Polynucleated Urban Landscapes (Batty 2001)

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DOI: 10.1080/00420980120035268

2001 38: 635Urban StudMichael Batty

Polynucleated Urban Landscapes

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Urban Studies, Vol. 38, No. 4, 635655, 2001

Polynucleated Urban Landscapes

Michael Batty

[Paper received in nal form, December 2000]

Summary. City systems show a degree of resilience and persistence that has rarely been

emphasised in urban theory. There is a fascination for recent and contemporary change which

suggests that phenomena such as the rise of the edge city, for example, comprise the predomi-

nant forces determining how a polynucleated landscape of cities is emerging. We argue here that

such explanations of polynucleation are largely f alse. Urban settlement structures from much

earlier times are persistent to a degree that is extraordinary. We show this in two ways: rst,

from empirical evidence of stable rank size relations in the urban settlement system for Great

Britain over the past 100 years; and, secondly, from simulations based on weak laws of

proportionate effect which produce aggregate patterns entirely consistent with these empirical

relations. We then propose various spatially disaggregate models of urban development which

generate an evolution of polynucleated settlement from initial, random distributions of urban

activity. These models simulate the repeated action of agents locating and trading in space which

illustrate how early settlement patterns are gradually reinforced by positive feedback. Theseproduce lognormally distributed settlement structures that are characteristic of city systems in

developed countries. In this way, we begin to explain how aggregate urban structures persist in

spite of rapid and volatile micro change at more local levels of locational decision-making.

Polynucleated urban landscapes are clear evidence of this phenomenon.

1. Continuity and Persistence in Urban

Systems

George Holmes in his Preface to The OxfordHistory of Medieval Europe says:

Most Europeans live in towns and villages

which existed in the lifetime of St.

Thomas Aquinas, many of them in the

shadow of churches built in the 13th cen-

tury. That simple physical identity is the

mark of a deeper continuity (Holmes,

1992, p. iii).

This deeper continuity referred to by

Holmes has not been central to the theory of

cities developed over the past half century.Theorists and commentators have preferred

to emphasise urban development as embody-

ing new events such as edge cities which

reect changing lifestyles and new technolo-

gies. There has been little attention given to

explaining urban development as patterns of

settlement which persist and whose contem-

porary form is the product of a myriad of

historical decisions, many of them buried

deep in the past. Yet the evidence is clear.

Cities and their structures are long-lived af-

Michael Batty is in the Centre for Advanced Spatial Analysis, University College London, 119 Torrington Place, London, WC1E6BT, UK. Fax: 0171 813 2843. E-mail: [email protected]. The author wishes to thank Danny Dorling of Leeds University for providingthe updated population series from 1901 to 1991. Sanjay Rana of CASA wrote the programme to scan the simulated images for theranksize computations for the agent-based models. This research has been partly nanced by the ESRC NEXSUS Project: Network

for Complexity and Sustainability (L326253048).

0042-0980 Print/1360-063X On-line/01/040635-21 2001 The Editors of Urban Studies

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MICHAEL BATTY636

fairs, where new locations which appear

from nowhere so to speak, are the exceptionrather than the rule (Hohenberg and Lees,

1985). Where new development does occur,

it is largely due to growth into virgin terri-

tory and even there, such development is

often built on an earlier, more stable and

lasting structure of villages and farmsteads.

If you look at populations in southern

England, everyone still lives within 4 miles

of churches which had been planted by the

15th century. In Buffalo, New York, every-

one comprising the population in the metro-

politan area lives within 3 miles of

farmsteads that existed in 1820 (Batty and

Howes, 1996). Batten, in his discussion ofnetwork citiescities which have grown to-

gether to form specialised polynucleated

formssuggests that:

The seeds of such a network economy

were sown as far back as the 11th century,

when safer trade routes triggered the re-

vival of many medieval cities in Europe

(Batten, 1995, p. 319).

In fact, quite simple explanations of the

growth of cities from initially randomly dis-

tributed rural populations to agriculturally

based market systems as reected in central

place theory, and thence to urban landscapes

structured around industrial resources, appear

quite adequate. Within such landscapes, re-

structuring takes place incessantly with new

nodes of specialised production and market-

ingedge cities in the current jargon (Gar-reau, 1991)clearly occurring. But in terms

of the volume of activity, these nodes are

small, with most of the developed urban

landscapeas in North America and western

Europerestructuring itself and growing

into its older, more established pattern of

settlement.Systems like this are path-dependent in

that history dictates how they evolve andhow they restructure (Arthur, 1988). Deci-

sions which are made early in the history of

urban development, often for entirely expedi-

ent reasons, determine future decisions

through positive feedbacks which in turn re-

inforce existing forms and functions. Given

enough time and the continual effect of such

feedback, systems such as cities exhibit adegree of persistence and continuity which is

increasingly difcult to break (Batty, 1998).

