and Automata

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Cities and Cellular Automata

ROGER WHITE

Departmentof^Geography,Memorial UniversityofNewfoundland,St.John’s,Newfoundland,A1B3X9, Canada (Received7 December1997)

Cellular automata provide a high-resolution representation of urban spatial dynamics.

Consequentlythey givethemostrealisticpredictions ofurban structuralevolution, and in particulartheyare able to replicatethe variousfractal dimensionalities ofactualcities.

However, since these modelsdo not readily incorporate certain phenomena like density measures and long-distance (as opposed to neighbourhood) spatial interactions, their performance may be enhanced by integrating them with other types ofurban models.

Cellular automatabased modelspromise deepertheoreticalinsightsinto thenature ofcities asself-organizing structures.

Keywords." Cellular automata, Urban spatialdynamics, Modelintegration, Fractaldimensions, Complexity

1 INTRODUCTION

As

computing power continues to expand, micro- simulationmodelsbecome increasingly interesting as ways to understand complex systems. The approachhas much to recommend it, since it gets to the level at which objects or agents act to structure a system, and thus allows us to explore the phenomenon realistically while imposing a minimum of arbitrary simplifying assumptions.

Nevertheless, the strength ofthe approach is also its weakness: as the scale of the map approaches 1:1, it no longer provides an overview of the country. In other words, the detailed nature of explanation inherent in micro-simulation models may make it difficult to understand the general natureof emergent properties. Onthe otherhand, macro-scale models may give extremely useful

111

descriptions of emergent behaviour, even if they do not show how these macroscopic structures arise. Clearly it is desirable to have both levels of description, sinceeachcomplementstheother,and themicro-levelprovidesadeeper understanding of the reasons for the macro-scale phenomena. But evenbetteristohaveanapproachthathas proper- tiesofbothlevels,and providesalinkbetween the two.Cellularautomata

(CA)

^{aresuchanapproach.}

Cellularautomata arewellsuited tourban model- ling, being inherently dynamic and intrinsically spatial,withgood spatialresolution.Although they are remarkably simple mechanisms theycan gen- erate spatio-temporal patterns of unlimited com- plexity

(Wolfram,

1984; Langton, 1990; 1992; Bak Chen and

Creutz, 1989),

and that is one reason why they are such powerful modelling tools.

Nevertheless, cities are even more complex, being

generated byalarge numberofprocessesoperating at avariety of spatial andtemporalscales. Thusin order to use CAfor modellingcities with auseful degree of realism, it is necessary to use more complex CA that is, CA with more states, a larger neighbourhood, and more complex transi- tion rules. Itis also necessaryto constrainthe CA in various ways in order to represent the effect of processesoperatingatscales thatcannotbe directly included in the CAtransition rules by ineffect modifying the CAtransitionrulesastheCAruns.

DEFININGCELLULAR AUTOMATA FOR

URBAN

MODELLING

A

CA may be defined as

(1)

^{a discrete} cell space, togetherwith

(2)

^{a set}

of

possible cell^statesand

(3)

^a

set

of

^{transitionrules that}determine thestate

of

^each

cellas a

function of

^thestates

of

^{all cellswithin}

⁽⁴⁾

^a

defined

cell-space neighbourhood

of

^{the cell;}

⁽⁵⁾

^time

is discrete and all cell states are updated simulta- neously at each iteration. While the concept is re- markablysimple, and many of themostintensively studied

CA,

like Game of Life are,infact, almost trivially simpleasmechanisms, thedefinitionleaves quite a bit of room for variety, and the CA that have been mostuseful inurban modelling tend to depart significantly from theidealof simplicity. It is useful, therefore, to elaborateon this definition from the point ofviewofurban modelling.

(1)

Cell Space. The most important character- isticsof cell spaceare dimensionality, cellsizeand cell shape, and homogeneity or heterogeneity. CA developed for urban modelling are, not sur- prisingly, typically defined on a two dimensional cell space, though it is easy to imagine using a three dimensionalcell spacein orderto be ableto represent building height, or, more generally, density of land use. One dimensional

CA,

while less common in modelling applications, are being used for urban traffic modelling (Nagel et al., 1997;DiGregorioetal.,

1996).

Cellsize in current applications ranges from 250m (e.g. White et al.,

1997)

down to tens of metres (Batty and Xie,

1994). At

the latter scale, cells may represent cadastral units (i.e. actual land

lots),

^which may have any shape; thus while it is often convenient forvarious reasons (e.g. ^runtime) ^{to use a}regular cell space, that is not necessary. In conventional CAthecellspace onwhichthe cellstate dynamics is defined is homogeneous that is, all cells are identicalexcept for theirstate. In modelling cities, however, where it is natural to let cell states represent land use or land cover, it is useful to work on a heterogeneous cell space, where each cell may have an intrinsic quality

(or

a vector of qualities) representing such characteristics of the real geographical space as soil quality, slope, or landuseregulations.

