

catchmentareas,especiallyintheirregularpatterns.

AccordingtoBrettelsanalyses,thecomputationprocessofstringscoulddelivermeaningfulcorrelationsfor

theperceivedroute.Nevertheless,theassumptionthatorderlinessislikelytobemorerelatedtocomplexity

measure and structure to entropy could not be proven in her study, possibly due to limits of the survey

setup.8

So are we measuringIntelligibilityversuscomplexityorintegrationversuscomplexity? The central

pointofattention istorealisethat intelligibility isasystemproperty;thecorrelationbetweenconnectivity

(C)andintegration(I).Complexity(CT)ofgraphsisalsoasystempropertythatreflectshowmanystepsare

requiredtoconstructastringofinformationforthesystem(orsubsystem).Consequently,valuesofthetwo

parameterscanbecomparedandcorrelatedtogether.Thesubgraphateachnodeisalsoasubsystemand

thesamemeasurescanbeusedtoinspectthecharacteristicwithinthewhole.

4.1TheTcodeMeasure

TheapplicationoftheTcodestringcomputationmethodisbasedonthedeterministicinformationtheory

that was developed by Titchener (1998a; 1998b; 1998c; 2004). In this method, an algorithmic process is

appliedtosetsofinformationtocomputethestringmeasures,denotedasTcomplexityandTentropy

(see also: Titchener, 2004; Titchener et al., 2005; Speidel et al., 2006; Speidel, 2008). The string signifies

varioustypesofinformationencodedintosymbols.

Ifthestring comprisesarepeatingsequenceofonesymbol only (oneattribute),thenentropydeclines to

zerovalueandthecomplexitystructureofthestringget lower,e.g.OOOOOOOOOOOOO,but ifastring is

composed of two or more symbols then the probability of appearance gets higher, e.g.

LROoRRoRLOoLLLORLOooOoR. This means the complexity of string increases according to the size of the

symbolsandthecomposition.

Thesizeofstringisacrucialfactorsince longerstringsgivemoreaccuratemeasurementsthanshortones.

The complexity of string depends on the number of production steps that are required to construct this

string (Titchener, 2004). Example 2 gives a tape of 100 digital codes that is spontaneously composed by

typicallyduplicatingonesymbolcodetoanotherone,byprocessingtheTcodemeasure:

7 Theprobabilityofchoicesthatcouldbemadeatdecisionpointsfordirectionalturns.Accordingly,entropydescribeshowmuch

informationisthereinasignalorevent.

8 Thisresultmaybelimitedowingtothesmallsizeofsamplesandshortstrings.

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Figure2:Avarietyofmeanstoillustrateasystem

network:Linkography, Archiography,andMarkov

Chain.

Figure3:Twodifferentrepresentationsforthesamelarge

system:adesignprotocolforonehourdesignsessionconsisting

of453nodes.

Aboveisthelinkographypatternandbelowisdisplayeduponthe

connectivitystrengthpernodes.

Archiograph

MarkovChain

Linkography

Thereare

avariety

of

means

to

illustrate

anetwork;

see

Figures

2and

3for

some

examples.

The

scope

of

this paper is not concerned with multiple representations to illustrate the system, but rather intends to

understandtheconstitutingforcethatattunesthecomponentsof it(suchaswhat isbeyondthe linksand

relationsbetweenthenodes).

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Before embarking on an analysis of the distribution of integration in each of the individual nodes in the

system,webeginwithanumberofcommonfeaturesofthesetof linkographies,whichgivesome ideaof

the nature of the processes envisaged. After a preliminary study on some samples of linkography, the

concludedpointsaretwofold:

First, since the total number of links in any system of size n is (n1), then the size of any nodes possible

relations equals (n1) as well. This means that at any node, the sheer number of links in the subgraph

createdfromthisnodetotheothersinthesystemhasthesamesizeeffectwitheverynode.Accordingly,all

the measures are relativised at every level in the system before embarking on comparisons. A second

featurethatdifferentiatesbetweensystemsisthevarieddistributionof links.Thisshouldbeconsidered in

theestimationprocessofstringsofinformationtoincludethesequencesofsetsinourinterpretationrather

thanviewingthesystematthenodelevelonly.