Although new events do deviate from the

long-term pattern, at any point in time these

are never sufcient to set the system on a

new trajectory. Cities do rise and fall, but

only over long periods where locational ad-

vantages change slowly. New settlement pat-

terns do emerge, but these are usually due to

growth into previously undeveloped re-

gionsthe frontieror to the gradual lling-

in of earlier patterns where the ultimate

structure which emerges is based upon rein-

forcing embryonic structures established inearlier times.

This is but only one plausible explanation

of urban evolution. In this paper, we will tell

a story which is consistent with the fact that

most urban development is rooted in his-

tory, bringing to the fore the argument thathistory matters. First, we will argue that

appropriate aggregate measures of settlement

which reveal continuity and persistence inurban development, should be based on vari-

ants of rank size relations associated with

city populations. These relations have been

shown to be stable over long periods of time

and we will present our empirical evidence

for such continuities using data from Britain

over the past 100 years. We complement this

by arguing that weak positive feedback is

both necessary and sufcient to persistent

urban structures and we illustrate this froman aggregate simulation of settlement pat-

terns using the standard growth model based

on random processes of proportionate effect.

We then introduce a class of spatial mod-

els operating at a ne scale in which individ-

ual agents of a population move, grow and

decline through processes of decision-making incorporating positive feedback. We

elaborate the model in various ways, nallydemonstrating how initial patterns of settle-

ment from random distributions, evolve to

generate rank size relations within which cit-

ies rise and fall through continual locational

change. These models are signicant in that

they demonstrate how long-lasting repetition

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POLYNUCLEATED URBAN LANDSCAPES 637

of routine locational processes generates spa-

tial structures which are hard to break, henceresilient. We then show how our earlier em-

pirical and theoretical measures of persist-

ence based on rank size are also borne out in

these model-based simulations. As a conse-

quence of this argument, the structures we

generate are inevitably polynucleated in a

spatial sense and accord well with more

causal observations of the way cities co-op-

erate, fuse and grow together.

2. The Empirical Evidence

To proceed, we require some formal

denitions appropriate to a simple measureof structure based on city size, Pi, the popu-

lation in area i. Assuming N cities or areas

which contain population and which exhaust

the space under consideration, then the sim-

plest measure of structure in the distribution

is based on ranking population by size, indescending order where the rank r of the

largest population is 1. We dene the popu-

lation in area i which has rank r as Pir. Therank order{Pi1.Pj2.Pk3..Plr..PmN}

thus represents the set of relations between

the areas that dene the settlement structure

while the degree of persistence can be calcu-

lated as some statistic of the difference be-

tween rank orders at two points in time, say

between t and t1 1. Such statistics are

dened on differences between Pir(t) and

Pir(t1 1) where the subscripts t and t1 1 have

been added to the rank order of each popu-lation by area to distinguish between changes

in rank and population volumes through

time.

Differences in ordering also reect absol-

ute changes in the population of each area,

which implies growth or decline. We calcu-

late the total population at time t as

Pt5

O

N

i5 1Pir(t) (1)

where the summation in (1) is over i or r, for

each area i is uniquely associated with a

single rank r(t) at time t.

In many situations, we are not interested in

the absolute change in populations from one

time-period to another, but in the relative

distribution and thus we dene the relativechange in population or market share as

Pir(t)5Pir(t)/Pt

Oi pir(t)

51 (2)

We can now calculate a series of simple

differences between the rank-ordered distri-

bution of population at two different points

in time. First, the absolute difference in

population shares for any area i is dened as

C t1 15Oi

|pir(t1 1)2pir(t) | (3)

and this statistic gives a measure of the per-centage shift in population between any two

time-periods. Another view of this shift is

given by changes in rank order. The average

number of ranks that change in a typical area

between t and t1 1 is given by

K t1 15Oi

| rit1 12 rit|/N (4)

where, rit1 1 is the rank of area i at time t1 1

and rit is the rank of the same area at time t.Note that the sum of each set of ranks is

