space syntax 8242 1
TRANSCRIPT
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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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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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