Thelinkographycontainsastructuralhierarchy:thefirstlevelstartswiththe"nodes"thataggregatetoform

the "network" or "subsystem". Nodes and networkstogether construct the linkography pattern. It might

happeninsomecasesthatnetworks(subgraphs)donotintersecttogetherbecausethetrainofthoughtsin

thisdesignvenueisdisconnectedandthechunksofideasareunrelated.However,inmostcasesnetworks

intersectinoneormorenodes.Thismeansthedesignthoughtsarestructurallyinterrelatedandbuiltup.In

thissense,linkographyhasdifferentpatternsandconfigurations:highlyordered/fullysaturated,structured,

or disordered. There are some other configurations beyond these intrinsic types such as the mechanistic

"sawtooth"patternthatreflectsahighlyorderedandsystematic(repetitive)process,orthe"fullysparse"

disconnectedonethatresemblesatotally"unrelated"discourse,andsoforth.

4.2ProcessingtheSystemasMultipleSubgraphs

Any system can be transferred into strings of information by coding the dependency relations between

nodes.Inthecaseoflinkography,abinarydigitformatisproposed:1forlinkedrelationshipsand0for

unlinked.Thepositionofeachsymbol inthestringrefersto itssequence inthepatternaccordingtotime

and thus the distribution of the dependency relationships is included while constructing the string. This

sectionsuggests a method to studythe subgraph atany nodeandextract the strings from it in order to

computetheTcodemeasureonthelinkographypatterns.

Method:ComputingtheTcodemeasureonthesubgraphforeachnode

Theestimationprocess isbasedontheconcatenationofbackandfore linkstogetherforeachnode in

thesystem(seeFigure5).Despitethesubgraphs(concatenationof linkspernode)havingequalsizesat

every

node,

Tcomplexity

and

Tentropy

values

fluctuate

along

the

linkography.

An

example

of

how

to

extractastringofinformationforacertainnodessubgraphisshowninFigure6.

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Figure5:ProcessingdynamicTcomplexityoncompletelinkographybyconcatenatingthebackandforelinkstogetherpernode

Example3:

AccordingtoFigure5,thesubgraphcreatedatnodeno.15consistsofthefollowingrelations:

Figure6:Exampleofextractingthestringofinformationpersubgraph

4.3Intelligibility,ComplexityandEntropy

Intelligibilityandcomplexityarepropertiesofsystem.Foragraphthatconsistsof100nodes,eachnodewill

havetwovalues:(1)intelligibility,whichisthecorrelationbetweentwovalues:connectivityandintegration;

and (2) complexity, which is measured for the subgraph of relations at this particular node. Both

measureshavesizeeffects.For intelligibility,whereasystemsizen(n

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Thesystem isintelligible ifthecorrelationvalue ismorethan0.5,and"unintelligible" ifthevalue is less

than0.5).

Stringmeasuressuchascomplexity,informationcontentandentropyalsohavesizeeffects.Forastringof

size n, (n20),theTcomplexitywearelookingatisasubgraphofthewholelinkography,namelythosedirectly

connectedtonodeweareestimating.

Subsequently, the Tcomplexity and Tentropy measures are comparable to the integration value at that

samenode.Hence,thehighlyconnectednodesatanysystemcouldbecorrelatedtothestringmeasuresat

the same node in order to investigate the proposed hypothesis. According to Brettel (2006), that

intelligibilityissignifiedthroughoutorderlysystems.

Asaruleofathumb,theshortestlinebetweentwopoints isastraight linethathasafirstordersynthesis

form.Apiazzaishighlyaccessiblefromallitssurroundingpoints(areas),theproposedpathofnavigationis

clearandeasiertotravel,andthustheexpectedness ishighandthecomplexity is low.Aculdesachasa

verylowintegrationvalueinthesystemandnotmanyoptionsexisttoapproachitonlyoneaccesspoint.Thatmakesitverycomplextoreach.