Oi

rit1 15Oi

rit5N(N1 1)/2

Lastly, the average shift of rank in (4) can be

expressed as a percentage shift which is

given as

kt1 15K t1 1/N5Oi

| rit1 12 rit| /(N2) (5)

It is possible to have considerable changes in

the percentage shift in populations, Ct5 1. 0,but no changes in rank, Kt1 15 0, especiallyif the population is concentrating or dispers-

ing systematically with larger areas growing

and smaller ones declining proportionately

in market share or vice versa. Zero change inmarket share of course implies no change in

rank from time-period to time-period.To illustrate the degree to which the settle-

ment pattern changes, we will examine the

distribution of population in Great Britain

(England, Scotland and Wales) at 10-yearly

periods from 1901 to 1991 for N5 458 stan-

dardised administrative areas which cover



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MICHAEL BATTY638

Figure 1. Rank size population distributions in Great Britain, 1901 91.

the entire land area. During this period which

is most of the 20th century, the population

grew from nearly 37 million to 54 million, a

change of some 46 per cent which occurred

mainly in the rst 50 years. This large

change masks the fact that the settlement

structure has remained remarkably stableduring this period. Because populations are

log-normally distributed (for reasons that we

will elaborate later), we can show this stabil-

ity by plotting the log distribution of popu-

lation and rank in descending order.

We have plotted log[Pir(t)]against log[rit]

for t5 1901, 1911, 1921, , 1991, in Figure

1, and this shows that the shape of the distri-

bution changes very little from decade todecade. If we correlate every distribution

Pir(t) against each of the others for the 10

different series, the lowest correlation is be-

tween Pir(1901) and Pir(1991) which is 0.79 (ac-

counting for some 0.62 of the variance).

Figure 1 demonstrates remarkable persist-

ence in the settlement system which from

more casual evidence at this scale of aggre-

gation, implies that no new major settlements

have appeared in Great Britain during the

past 100 years. During this period, the largest

settlements appear to have fallen in popu-

lation with the smallest increasing, thus rep-resenting a mild dispersion of populations.

We can examine the differences implied in

Figure 1 more effectively if we compute the

shift statistics given above in equations (3)

(5) and although we can do this for every

pair of distributions, we choose simply to

concentrate on the differences between 1901

and 1991. In Figure 2, we show the logged

rank size distributions for these years, thistime based on the relative population shares

pir(1901) and pir(1991) which enable us to remove

the absolute growth effects. We also show

the distribution of the 1991 population or-

dered according to the ranks that the same

areas i had in 1901that is, the distribution



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Figure 2. Rank size and population shifts in Great Britain between 1901 and 1991.

log[pir(1991)] against log[ri1901]. Although there

are substantial shifts in the populations of the

largest areas, the rank order remains fairly

stable at the upper ends of the distributions

while there is more change at the lower ends.

Over the period, the percentage shift in

population C 1991 has been in the order of 46

per cent, much the same as the rate of

growth. In fact, this reects shifts up or down

which mean that the absolute percentage dif-

ference is half of this, some 23 per cent. Thisis consistent with the shifts in rank order

measured by K 1991 and k1991. The average

number of ranks that are changed for each

area K 1991 is 86 from 458 and this implies a

percentage rank shift k1991 of almost 19 per

cent. This bears out the fact that during the20th century at this scale, the settlement pat-

tern has been extremely stable with the pat-

tern established by 1901 dominating changein the following 100 years.

We have been extremely careful so far to

avoid any discussion of the rank size rule

for two reasons. First, as the theory has

developed since Zipf (1949) presented the

rst general treatment, the events that have

been the subject of inquiry have been re-

garded as distinct cities rather than areas

which completely and perhaps arbitrarilysub-divide an entire territory. We would ar-

gue here that our sub-divisions are not cities

and that rank size and other scaling relations

are as appropriate, if not more so, to arbitrary

disaggregations of any spatial system. Sec-

ondly, most of the work on rank size has

chosen to examine not the entire distribution

of city sizes, but the long tail of the city sizedistribution which can be approximated by a

power law (see Carroll, 1982). Nevertheless,

it is instructive in this context to examine the

relationship to the mainstream as there is

much commentary on the persistence of city

systems over time which supports the analy-

sis here.