(2)

Cell State. In most urban modelling appli- cations, cell states represent land use and land cover, but depending on the focus of the model, they may represent other features of the urban area; Portugali and Benenson

(1995; 1996),

for example, use cell states to represent social cate- gories, e.g. immigrantvs.nativepopulations.And, broadening the idea of

CA,

the conventional Boolean cell state could acquire a cardinal mea- sure in order to represent such characteristics as building qualityordensity. Finally,it isoftencon- venient to define some cell states asfixed, so that while cells in these states can influence the state transitions ofother cells, they are not themselves subjecttochanges ofstate(e.g.Whiteetal.,

1997).

Rivers and parks are examples of land uses that could well be represented byfixedstates.

(3)

Transition Rules. These arethe heart ofCA since they represent the process that is being modelled; theyarethus thekeyto agoodapplica- tion.Theymay be deterministicorstochastic; very simple (e.g. a cell changes to the modal state of its neighbourhood), or quite elaborate, like the transition rules of the urban model described below in Section 4. Typically the rules are fixed, but some work has been done with simple CA that evolve to solve a test problem, and there is potential for use of CA with rules that evolve endogenously to represent such phenomena as urban realestatespeculation.

(4)

Neighbourhood. Conventionally, the neigh- bourhoodto whichthetransitionrulesareapplied is small. In two dimensional rectilinear CA the most commonly used neighbourhoods are the Von Neumann

(4-cell)

^{and the} Moore

(8-cell).

These are convenient, and for many physical processes, where fieldeffects areabsentand inter- action occurs only through contiguity, they are entirely appropriate. They are occasionally used for urban modelling (e.g. Cecchini and Viola, 1990;

1992);

but ingeneral larger neighbourhoods would seem to be more appropriate, since land use decisions are made by people who are aware of and able to take into account conditions over a larger area than that represented by a Moore neighbourhood. In general, the size ofthe neigh- bourhood should be defined withreference to the distance over which the processes represented by thetransitionrules operate.

(5)

^Time. Discrete time with simultaneous cell state transitions is standard.

However,

there may be situations where it is convenient to nest time scales, ^{with some} parts of the CA running a number oftime steps for each step in the rest of the model. For example,in amodel ofaregionin which some areas are subject ^to seasonal inunda- tion, the low-lying areas are modelled on a monthly basis to capture the dynamics of the flooding, whileupland areas are modelled with a time step ofoneyear(Uljeeet al.,

1996). A

rather different approach to time would be to update cell states sequentially; such an approach would depend on having a method for choosing the sequence of cells that was inherently meaningful in terms of the modellingproblem.

CELLULAR AUTOMATA BASED URBAN MODELS

A number of CA models of urban systems have been developed, together with several modelling systems. Tobler

(1979)

was the first to propose a cellular approach ^to geographical modelling, and his idea wasfollowed up byCouclelis

(1985;

1988;

1989;

1996)

and later Takeyama

(1996).

^Couclelis

uses the ideas of CA modelling to explore the natureof space and spatialrelations inthecontext of dynamics, but doesnot attempt applicationsto specific cities.

However,

she points ^out thatwhile standard spatial interaction based approaches to the modelling of urban and regional systems pos- tulate arelationalspace, usually oflowresolution, in which absolute location is virtually irrelevant, GIS (Geographical Information Systems), on the other hand, presuppose ^an absolute space, and one of very high resolution. Both approaches are appropriate forcertainproblems, butanadequate representation ofacitywill involvebothrelational and absolute spaces.

As

she says,

On theone handisabsolutespace andtheconcrete geo-referenced location or object with attributes, thatknows nothing

of

^itssurroundings,"on the other is relative space and the

varified

complex spatial relationthatisso innocent

of

^thespecifics

of

^placeas

tobe dubbed "thegeography

of

nowhere".Inbetween isproximalspace and the generalized CA models, thatcannow...partake

of

the earthy datarichness

of

^{GIS while} probing hypotheses about the large scale

effects of

micro-scale interactions. That these models can now..,be extended into the realm

of

relative space, where the most robust theories

of

urban and regionalgrowth are still to befound, ^is a[n]..,excitingprospect. (Couclelis, 1996,p.

174)

Couclelis

(1996)

^{and Takeyama}

(1996)

^{thus pro-} pose^ageneralized modellinglanguagewhichwould permit integrated dynamic spatial modellingatall scaleswithin aGIS framework.

Batty

andXie

(1996)

andXie

(1996)

have devel- oped a CA for urban modelling which generates both landusepatterns andanassociatedtranspor- tation network. Their use ofa CA to generate a network is interesting and novel, but not yet entirely successful; ^one problem may be that

CA

processes are inherently local, whereas the evolu- tionofatransportation networkmustreflect both localand long rangeinteractions. Theirmodelsare not developed to model actual cities, nor do they carryout sensitivity analysis to determinegeneral

characteristicsofmodelbehaviour. They conclude that

It is unlikely that current approaches can really generateacceptablelevels

of

performancein terms

of

thegoodness-of-fit ^to existing patterns in the same way ⁱⁿ which less complex, traditionalmodels are able, andthusthe emphasisinapplication should^not be on

fit

^{but on}feasibility andplausibility. (Batty and Xie, 1996, p.

203)

It is significant that, like Couclelis, they discuss theirCAmodelsinthecontextof"traditional"(i.e.