Giving an example of a particular spatial structure, Figure 7 illustrates two hypothetical network systems

that are connected via only one node (resembling a bridge between two riverbanks), the real relative

asymmetry value of this single node equals zero. Since integration and real relative asymmetry (RRA) are

inverselycorrelated,thismeansthatthemostintegratedpointinthesystemisthehighlylinkednode.Other

nodesineachsideareequivalentinintegrationandRRAvalues(seeFigure7).

Thisnetworkwillstillberepresentedinseveralways.Itisobvioustoinferthatbothnetworksidesarehighly

ordered. However the string of information for each node in the system contains repeated symbols that

indicateonlyonepossibleoption(symbol)ofinterconnectivityinsideeachside.Thiswillsignificantlyaffect

thecomputedbarcodemeasuresforeachnodeinthesystem.Forexample,takingnode9:

String proc essed: 11111111111111111111

Length (c ha rs): 20

T-CO MPLEXITY T-INFORMATION T-ENTROPY

4.32 (ta ugs) 5.4 (nat s) 0.274 (nat s/ ch ar)

4.32 (ta ug s) 7.9 (b its) 0.396 (b its/ c ha r)

(N.B.Accuracyfor information and entropy is limitedfor short strings, due to approximation of bound by the logarithmic integral

function,li().

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Example4:

Figure7:RRAvaluesoftwohypotheticalnetworksystems(connectedthroughasinglenoderesemblingabridgebetweentwo

riverbanks)

5.APPLICATIONS:EXAMPLESANDCASEMATERIAL

5.1HypotheticalCasesofShortStrings

Thefollowinghypotheticalcasesaregeneratedtoinspecttherelationbetweenhighlyconnectednodesand

the Tcomplexity and Tentropy measures. The patterns vary between orderliness and structured

configurations. The RRA value is utilised to search for the most integrated node(s) in each pattern and

processthecomparisonwiththestringmeasures.

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15 13 11 9

1 3 5 7

2 4 6 8

16 14 12 10

91 3 5 7 11 13

102 4 6 8 12 14

9

1 3 5 7

2 4 6 8

3

5

4

6

7

2

13

5

4

6

7

21

8

91

2

3

54

6

7

89

Figure8:ValuesofRRAandStringofInformationforHypotheticalCasesofSmallSystems

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Accordingtotheresultsofshortstrings,thefollowingpointscanbeconcludedatthispreliminarystage:

1. The high certainty of prediction in some networks might deliver only one choice (100% choice); thusentropy equals zero if applying Shannons equation, and Tentropy decreases if applying Tcode

algorithms.

2. First, the total number of relations in any system of size n is n(n1)/2. However, the size of anysubgraphstringequals(n1).(SeeFigure6)

3. DespitethedifferencesbetweentheRRAvaluesinanysystem,itmighthappenthatallnodeshavethesamestringmeasuressinceallhavethesamepercentagechoices(numberoflinks).

4. ThefallofTcomplexityandTentropy indiceswiththeriseofRRA inthecaseslookedatmisleadsourhypothesisandcausesdisruptiontothecorrelationvalues.Thereason forthis isthe lackofaccuracy

experiencedwithshortstringsofinformation(lessthan20codes).

5. Either Shannon entropy or the deterministic entropy is inversely proportional with RRA, butsinceintegrationequals(1/RRA),thequestionarisesofwhetherthisconfusioncomesaboutbecauseof

inaccuratecomputationofshortstrings,ormighttherebeanotherparameterthathasitseffectonboth

measures?