The rank size relation is based on tting a

power law to the long tail of distributionssuch as those shown in Figures 1 and 2. This

means that there is an arbitrary cut-off

imposed for those areas that do not vary

log-linearly with rank, which are the smallest

settlements. This is itself problematic as it

effectively excludes the origins of urban



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MICHAEL BATTY640

Table 1. The rank size relation tted to the 1901 91 population data

Year t Correlation R2 Intercept Kt P*1t510Kt Slope at

1901 0.879 6.547 3526157.772 2 0.8171911 0.880 6.579 3801260.554 2 0.8101921 0.887 6.604 4025650.857 2 0.812

1931 0.892 6.607 4046932.207 2 0.8021941 0.865 6.532 3410371.276 2 0.7401951 0.869 6.482 3034245.953 2 0.7001961 0.830 6.414 2595897.640 2 0.6511971 0.815 6.322 2101166.738 2 0.6011981 0.816 6.321 2095242.746 2 0.6011991 0.791 6.272 1872348.019 2 0.577

growth. Nevertheless, the typical relation is

based on

Pr5Kr2 a (6)

where, Pr is the population of the area ranked

rwhich is the same as P*r* with the area i and

time t subscripts suppressed; K is a constant

of proportionality and a is a parameter thatcontrols the change in population between

ranks.

The strong form of the rank size rule

which was argued by Zipf (1949)as well as

Aubach, Lotka, and others before, see Carroll

(1982)supposes that a5 1 which, fromequation (6), means that the rst-ranked area

is P15K. Then for any rank r

Pr5P1/r (7)

The implications of equation (7) seem rather

articial in that the second-ranked city is half

the size of the rst, the third one-third thesize, the fourth one-quarter and so on down

the hierarchy. But the empirical evidence is

so strong for many city systems in different

parts of the world, and the persistence of the

relation through time so clear, that Krugman

remarks the regularity is

so exact that I nd it spooky. The picture

gets even spookier when you nd it is not

something new ( Krugman, 1996, p. 40).

There are many problems with this rule.

First, some countries have one dominating

large citythe so-called primate city

which stretches the long tail at its very top.

We have already remarked on the fact that

the short tail of such distributions is effec-

tively excluded. The denition of the discreteevents that comprise the objects in the distri-

butioncitiesis also in doubt. These are

continually changing in size and area, and

there has been little sensitivity analysis of

how their denition affects the parameters of

the power law. There seems also to be a

tendency for the value of the parameter a in

equation (6) to increase slightly through

time, implying that city systems are concen-trating, with some of the best evidence for

primacy and increasing concentration devel-

oped for the European system by de Vries

(1979) and for the French urban system by

Pumain (2000) and others in her group such

as Guerin-Pace (1995). To show, however,

that the arbitrary sub-division of the urban

space demonstrates the persistence of the

British urban system, we have tted the scal-ing relation P*rt5Ktr

2 at

t(or rather its loglin-

ear form: P*rt5 logKt2 at log rt) to the datain Figure 1 and this yields the parameter

estimates shown in Table 1.

From Table 1, it is clear that the slope

parameter at is much bigger than that whichwould have resulted had the relation been

tted without the short tail, and that the

system is getting less concentrated throughtime rather than more concentrated. None of

this detracts from the scaling that is implicit

in the data or the persistence of the system

through time. In fact, although we will not

present the results for tting the truncated

distribution because it is not our purpose to



8/22


justify the rank size rule in this paper, it is

worth noting that the rank size relation fromthese data is closer to Pr5P1r than theclassic form. This provides another tantalis-

ing possibility that continues to fuel research

into this intriguing relation.

3. Theoretical Imperatives

Central to change and growth in any system

is positive feedback. At a purely phenomeno-

logical level, if the rate of growth (or de-

cline) is the same everywhere, the system

simply expands (or contracts) uniformly

without any change to its internal organis-

ation. This situation is rare. If the rate ofgrowth varies as a function of the size of

elements comprising the system, then the

largest elements will grow more than propor-

tionately, eventually dominating the system.

In city systems, such growth implies that the

largest city eventually dominates all the oth-ers; in short, it is as though the largest city

sucks all the growth away from the other

cities. This kind of situation is also rare, inthat the resulting steady state is simple and

unchanging, hardly characteristic of the kind

of growth that is implied by the empirical

evidence of the last section.