"Lowry-type") spatial interaction based models, thus recognizing a complementarity. The issue of

"goodness-of-fit"istwopronged. Ontheonehand,

CA,

by virtue of their spatial detail, permit ex- tremelyrealisticresults;thetraditionalapproaches captureonly highly aggregated^measures of urban structure, though they may be "realistic" in the sense that an appropriately specified and well- calibrated model may yield avery good value for ameasureof goodness-of-fit.

However,

thismerely brings into stark relief the fact that measures of goodness-of-fit are highly generalized measures, whereas a city is an extremely complex entity.

A

model capable of producing truly realistic results will thus produce a complex output, and at present there are no appropriate measures of goodness-of-fit.

As

modelling techniques advance, and with them the quality ofthe model output, thisproblemisbecoming critical;wewill return to itbelow.

Portugali and Benenson

(1995; 1997)

and Portugali etal.

(1994),

^while investigating general principles ofurban self-organization bymeans of CA models, have gone farther in the search for empirical realism. They are concerned primarily withurbansocialstructure, and theattitudeswhich engender andarein turn modifiedby thatstructure.

They workwithmodelswhichtheycharacterize as modelsofhuman agentsin cellspace.Thecell space representsindividual residentiallots,and cellstates are actually vectors representing such qualities as statusand value of the dwellingaswellasthe char- acteristics of the family inhabiting it. Transition

rules relate changes in dwelling characteristics to the characteristics of families in the cell neigh- bourhood, and also relate changes in family characteristics (e.g. ethnicity or attitude toward neighbours) to the characteristics offamilies and dwellings in the neighbourhood. These models clearly demonstrate the close link between basic theoretical ideas and empiricalrealism that is one of the useful characteristics of cellular automata modelling.

Other applications of CA to urban structure (Cecchini, ^{1996; Cecchini and Viola 1990;}

1992)

have emphasized simple automata, ^for example two state (urban,

rural)

or small neighbourhood

(Von

Neumann or

Moore)

formulations. Makse et al.

(1995)

have adapted a physical model, ^a modelofcorrelated percolationinthe presence ofa density gradient, to the problem of urban form.

They use the two state model to simulate the growth of an urbanized area, and show that the result has a fractal structure. The fractal dimen- sions they generate are almost identical to those characterizing the output of other cellular models

(see

^{Section 4}

below)

(White ^and Engelen, 1993b;

White etal.,

1997).

Since the lattermodels do not imposeadensity gradient, itwould seemthat that feature ofthe correlated percolation model ^{is not} necessary: aglobal density gradientcan be gener- ated by thelocal dynamics of the CAitself, anot uncommonphenomenon.

CONSTRAINED CA-BASED URBAN LAND USE MODELS

White and Engelen

(1993a,b;

1994; 1997b), ^White

et al.

(1997),

and Engelenet al.

(1997a)

^{have also} developedCAbased urban models thatjoin theoret- ical concerns withempiricalrealism. These arethe only CAbased urban models that have been sub- jectedtoextensivesensitivity analysis, and also^{to a} degree of empirical testing. Itisthus usefultofocus onthese models bothinordertoget anindication of the quality of the results possiblewithCAbased urban models, and also to examine more closely

some of the major ^issues that arise in urban modellingwithCA.

Unlike the models of Portugali and

Benenson,

which focus on socio-economic phenomena, or those of

Batty

andXiewhich use acellular mechan- ism to generate a transport network, these are essentially landusemodels. The focusisspecifically onthe dynamics of landuseand land cover, while the transport network istaken as given (thoughit may be altered or extended exogenously duringa simulation). ^{Land use and land cover} types are divided into two categories:

fixed

^{features, like}

water bodies and parks, and active land use

functions

^like housing, commerce, and industry.

In the neighbourhood dynamics ^defined for the

CA,

^fixed features appear as arguments in the transitionrule functions, and so affect thechange of state from one active land use function to another, but they are not themselves subject to state changes

(other

than changes determined exogenously to the CA mechanism itself). ^Like thefixedfeatures, the transportation network may affect thetransitionprobabilities ofactivecells,but is not itselfchanged bythe cellular dynamics.

The transition rules applied to active cells incorporatefivetypes of factors:

intrinsic suitabilities representing inhomogene- ities in the geographic space being modelled (e.g. soilquality orlegalrestrictions);

the neighbourhood

effect,

representing theattrac- tive orrepulsive effects of thevarious land uses

(cell states)

ⁱⁿ the neighbourhood consisting of the 113 cells lyingwithin a radiusof 6cells of the targetcell;

a local accessibility effect, representing ease of accessto the transportnetwork;

a stochasticperturbation capturing the effect of imperfect knowledge and varying needs and tastesamong theimplicitactorswhosedecisions arerepresentedby cellstatetransitions;

globalconstraintsonthe numberof cellsineach state, reflecting the fact that the requirements for land forvarious activities arelargelydetermined exogenouslytotheinternallandusedynamics of

the city being modelled, depending rather on such factors as the size of the city and the need fora balance among^various activities. Thus at each iteration, atarget number of cells for each state isspecified, and theCAdeterminesnotthe number of cells in each state, but only ^their location. The target numbers for each cell state may simply be read fromafile,orgenerated bya linkedmodelas describedbelow.