The application of Tcomplexity and Tentropy is tricky in this sense. Two points can be made from our

experienceofprocessingthecomputationmethod:(1)Thepositionofnodeswithinthesystemdetermines

thesynthesis(structureofsymbols)oftheextractedstringsincetheconnections(links)thatcouldbemade

fromacertainnodetotheother(s)arebasedonthechoicesofroutes/links;(2)Sinceeachnode'sforelink

is another point's backlink within the system, then an introduction to some redundancy in this way

should be considered inthe estimation processtoavoid replications (incase of concatenating the overall

strings intoone forthewholesystem).Toreconcilethese findings,anotherseriesof longstringcasesare

analysedinthenextsection.

5.2HypotheticalCasesonLargeSystems

Thecasestudiesareextendedtoincludetheanalysesofeightexamplesoflongerlengthinordertofurther

test the hypothesis and to overcome the inaccuracy experienced with short strings. These hypothetical

systems are divided into two categories: modular order and structural, where the former is known by its

repetitive andrhythmic patterns andthe latter is distinguishedby itsvariationofchoices. Syntactical and

string measurements are applied to study the degree of correlation between integration and dynamic

TcomplexityanddynamicTentropy.9 SeeFigures9aand9b.

Table2showsthecorrelationvaluesforallthehypotheticalcases.AccordingtoFigure10(drawnfromthe

results of Table 2), a strong inverse correlation between integration and Tcomplexity and Tentropy is

proved in all the ordered cases. Figure 9a shows the highlighted nodes in each case. The shallower the

system(e.g.case4),thehigherthedegreeofcorrelationbetweenintegrationandTcomplexity.Thedenser

the system (e.g. case 3), the lower the degree of correlation between integration and Tcomplexity.

However, in Figure 9b, only one case delivers a high correlation between integration and Tcomplexity,

reaching0.55 incase3.This lackofevidence isduetothe lowdegreeofdiversification inthestructureof

thesystem(thepatternisshallow)thatwasnotthecaseintheotherpatterns.TcomplexityandTentropy

areuncorrelatedalongorderedandstructuredcases.

9 Thetermdynamicentropy,introducedbyGeroetal.(2011)toindicatethateachnodeinthesystemhasitsownentropicmeasure

andthereforethevaluesfluctuatealongthelinkography.Seealso:KanandGero(2011a;2011b).

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Integration

T-entro py

T-c omp lexity

Integration

T-ent ropy

T-co mplexity

Integration

T-ent ropy

T-co mplexity

Integration

T-entro py

T-c omp lexity

Case2:OrderForm Three

NetworksSystemwithFourPivotal

Nodes

Case1:OrderForm

TwoNetworksSystemwithOne

PivotalNode

Case4:OrderedLinkography

SystemComprisingSeven

NetworksSixPivotalNodes

Case3:OrderForm Three

NetworksSystemwithTwelve

PivotalNodes

Figure9a:TherelationshipbetweenintegrationvaluesandTcomplexityandTentropyinOrderedLinkographysystem

VariablesofCorrelation ModularOrderSystems StructuredSystems

Case1 Case2 Case3 Case4 Case1 Case2 Case3 Case4

Integration:Tcomplexity 0.76 0.22 0.86 0.39 0.44 0.21 0.55 0.01

Integration:Tentropy 0.17 0.13 0.022 0.04 0.21 0.24 0.28 0.09

Tcomplexity:Tentropy 0.25 0.07 0.01 0.20 0.25 0.07 0.24 0.18

Table2:Valuesofcorrelationforthehypotheticalsystems

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Integration

T-entro py

T-c omp lexity

Integration

T-entro py

T-c omp lexity

Integration

T-c omp lexity

T-entro py

Integration

T-c omp lexity

T-entro py

Case1:LowDenseStructured

LinkographySystem;

ComprisingFiveNetworksSix

PivotalNodes

Case2:SemiDenseStructured

LinkographySystemComprising

FiveNetworksSixPivotal

Nodes

Case3:StructureForm Diffused

NetworksLinkographySystem

Case4:StructureForm Highly

denseNetworkLinkography

System

Figure9b:TheRelationshipBetweenIntegrationValuesandTcomplexityandTentropyinStructuredLinkographySystem