Arthur (1988) has examined these types of

growth through positive feedback, dening

three distinct cases. First, where the elements

of the system grow in proportion to some

natural advantage, even in the presence of

noise, this natural advantage reinforces itselfthrough positive feedback. There is no path-

dependence in that any deviations from the

pattern of natural advantage are eventually

removed. Arthurs (1988) second case is

quite the opposite in that growth is based on

chance rather than necessity. Growth occurs

in proportion to what is already there but inthe presence of noise, and although this pro-

cess settles to a steady state, noise dictatesentirely what this state will be. The third case

is one where growth depends both on natural

advantage and on economies of scale in that

as a city grows its future growth depends on

what is already there. If one region gets

ahead, then this growth is reinforced and

eventually, the region or city will dominate.

This is akin to the situation where all growthis attracted towards the biggest city as the

rate of growth is largely a function of the size

of each city.

In the last two cases, the ultimate state of

the system depends on the way random

events or noise dictate its historical evol-

ution, but it is the second case that appears

most realistic. From a random distribution,

some pattern ultimately emerges which is not

based on simply one city asserting itself.

However, Arthur does not deal with cases

where the growth rate itself is random, f or

his model is based on the randomness of

location. Thus a fourth, more persuasive,case might be considered where growth is in

proportion to size but the rate of growth is

random, thus ensuring that there will be no

ultimate unchanging steady state. To demon-

strate this, we dene population in area i and

time t as Pit where for the moment we sup-press rank r. Then the change in population

between any two time-periods t and t1 1 is

dened asD

Pit5

Pit1

12

Pit, with the rateDPit/Pit. We are now in a position to formal-

ise the process of growth.

First, we state that the rate of growth is

random, dened as

DPit

Pit5 e it (8)

where, it is the random rate of growth

associated with area i from time t to t1 1.

Using an appropriate form for equation (8)and integrating from the initial distribution

Pi0 to the current Pit gives

log(Pit)2 log(Pi0)5Ot

s5 0

e is (9)

The model thus becomes

Pit5Pi0Pt

s5 0

e is (10)

where the change from one time-period to

the next is

Pit1 15 e it Pit (11)

The process implied by equations (10) and

(11) is one of proportionate random growth



9/22

MICHAEL BATTY642

where the increments of growth are dis-

tributed lognormally. From any initial distri-bution Pi0 which may in itself be random (or

uniform), the limit of the process is lognor-

mal. If the growth rates were all the same

that is, there were no randomness and it5 c,

"itthen there would be no redistribution of

activity from the initial distribution, that is,

Pit/Pkt5Pi0/Pk0. Randomness is thus essential

in generating a distribution that is continually

changing.

To show that this model generates distri-

butions that are persistent and similar to

those which have been widely observed for

city systems (as in Figures 1 and 2), we have

run the model in equations (10) and (11) formany iterations, with parameters that loosely

approximate those of the population of Great

Britain between 1901 and 1991. We have set

the number of areas as N5 458, the total

time of the simulation as T5 1000, and we

have begun the process with a uniform distri-bution Pi05 1/N. We have run the model for

1000 time-periods which provides a rough

comparator with the settlement history ofBritain from 1000 to 2000. As with the em-

pirical data which we illustrated in Figures 1

and 2, we show the rank size relations pre-

dicted from the proportionate growth model

in Figure 3 for Pir(900) and Pir(1000) against their

respective ranks ri900 and ri1000. We also show

the distribution Pir(1000) against the previous

rank ri900 which is measure of the displace-

ment of ranks over 100 time-periods.

During this period, the shift in populationK1000 is about 22 per cent which is about halfthe actual shift in the real data for Great

Britain. The average number of ranks which

are altered during this time is K10005 34which is about 7 per cent of the total number

N. These gures are less than half those

associated with the real change and this sim-ply implies that we need to redimension the

time over which the process operates to bringthe simulation nearer the observed changes

for Great Britain. In fact, it is not the actual

values that are important here but the

congurational nature of the simulated rela-

tions. The forms of the rank size relations

generated in this way are extremely close to

those observed in Figure 2 for Great Britain.

Lognormal distributions do emerge from theinitial uniform distribution of settlement as

expected and the shift in ranks over 100

time-periods (t5 900 and t5 1000) has a

similar form to that shown in Figure 2. In

fact, comparison of Figures 2 and 3 which

are dimensioned on the same axes, imply that

the observed and simulated distributions are

from the same class.