The first four factors are incorporated into the calculation ofavector oftransition potentials for each active cell that is, apotentialis calculated for each possible (active) cell state. Then the transition rule is applied: each cell is changed to the state for which it has the highest potential, subjectto theglobalconstraints.

Thepotentialsarecalculatedas follows:

Phj

vajsj(1 + Z Z Z

^mkdlia

⁺

^Hj

⁽¹⁾

k d

where:

Phi

^{is the transition} potential from state h to state j;

rnkd--aweighting parameter appliedtocellswith state kin distance zone d

(cells

that are closer usually are weighted more heavily; but weights may bepositive, representing^anattractiveeffect, ornegative, if two states areincompatible);

Iid--

^{ifthe state} of cell i-k, otherwise

Iid=

^0, where is the index of cellswithin the distance zoned

(the

role of

Iid

^{issimplyto ensurethatonly} the relevant weight _rnkafor the cell at location i,d- i.e. the weight corresponding to cell’s actual state isincluded inthecalculation);

/-/j.-^{an inertia parameter, with} /-/j.

>

0 if j-h;

otherwise, /-/j.-O

(/-/j.

^{increases the likelihood} thatacellwill remain in its current

state);

sj-the

suitability of the cell state for j, where O_<sj_< 1;

aj.-the

accessibility parameter,

aj

(1 + ^O/(j)

^-1

(2)

whereDistheEuclidean distancefrom the cellto

the nearest link of the network and is a coefficient expressing the importance of accessibility ^on the desirability of the cell for land use activityj. Accessibility is thus defined strictly locally, in terms of distance to the network, rather than globally in terms of distancesthroughthe networkto distantpoints;

v--the stochastic disturbance term,

v

+ (-ln(r)) (3)

where

(0 <

^r

< 1)

^{is a uniform} random variate and c is a parameter that allows the size ofthe perturbation to be adjusted.

The random term permits the cellular ^city to explore its state space, since cells that are appar- ently unsuited to a particular activity will never- thelessoccasionallybeconvertedtothat state. Since the inertia term will normally prevent a cell from reverting immediately to its previous state, these

cells may function as nuclei of new developments by increasing thetransitionpotentialsofother cells in theirneighbourhood.

This generic cellular model has been used to generatetest cities(Fig.

1)

onbothahomogeneous cellspace (i.e. space with no suitabilities or trans- portation

network)

and acell space homogeneous exceptfor thepresence^{of a}transportationnetwork, for purposesofsensitivity analysis.Ithas also been used as the basis for simulations of actual cities and regions (Figs. 2,

4).

The resultsaresurprisingly realistic for such a simple model, especially for one lacking any representation of long-distance interactions.

Themoststriking featureisthatallcellularcities generated with versions of this generic model exhibitafractal form:

The perimeter ofthe urbanized area

(as

^defined bycells inany staterepresentingan urban land

FIGURE Simulation ofagenericcity generated onahomogeneouscell space, withatransportsystemrepresented by"road"

cells. Landuses: dark, commerce; medium,industry; light, housing.

use like housing or

commerce)

^{is a} fractal (Whiteand Engelen,

1993b).

The radialdimensions, calculated asthe deriva- tive of log(area) with respect to log(radius), of both theurbanized areaand ofindividualurban landuses arebi-fractals.Forexample,the urban- ized area is characterizedby aradial dimension of approximately 1.92 out to a critical radius;

beyond thatradiusthe dimensionfallstoavalue in the neighbourhood of 1.1. Inside the critical radius the cellular pattern is relatively stable, representing ^an urbanization process that is essentially complete; outside it, the cell state configurations change rapidly, ^as the CA generates the future urban form (White ^and Engelen, 1993b;Whiteetal.,

1997).

The radial dimensions of the individual land.

uses are significantly different from each other, reflectingarelative concentrationofthevarious land uses in successive concentric zones, with, for example, commerce relatively concentrated toward the centre and housing toward the periphery (White and Engelen, 1993b; White etal.,

1997).

The size-frequency spectra ofland use clusters arelog-linear, another measure of fractal struc- ture (White and Engelen 1993b; White et al.,

1997).

Inall cases, actualcities arefoundtoexhibitthe same fractal dimensionality (White and Engelen, 1.993b; White et al., 1997; Frankhauser, 1991;

1994).

Frankhauser

(1994),

for example, found thatfor 17 of19world cities, theradial dimension of the urbanized area within the critical radius ("rayon de s{grgation") ranged in value between 1.94 and 1.99. 1960 Landuse data for Cincinnati,

USA,

yielded^alog-linear size-frequency spectrum withaslopeof 1.29 forclustersof commercialland;

a cellular simulation of the city to the same year alsogavealog-linear spectrum,withaslopeof 1.44 (White^etal.,

1997).

AndBattyandLongley

(1994)

determinedthe fractal dimension ofthe perimeter of the Cardiff,

UK,

urbanized area to lie in the range of 1.2-1.3. Thus in terms of their various fractal dimensions, cellularcities would appearto behighlysimilar to actual cities.