Integration:Tcomplexity

Integration:Tentropy

Figure10:TheCorrelationValuesOfIntegration:TcomplexityandIntegration:Tentropy

Case2

Order

Case4

Order

Case1

Structure

Case2 Case3 Case4Case1

Order

Case3

Order

AccordingtoFigure10(drawnfromtheresultsofTable2),astronginversecorrelationbetweenintegration

andTcomplexityandTentropyisprovedinalltheorderedcases.AccordingtoTable2,Figure10illustrates

the correlation values for each case study. It is apparent that some values are negative: negative

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correlation.Thismeansthatinarelationshipbetweenthetwovariablesonevariableincreasesastheother

decreasesandvice versa.Aperfectnegativecorrelationmeansthatthe relationshipthatappearsto exist

between two variables is highly negative (might reach 1) of the time.10

Nevertheless, the inverse

correlationbetweenTcomplexityandTentropybringsoutanotherpointtotest.Itishypothesisedthatthe

more complex a string (the variety of symbols), the higher the probability of uncertainty. In short, would

entropy increase with higher complexity measures? How would the hypothesis of a converse correlation

withintegrationbeaffected?

5.3ApplicationToRealLinkographies

Figure11presentstwodifferent linkographiesbasedonrealdesignprocesses.Themost integratednodes

are identifiedandcorrelatedwithTcomplexityandTentropy inadditiontotwoothergraphparameters,

betweenness and closeness centrality. The values of correlation are listed in Table 3. Both systems

depictthefollowingoutcomes:

1. Asignificantcorrelationbetweenintegrationandclosenesscentrality.2. AsignificantcorrelationbetweenTcomplexityandTentropyparticularlyprovestheearlierresultthat

shortstringscomputationsareinaccurateandrequiretobeinspectedthroughlargesystems.

3. Adirectcorrelationbetweenintegrationandclosenesscentrality.4. AninversecorrelationbetweenintegrationandeachofTcomplexity,Tentropyandbetweenness.

VariablesofCorrelation Linkography1(sizen=328) Linkography2(sizen=453)

Integration Tcomplexity 0.23 0.23

Integration Tentropy 0.22 0.23

Integration ClosenessCentrality 0.73 0.85

Integration Betweenness 0.07 0.21

Tcomplexity Tentropy 0.99 0.98

Tcomplexity ClosenessCentrality 0.46 0.39

Tcomplexity Betweenness 0.37 0.24

Tentropy ClosenessCentrality 0.46 0.39

ClosenessCentrality Betweenness 0.37 0.41

Table3:Valuesofcorrelationfortwolargelinkographies

10 Acorrelationinwhichlargevaluesofonevariableareassociatedwithsmallvaluesoftheother;thecorrelationcoefficientis

between0and 1.Itisalsopossiblethattwovariablesmaybenegativelycorrelatedinsome,butnotall,cases.Aperfectnegative

correlationisrepresentedbythevalue 1.00,whilea0.00indicatesnocorrelationanda+1.00indicatesaperfectpositivecorrelation

(definitionfromhttp://www.investopedia.com/terms/n/negativecorrelation.asp#ixzz1ceYmvKXE).

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Integration

-complexity

T-entro py

Betweenness

Closeness

Centrality

Integration

T-c omp lexity

T-en tropy

Betweenness

Closeness

Centrality

Figure11:Differentquantitativemeasurementsforadesigncasestudy

Integration isco nversely related

with RRA,

T-co mp lexity,T-entrop y, and

T-com plexity andT-entropy a re directly

related. T-com plexity is

co nversely related

with integration and

Betweenness.

T-entrop y is co nve rselyrelated with

integration, closeness

centrality, and

bet wee nness, and isdirectly related with

T-co mp lexity

Betw een ness isco nversely related

with integration,

closeness centrality,

T-com plexity andT-entrop y

Closeness Centrality isco nversely related

with Betweenness,

T-com plexity and

T-entropy b ut d irec tly

related with

integration.