To impress this similarity even further, we

have tted straight lines to the simulated

rank size relations in Figure 3. The initial

distribution here at t5 1 is uniform and the

variation with respect to rank is thus arbi-

trary. For comparison, however, we haveincluded this in the estimation which we

show in Table 2. As the total populations are

not comparable in any way with the observed

results, then we have regressed the market-

share equivalents pir(900) and pir(1000) against

their respective ranks. The results are givenin Table 2.

It is worth noting how similar these results

are to those for Great Britain. Although theslope increases slightly as the simulation

continues between t5 900 and t5 1000, im-

plying a slight increase in concentration, the

size of the slope and the correlations are

consistent with those in Table 1 for the 1901

and 1991 British regressions. This, combined

with the visual consistency between Figures

2 and 3, implies that the law of proportionate

effect which is based on random growth

appears extremely appropriate for explainingthe persistence of settlement structure and

the way cities or areas can slowly change

their relationships to one another through

temporal growth.

In fact, this model was rst suggested by

Gibrat (1931). Surprisingly, although widely

referenced in the rank size literature, it hasnot been used very much. A variant due to

Simon (1955) has been recently popularisedby Krugman (1998), and there was some

considerable discussion in the literature of

the Gibrat Simon model in the 1970s (see

Parr, 1976). More recently, there has been a

reawakening of interest in Zipfs law and

proportionate effect in theoretical physics



10/22


Figure 3. Simulated ranksize and population shifts between t5 900 and t5 1000.

(see Marsili and Zhang, 1998). But the single

most important issue here is the fact that

settlement systems seem to persist over long

periods with few changes in the rank of areas

and that such persistence is due to random

but proportionate growth, negating any econ-

omies of scale that might be associated with

larger cities. This means that over very long

time-spans, this process can change the rank

and size of cities dramatically. Thus the pro-

cess does provide opportunities for radical

change in structure while at the same timeimplying the persistence of well-established

patterns.

4. Urban Landscapes from the Bottom-up:

Agents, Actions and Interactions

To demonstrate the argument that urban sys-tems evolve spatially through the incessant

application of weak positive feedback overlong periods of time, we require models

which meet three criteria. First, the models

must treat activities at as disaggregate a scale

as possible, ideally at the individual scale so

that the greatest possible micro-diversity

occurs consistent with random growth. Sec-

ondly, growth (or decline) must occur

through the routine action of many decisions

slowly building on one another over long

periods of time; while, thirdly, to evolve city

systems where different clusters of cities are

highly interdependentwhich is the modern

mark of polynucleation or network cities

such models must simulate movement be-

tween individuals and the resources they

consume at the micro-level. Agent-based

models of this kind have only recently made

their appearance in the social sciences,largely due to advances in computation and

data which enable individual objects or

events to be simulated explicitly and, to date,

most applications have been to theoretical

situations (Epstein and Axtell, 1996). Here

we will develop a series of such models

which reect a plausible way of unpacking

the processes that lead to the kinds of persist-

ence and polynucleation representative of thechanges in settlement patterns that we mea-

sured earlier in an empirically and theoreti-

cally aggregate manner using rank size

relations. Once we demonstrate how these

models work, and what they are able to

produce, we will also show that these models



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MICHAEL BATTY644

Table 2. The rank size relation tted to the data from the random growth model

Year t Correlation R2 Intercept Kt P*1t5 10Kt Slope at

1 1 0 1 0900 0.840 2 1.077 0.083 2 0.777

1000 0.844 2 0.995 0.101 2 0.824

are able to generate persistent polynucleated

structures with similar rank size relations.

Agents A are now dened by the subscript

i and resources R by the subscript k. At the

most basic level, agents move within the

system f rom their origins given by co-ordi-

nates Xi, Yi in search of resources (to keepthem alive!) which are located at destinations

Xk, Yk. The location of agents as they move to

capture resources is signicant in that their

location at time t is given by A(xit, yit while

resources do not move and hence are xed at

R(Xk, Yk). The resolution at which the spatial

system is represented, and at which move-

ment takes place, is given by a series of

ne-scale cells dened by distances Dx and

Dy, and the movement of any agent i from

time tto t1 1 is always computed in terms of

their co-ordinates as

xit1 15xit1 Dxit1 ex tt

and

yit1 15yit1 Dyit1 eytt (12)

Note that the changes in co-ordinates Dxit and

Dyit as well as the small random errors inmovement xit and yit are specic to agent i

at time t. The computational mechanics re-

quired to simulate such movement are not

central to this paper, but their details can be

found elsewhere if required (see Batty and

Jiang, 2000).