Fractaldimensionsare,however, highlygeneral measures.One ofthepotentialstrengthsofcellular modelling lies in the spatial detail, and hence the realism, thatitcanprovide; and whilefractilityisa measure of thecomplexityof theurbanform,it isa very abstract one, and thus does not capture the particular form of specific cities or their cellular simulations. Cellular simulations of actual cities, for example Cincinnati (Fig.

2),

using the actual transportation network of major roads and

FIGURE2 Simulation of Cincinnati(left) and actuallanduse(simplified), 1960 (right); the road systemusedinthe simulation isnotshown.SeeColourPlateI.

suitability maps representing theslopeoftheland, give resultsthatarevisually quitesimilar informto the landusemap of the actual city.

Inthecaseof Cincinnati, the correspondence of the simulatedmapto the actual map of thecityis confirmed by Kappa index values

(see

Monserud and

Leemans, 1992)

ranging from 0.51 (fair) ^for commerce to 0.63-0.69 (good) for the other land uses.

However,

Kappa is based on a cell-by-cell coincidence matrix, and so frequently does not capture the more general similarity of patterns.

This problem is evident in the comparison of the simulated and actual locations of commercial activity (Fig. 3),whereit isclearthatthetwo loca- tionalpatterns are quite similar,even though very few ofthe cellscoincide.

A

recentlydevelopedmap comparison technique making use of fuzzy set theory

(Power, 1998)

is basedon acomparison of local configurations as opposed to individual cell coincidences.

An

applicationtothe Cincinnati case gives similarity values about 15% higher than the Kappa index. Thus while the various quantitative measures of the modelling results support the

FIGURE 3 Comparison of simulated and actual locations of commerce, for simulation shown in Fig. 2. See Colour PlateII.

conclusion that the cellular approach is worth- while, theyserveprimarilytorevealtheinadequacy of thecurrentkit oftechniquesforassessing spatial patterns aproblem noted by Mandelbrot

(1983)

among others.Inotherwords,ourabilitytomodel reality is outrunning our ability to evaluate the results.

LIMITATIONSOF CADUETO SHORTRANGEINTERACTIONS

While it isgratifying that the results of modelling withcellularautomataareasrealistic astheyappear tobe,^{it is}also,^{in one}respect, somewhat puzzling.

The fractal nature of the patterns generated by calibrated CA models implies spatial structure at all scales i.e. spatial autocorrelation up to the dimension of the grid (e.g. 80 x 80

cells)

^even though the processes that generate the structure operate onlyover short distances(e.g. 6cellunits).

There is no question about howthis happens: the iterative CA process can, given sufficient time, transmit a signal from any cell onthe grid to any other. Rather,the mysteryiswhy the resultsare so realistic, since in real citiesthe spatialinteractions that generate spatialstructure occur not only ata local scale corresponding to the CA neighbour- hood, but also atregional or long distance scales which are notrepresentedatall in theCAmodels.

Mustweto conclude, counter intuitively, that the enormous volume of long distance interactions within cities shopping trips, for example, or the daily trips to work are irrelevant to the spatial structuring of the city?

Mostexisting models of the dynamics of urban spatial structure (e.g. Allen, 1983; 1997; White, 1977; 1978; Wilson, 1974;

1976)

arebasedonrepre- sentations ofthese longdistance flows; and as we have seen,Couclelis

(1996),

Takeyama(1996),and BattyandXie

(1996)

havereaffirmed theneed for models of this sort, as have White and Engelen

(1994).

Fortunatelyfor traditionand intuition, there are indications that the cellular models do not handle all aspectsofurban spatialstructure aswell

as the spatial interaction based models. For example, cellular models of the sort characterized by

Eqs. (1)-(3)

represent landusequalitatively,in terms of land use type, and do so with excellent spatial resolution; but they do not represent quantities suchas density. In otherwords, lacking ameasure of intensity, the cellularmodel, in spite of itsspatial resolution,cannotgiveagood quanti- tative representation of the location of activity.

Thus, if only for such mundane reasons, ^{it will} frequently bedesirable to supplementaCAmodel with a traditional spatial interaction based repre- sentation of urban structural dynamics. But in addition, there is some evidence that intuition is correct, and that long rangeinteractions areimpor- tant in shaping the urban spatial structure; this is seen most clearlyinthecommercial sector.

Majorretail centrescompete forbusiness atthe scale of the entire urban area. The process of competition for customers, which determines the revenues of the centres and thus ultimately their success orfailure,must therefore be modelledin a way that captures the long distance spatial ^inter- actioninvolved, forexample bymeansofaLowry- type model, as mentioned by

Batty

and Xie. The models of urban retail centre dynamics due to Wilson

(1974; 1976)

^{and White}

(1977; 1978)

^are both based on long distance spatial interaction models. Extensive simulation experiments with a similarmodel(White^andEngelen,

1997b)

^indicate that the system of majorretail centres ischaracter- ized by two qualitatively different types of stable states, depending onthevalues of relevant param- eters: in oneof theseasinglecentre isdominant;in the other

(convex)

state, a number ofcentres of similar size dominatethe hierarchy. If the systemis caused to pass through this bifurcation, it may spend a significant time in a meta-stable phase transition state in which the distributionofcentre sizes is log linear, representing a fractal structure (Nicolis,

1989).