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6.INCONCLUSION

Inthispaper,wehavebeeninvestigatingtheapplicationsofcertainmeasuresthatcomefromspacesyntax

analysesofurbangraphstolookatlinkographysystems.Onehypothesis isthatcomplexityiscreatedfrom

the localsubgraphatdifferentscales inthegraphsystemthanfromthewholesystem.Since linkography

andurbansystemsdealwithmultilevelcomplexities,theoverallgoaloftheproposedanalyticalmethodis

torevealtherelationshipbetweentheparts(subsystems)thatconstitutesthesystemandthewhole.

Twoperspectivesaregiven:theentropytheoristwho looksattheoveralldistributionofsetsof itemsthat

formthesystemwhiletheinformationtheoristlooksattheindividualsequenceofitemsorthearrangement

ofsetsthatwillprobablyoccur.Theapplicationtolinkographyandthepointdepthentropyareexamplesof

the formerwhiletheTcodecomputationofstringsof information isadopted inthispaperto lookatthe

latter. Two different contexts are given in the case studies. Since urban configurations and linkography

systems are drawn from different characteristics, the assumption is thus made to examine whether the

syntacticalandstringparametersreceivesimilarcorrelationresponsesinbothcontextsornot.

The methodology merges syntactical and string measures to highlight the significant nodes in any system

and investigate the proposed hypothesis: are highlyintelligible

systems associated

with

complexity

and

entropy?Since intelligibility,complexity, andentropy are system properties, the method to process any

systemofnsize isanaggregationofsubgraphsforeachnode inthesystem.Thecasestudies include

small and large systems, hypothetical and real. In order to highlight the significant nodes further, other

parametersareaddedintothecorrelation:realrelativeasymmetry,closenesscentralityandbetweenness.

The relationships between string measures (Tcomplexity and Tentropy) and syntactical measures

(integrationandrealrelativeasymmetryRRA)arenotclearlydefinedbecauseofthe inaccuracyofshort

barcodes.Theassumption isthenmadethatvariable lengthbarcodeholdswithin itmanypossibilitiesand

choices.Provingthishypothesisrequiresfurtherinvestigationwithlargersystems.

Themoreanodeisconnectedtothesurroundings,thegreatertherepetitionfrequencyinthebarcodes,the

lesspredictabletheinformation,andthereforelowstringcomplexityresults.Theasymmetryoftheoverall

distributionofnodeswithinthesystemaccountsfortheassociativeness inthesystemandconsequently

givesanindicationofthestructure.RRAandintegrationvalues(inversemeasures)canbetrackedtotrigger

thedegreeofassociativenessandincubationwithinthesystem.

The importance of this study lies, on one hand, from the definition it purveys about the responsiveness

between the configuration of a system and the internal structure. On the other hand, it provides an

analyticalframeworktoacknowledgethedegreeofhomogeneitybetweenthepartsandthewhole.

To study aconfiguration that underlies arrangementsof nodes isabout the exposition of factsthat are

called orderly when the observer can grasp both their overall structure and the ramifications in some

details.

ACKNOWLEDGEMENT

TheauthorwishestoacknowledgethecontributionbyDrUlrichSpeidel,whohasprovidedcleardirectionsin

the study of deterministic information theory and the computation of string measures: entropy and

complexity. I am gratefulfor his generous advice and comments. I am indebted also to Mr. Mohamed

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AbdullahforhisthoughtfulsuggestionsandtechnicalsupportandtoDrSheepDaltonforprovidingsoftware

to compute the integration values. Last but not least, I wish to thank Prof. Alan Penn, Prof. Ruth

ConroyDaltonandMr.SeanHannafor theconstructivesupervision,criticaland inspiringthroughout.The

authortakesfullresponsibilityforallremainingshortcomingsinthispaper.

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