In essence, the structure we are suggestingis another way of representing the movement

of individuals from, say, home to work, orany other kind of trip-making decision, and

the basic model we will begin withwhich

is at the basis of all our subsequent models

can be conceived in these terms. Origins and

destinations do not vary and the dynamics

that are implicit in this basic structure are

those of routine movement. However, be-

cause we are simulating an interaction sys-

tem between cities that evolves through time,

we assume that when new agents enter the

system (through migration or growth), then

they need to learn where the resources are

located. They do this by responding to thesignals that have been left in the system by

preceding or current agents. These signals

form a landscape of routes or tracks which

we dene as a continuous function W(x ,y).

Now, the movement dynamics for any agentcan be separated into two phases: rst, in

terms of responding to where resources are

available, which involves learning in some

way about their location; and, secondly, in

terms of their return to origins once resources

have been located, so that the resources

might be consumed in some way. The learn-

ing phase of this dynamic proceeds by agents

moving in direct response to the local gradi-

ent of the landscape which records the local

density of interaction already established by

the behaviour of previous agents. Each agent

works out their local gradient which is given

as

gradW (xi, yi)


12/22


R(Xk, Yk), then it engages with these re-

sources in some way and thence returns to itsorigin using a different procedure. Agents

remember where they are located at

A(Xi, Yi) and thus x their changes in direc-

tion Dxi and Dyi as function of the origin

location, that is

Dxi5 f(Xi) and Dyi5 f(Yi) (14)

However, once the resources have been dis-

covered, the agent also remembers this lo-

cation. As the agent returns to the origin, it

lays a trail or updates the landscape func-

tion to reect this new knowledge by

W (xit1 1, yi t1 1)5W(xit, yit)1 / (15)

where, the constant / is a pulse reecting

the increased interactivity of the landscape at

this point, thus adding to the attraction of that

location as a signal for future movement.

These models are increasingly popular for

simulating movement where learning is aprerequisite to establishing stable structures

which emerge from no knowledge of the

landscape whatsoever. They have been de-veloped for simulating animal populations

such as ants (Resnick, 1994), but are

nding wide use in modelling movement

systems which are too complicated to rep-

resent in formal frameworks such as

telecommunications trafc, travelling sales-

men problems and the like (Bonabeau et al.,

1999). They are also being employed in the

micro-dynamics of trafc simulation, crystal

growth and related problems in physicswhere they are referred to as active walker

models (Lam and Pochy, 1993). They are

being used to model actual behaviour based

on walking in cities and buildings where they

are forming a new basis for pedestrian mod-

elling (Helbing et al., 2001). In this context,

they are essential in building up spatial struc-tures, from the beginning of time so to speak,

where the original landscape is uniform withno settlement, something which is implicit in

the long time-scales adopted by the approach

presented in this paper.

In the rest of this section, we will simply

illustrate the working of the model, noting

that the local dynamics of movement simu-

lated here will eventually be embedded in

broader dynamics where origin and desti-nation activity reect growth of the system.

Our rst example is based on one origin and

a regular small compact cluster of destina-

tions with all our examples being simulated

on a 2003 200 lattice. We have located 1000

walkers at the centre of this lattice and

located a square area of resources some 80

units distant to the north-east of the origin.

We can display three key patterns from this

model: rst, the actual position of each of the

agents in the space; secondly, all the paths

that the agents have taken so far; and the

landscape function (which represents cumu-

lative laying down of permanent tracks) atany time t. The 1000 walkers rst move out

from the origin in random direction because

the landscape function w(xi, yi) is uniform

everywhere; no resources have yet been

found and thus the pattern of movement

computed from equation (12) is random. Weshow this initial situation in the top two maps

in Figure 4 where we omit the landscape

function as it is uniform. As agents locate thearea of resources through this random

Brownianmotion, they then return to the

origin, laying trackswhich, in this case of

one origin and one destination, is a straight

line between the centre of the space and the

square patch of resources. After 200 time-

periods, the agents have covered the entire

space and, although the landscape function is

now quite distinct, there are still many agents

in the space who have not yet discovered theresources. By time t5 2000, the agents

within space show a denite tendency to

follow the tracks which now dominate both

the pattern of all tracks to date as well as the

landscape function. Figure 4 represents a

clear pattern of spatial learning and, in this

case, the simplest of networks where there isno competition between nodes.