Betweenthe late1950sand theearly 1980s, mostUScities passedthrough this bifurca- tion, spending a decade or two in the log linear, fractal transition, and then ending up clearlyinthe convex state (White and Engelen,

1997b). By

contrast, in all cases where the CA model based on

Eqs. (1)-(3)

was realistically calibrated, com- merce wasfractallydistributed indeed,with this model it is difficult to generate distributions that matcheitherof the stable configurations. Thegood agreement between the simulated and actual size distributions ofcommerce clusters for Cincinnati would thusseem to beafluke,dueto the fact that the simulation wastested against 1960data, when the citywascoincidentallyinthetransitionbetween steady states.

INTEGRATING CELLULAR AUTOMATA WITH OTHER MODELS

Inviewof theseproblems,andinrecognition of the fact that activity patternsin cities are dominatedby interactions beyond the scale ofa cellular neigh- bourhood, it would seem obvious to link CA models of urban structure with other, standard dynamical models (Uljee ^et al., 1996; White and Engelen,

1997b).

In this approach the dynamical model operates at amacro-scale, in contrast with the micro-scale of the cellular model. It may be either non-spatial, consisting, for example, of a model ofeconomicand demographicchangeinthe urbanarea as awhole, ^or spatial,as inthe caseof the standard spatial interaction based models operating on statistical units like census tracts.

Only in the latter case, of course, can it possibly make up for thedeficiencies of the cellular model described in the previous section. In either case, however, the macro-scale model constrains the cellular dynamics by providing target numbers for cells ofeach state, butⁱⁿthecaseofthe spatial model the targets may be regionally specific.

For example,amacro-scale model may generate a demand for a certain number of housing cells, with the actual locations to be determined by the cellular model; but if the macro-scale model is regionalized andsocontains its ownrepresentation of spatial dynamics, then the model will allocate the total predicted demand among the regions, so that the cellular model then works with regional rather thanglobaltargets.

The discussion in this section is based on experiencewith an integrated model consisting of the cellular model described above (Section

4)

linked to a dynamic, regionalized spatial interac- tion basedmodel ofurban structure (Fig.

4).

The interestingproblems arise, of course,inlinkingthe two levels. While sets of cells can be associated unambiguously with regions, the ^same is not necessarily true of the activities associated with the spatial units. State variables and parameters associated withcellscangenerallybeaggregatedin some appropriate wayover allcells corresponding to aregion and thenassignedto that regioninthe macro-scale model. But if regional data is to be passed to the cellular model, ^{it is} frequently not possible to assign it exclusively to the set ofcells corresponding to the region, ^aswe will seebelow.

Thus in generaltherelationship between an activ- ity variable in the regional model and the ^re- gional expression of that variable in cell space is one-to-many. Furthermore, while in simple

models there maybe aone-to-one correspondence between the activity variables as defined in the regional model and the cellstatesdefined in theCA model(e.g. population housing), ingeneralthis is not the case: the population variable in the regional modelmaycorrespondto severalCA cell states(e.g.population

[(1)

single family housing,

(2)

low riseapartments,

(3)

high-riseapartments]), andvice versa.

A typical macro-scale model, specifically ^a re- gionalized, spatial interaction based model of urban structure dynamics, will re-allocate activity

(X0.)

^e.g.population, industry,offices, jobs,etc.

fromone region (i) ^to another (j) on thebasis of the quantity of activityineach of the tworegions (Xi,

.),

^{the distance} between them

(d0-),

^{and the}

distance weighted attraction of all competing regions

(Vk):

x0. ^{f(x;, d0,}

i,j,k 1,..., R number of regions.

(4)

FIGURE4 Simulation ofSt. John’s (note thecellular structure of the land usemap), with inset showingthe 40 regions on whichthe macro-scale regional modeloperates. SeeColour Plate llI.

But as Couclelis pointed out, one of the major shortcomings of such a model is that its space is purely relational; the model makes no use of data with absolute spatial coordinates i.e.

the sort of data that is stored in a GIS and is used to generate suitabilities for the CA model described above. Neither does the model incorporate possible effects of the micro-scale configurationsgeneratedby the cellular dynamics, nor does it take into account possible effects of crowding or land shortages. Thus when such a model is linked to a CA model there is an immediate potential for reformulating it so that it includes additional variables that reflect these cellularlevel phenomena:

Xij g(Xi, Xj,

do., VO.,

wji,Pji,

sji) (5)

where _wjiis the ratio of the density of activity in thedestinationregionj tothatinthe origin region i, Pji is the ratio of the means of the cellular potentials for the activity

(as

ⁱⁿEq.

(1))

^inthetwo regions, andsjiistheratioof themeansuitabilities of cellsin thetwo regions for the activity.

What is gained by introducing these new vari- ables thatsummarizeinformationfromthe cellular model? Typical urban dynamics models as repre- sented by

Eq. (4)

embody only positive feedback effects: in most formulations the movement of activity into a region is positively related to the existing level of activity, with no negative or inhibitory terms included. In particular, with the exception of the models due to Allen

(1997),

they typically donotinclude adensityterm.Yetdensity is probably one of the most important controls onurban dynamics, since as density increases, the associatedcosts, particularly the risingcostofland andtheindirect costsof congestion,willeventually inhibitthecontinuedgrowthofthemostattractive regions.

Of course such effects can be included in the model,asAllen

(1997)

^hasdemonstrated;buttodo soin morethan themostarbitrary way requires the sort of detailed spatial data which resides at the cellular level.For example,ifdensityis obtained in theobviouswaybydividing the level of activityby

the area of the region, the measure will not be comparable from one region to another: in one region the entire area may be available to the activity, while in another only a small part may effectively beavailable, the restbeingtoosteep,to wet,oroccupiedbyanactivity thatcannoteasily be dislodged. Thus meaningful densities must be defined atthe cellularlevel;forexample,thedensi- tiesUsedtodefinewiⁱⁿ

Eq. (5)

could be calculated asthe level of activityinthe regiondivided by the number of cells occupied by the activity, giving ^a measureofactual operating density that would be closelycorrelatedwithdensityassociated costs.

Similarly, theterms_Pjiand_sjirepresent thecom- parative attractiveness of^two regions in terms of localorcellscalecharacteristics withinthe regions.

Therelative meanpotentials

(Pi)

of the cellsinthe two regions provide a summary measure of the relative attractiveness ofthe local cell neighbour- hoods in the two regions, while the relative mean suitabilities

(si)

^{of the} cells generalize the ^site- specific qualities of^Couclelis’absolute spacetothe regional level. Of course, all three of thesevariables summarizing cellular level propertiescanbe speci- fied in avariety of ways, and others could bedefined aswell.

The linkfrom themacro-scale model tothe cel- lular levelservesprimarilytoprovide the numbers ofcells required for each activity. But in the case of the regionalized models, this is also the means bywhich the effects of long range interactions on the spatial restructuring of the cityare transmitted to the cellular model, so that regional scale inhomogeneities are introduced into the cellular dynamics. There are a variety of ways in which the link can be specified, but all of them involve treating the regional activity levelsgenerated bythe macro-level model as demands

(t+ 1XD,.

rather than as actual new levels of activity, and then passing these demands to the cellular model for allocation incell space and hence implicitly among regions.

A

relatively loose connection between the two levels can be established by using the changes in demand in each region to re-scale the cellular

potentials calculatedinEq.

(1).

Thusthepotential valuesfor cellsin aregionwith agrowing demand foran activityarerescaled upward relative topo- tentialsfor cells inregions withlowerincremental demand,buttheCAthenruns as anunregionalized whole, constrained only by the total number of cells required for each sector. Thusingeneral the amount of activity implicitly allocated to each region by the cellular model will notbe the same as the amount called for by the regional model, although regionswithhigherincrementaldemands

(/t+ 1XDi)

^willbe relatively more attractive atthe cellular level and thus will typically receive rela- tively more cells, and consequentlymore activity.

The level of activity ^{in a} region is determined implicitly by the numberofcellstogetherwiththe densityofactivity percell,so afinalregionalactiv- ity level

(+ 1X

i) ^as established in the CA can be calculated and returnedto the regional model.

A

tighterconnection canbe madebytreating the regional demands foractivities as regional targets at the cellular level. Regional activity demands

(t-t-1XDi)

^{are converted to} regional demands for cells, and a regionalized CA attempts to allocate the cellswithinthe specified regions. While in this approach the cellular model is regionalized with respect to target cell numbers, the calculation of cellular potentials(Eq.

(1))

^ismade without regard toregional boundaries:theneighbourhoodof acell may include cells in another region. Thus it is possible, for example, for clusters of activity to developwhich crossregional boundaries (Engelen etal.,

1997a).

The regional targetsapproachis not sostraight- forward as it may seem, however, since it is not always possible to satisfy all of the demands for cells within aregion. Thus theremustbe mechan- isms for re-allocating some ofthe cell demand to other regions. These may take many forms, but typicallytheywillbebasedonsuch factorsasspace available, existing regional demand for the activity

(t+IxDj,

ji), density, and suitabilities. The intensity of the competition for cells in a region asmeasuredbytheratioof cells demandedtocells available canbe interpretedas asurrogate for land

price. Sincehigher land prices leadto higher den- sities,densities areadjustedateachiterationof the cellular model to reflect the changing demand for land. Both thenew densitiesand the derived final regional activity levels

(t+ 1X

i) ^{are returnedto the} macro-scale model.

We do not yet have ^sufficient experience with these linked models to know to what degree the performance of the component modelsisimproved by the integration. Cellular models generally per- form very wellontheirown,so it is tobeexpected that linking them with regionalized macro-scale modelswouldnotresultin dramaticchangesin their performance. The most noticeable improvements seem tobein sectors like commerce which arehighly dependent on long range interactions and which have provedmostdifficult tomodel wellwithstand- aloneCA. Ontheotherhand,the spatialinteraction based regional modelsmaybegreatlyenhancedby integrationwithcellularmodels,primarily because the cellular models, taking into account a vast amount of detailed, high ^resolution spatial data concerning factors important to location, ^{act to} constrain the regional dynamics. The traditional spatial interaction based models, popular among plannersinthe 1950s and1960s, have fallenoutof favour,inpart,perhapsbecausetheirresultsarenot locationally specific; i.e. theyhave extremelypoor resolution, and thus cannot easily be linked to specific urban phenomena. Potentially one of the major effects of linking these models to cellular modelswillbeto revivetheirpopularity.

More generally, there is no reason to limit integrationto urban models. Cities are inherently complexentities, with socio-economicphenomena unfolding in an environment that is both con- structed and natural. In the CA discussed so far, thebuilt environment ismodelledonlyin ageneral way

(as

land

use),

and the natural environment is treated as a set of boundary conditions, repre- sented inthe land cover features andthe suitabil- ities. But the natural environment has its own dynamics, andin anurbanareathese dynamicsare intimately linked to the socio-economic and land usedynamics of the constructed city.For example,

a regional hydrological model may represent the dynamics ofinfiltration, runoff, ^streamdischarge, and floodingasfunctionsof precipitation and land cover. Butthe balance ofinfiltrationand runoffis largelydeterminedbylandcover orland use, so as urbanizationproceeds and landusechanges, infil- tration and runoffwill be altered; ⁱⁿ consequence groundwater supplies will be affected and so will thefrequencyandextentof flooding.Butthelatter effects may in turn feed back onthe evolution of the urban area, forexampleby discouraging devel- opment of the areas that are newly at risk of flooding,orbyinstigatingaplanningintervention establishinganaturalreservededicated torecharge of thelocal aquifer. Thusit isnaturaltolinkcellular urban modelstomodels of natural processes (Uljee etal., 1996;White and Engelen,

1997a).

Inspite of the specificdifficultiesthatarealways encountered, one of the great strengths of CA based models is the relative ease with which they can be linked to other models. Because process orientedmodels of manyphenomenaareinherently spatial and operate^atlocalscales,asdo

CA,

there is frequently a basic commensurability which facilitates integration. In this situationthe output ofone model may bepassedto the other where it modifies a boundarycondition; for example, in a case where forested land was being converted to commercialuse, the output ofan urban land use model would modify the local infiltration rate parameters ofahydrological model, which would thus express the effect of the landusechange.

Evenwhen the models are notcommensurable, as is the case with regional models, or with non-spatial economic models (e.g. input-output models; see Engelen et al., 1997b; ^White and Engelen,

1997a),

it is stillpossible,as wehave seen, to establish strong linksbetween the models. The reasonlies inthe rule-basednatureof

CA.

Cellular modelsdifferfrom each other primarilyin termsof their transitionrules, ^{which can} take onvirtually anyform,including hierarchical rule sets,in which one rule may over-ride another. This flexibility permits modelstobelinkedthrough^theirrule sets, where the output of ^one model modifies the

transition rules of the other.

As

the need for integrated modelling of human-natural systems becomes more obvious, it seems likely that the flexibility ofCA willbe seen as one oftheir most importantcharacteristics.

7

CONCLUSIONS

Cellular automata constitute a powerful tool for understandingcities at morethan onelevel.

At

the most mundane level, ^their high spatial resolution and micro-scale process modelling permit a high degree of verisimilitude. Such highly specific models, linked to the GIS that currently serve as the primary support for urban spatial planning, would radically extend the capabilities of the GIS byaddingatemporal,predictivedimensionto their high-resolution spatial database, thus enabling plannerstoexperimentwithpossible urban futures.

But thereis a deeper significanceto therealism and practicality ofCAbased models. The history of urban modelling is plagued by the legacy of oversimplification. Starting with the standard land use models, which assumed a monocentric city lyingon anisotropic plain, and continuing with the macro-scale urban dynamics models discussed above, ^{which are defined on} highly aggregated regions and thus lack spatial resolution, the results have been too general or too unrealistic to be of practicaluse.

Moreimportantly, these traditional approaches havefurnished relatively few fundamental insights intothenatureof urban systems, primarily,itwould seem, because theyaretoodivorced from thenature of cities.

As

self-organizing structures, cities are inherentlycomplexobjects.Eventhe "simple" pro- perties ofcities cannotbeunderstoodiftheunder- lying complexity is ignored. For example, the monocentric structure exhibited by many cities cannot be explained by a model that assumes monocentricity;nor canthefractalnatureofcities be discerned intheabsence ofarepresentation of the underlying structural complexity. Thus^adeeper theoretical understanding can only be achieved

through methods that bring together the general and the specific methods that capture and work with the detail and complexity that characterize realcities.

Extensive theoretical work on the principles of self-organization by

Langton (1990;

1992), Kauffman

(1989;

1990; 1993), and others

(see,

e.g.

Forrest,

1991), using CA and related techniques, hasbeguntoprovide deepinsightsintothe behav- iour of whole classes ofcomplex systems. There are, for example, strong indications that evolved complex systems will typically have fractal char- acteristics. In this light, then, ^{it is not} surprising that cities have afractal structure, ^or that cellular models of cities generate such a structure

(and

instructive that traditional urbanmodels do

not).

The point is that cellular techniques dramatically shrinkthe distancebetween highly specific models ofactual cities and models formulated to investi- gate fundamental theoretical issues. They may be expectedto leadto richerand more useful applied models co-evolving^{with a}deeperunderstanding of urban systems.

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