We now relax the problem by clusteringagents in 10 origin locations themselves ran-

domly located in the space, and 13 desti-

nation locations. The 1000 agents are

uniformly but randomly assigned to origins.

Destinations, as before, do not have volume

but simply area. We show these data in the



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MICHAEL BATTY646

Figure 4. The simplest settlement: one origin, one destination.

top row of Figure 5 which also illustrates the

evolution of the structure over 5000 time-pe-

riods. In this simulation, because the agents

have greater opportunities for discovering re-

sourcesthe densities are higher and the

locations more variedby t5 50, tracks arebeing marked out on the landscape even

though the entire space has not yet been

explored. By t5 500, the nodal structure

essentially the main links between origins

and destinationsis being stamped out in the

landscape and, by t5 5000, the structure is

extremely clear with most of the agents mov-

ing on the main links in the system.

What is interesting about these simulations

is how the transport structure becomes struc-

tured through random discovery with the

densest links in the central part of the spaceand links between any origin and destination

reecting distance in a simple way. In fact,

competition between nodes is clearly evident

in this structure with nodes and links that get

ahead faster in the process clearly retaining

this superiority. The pattern which is marked



14/22


Figure 5. Emergence of interactions between clusters of origins and destinations.



15/22

MICHAEL BATTY648

Figure 6. Settlement and interaction based on randomly located agents.

out by this system is polynucleated of course,

but is largely fashioned by the assumptions

made about xed origins and destinations.

We should also note that once an agent nds

a resource, the resource is consumed with a

given probability ensuring that agents are

able to exercise choice in their acquisition.

Our last example relaxes the effect of prior

structure. In Figure 6, we show the simu-

lation where the original distribution of

agents is entirely random but where there are

a small xed number of resource locations

which dominate the structure. The structure

which eventually emerges by t55000, repre-

sents the simplest star-shaped patterns of in-

teraction around resources. In fact, what

happens in this simulation is that the tracks

gradually begin to reduce in number as trackscompete against each other. For example, as

a track in the landscape begins to build up, it

attracts walkers from other tracks and thus

nearby tracks which run almost parallel be-

gin to compete for walkers. This process is

entirely conditioned by the degree of ran-

domness in path selection as reected in

equations (12) and (14). Although this is not

the clearest of examples, these effects doreveal that, even at this level in terms of

these models, history matters in that if one

location gets ahead, it tends to keep this

advantage through positive feedback. The

structure thus persists but it is held in a

precarious balance in that, over the much

longer term, such feedbacks can gradually

decay in their impact.

5. Evolving Polynucleated Urban Struc-

tures

So far, our origins and destinations have

been xed, there has been no growth, and allwe have shown is the way positive feedback

enables interactions to be established be-

tween these xed activities. We can now

relax the model and embed the movement

dynamics within a wider process of urban

dynamics which enables new agents and new

resources to be located while retaining the

learning capabilities of the interaction behav-

iour. In our models, we will assume a uni-

form growth process, locating one new agentat each time-period, introducing two possi-

bilities: rst, we will simply locate new

agents keeping the initial distribution of re-

sources xed; secondly, we will introduce

both new agents and new resources which

the agents generate, but again keeping the

growth process uniform. We could generatechanges to the location of agents through

internal movement of these locations in re-sponse to how the system is developingin

much the same way, for example, as agricul-

tural populations were attracted to the grow-

ing cities during the industrial revolution.

However, we prefer here to generate new

agents by locating them with respect to some



16/22


measured structure of the system, based on

some potential P(x, y, t) which reects the

spatial distribution of the agents so far and

which, like the landscape function introduced

above, responds to changes in new agent

location as a positive feedback.

We dene this potential function as

P(x, y, t1 1)5P(x, y, t)1 h=2 P(x, y, t)1

o(x, y, t)1 e(x, y, t) (16)

where, the diffusion term and its coefcient h

represent the gradual spreading of the poten-tial through time; the term o(x, y, t) repre-sents the additive effect on the potential of

one agent locating in each time-period; and

(x, y, t) is an error term that enables a de-

gree of diversity to be introduced in the

process.

Note that, although potential is computed

for every point or cell in the space at eachtime, only one location is associated with a

new agent o(x, y, t). The extent to whichpotential is reinforced through the continuedapplication of equation (16), depends on the

balance of forces within the equation. Inthese simulations, the effect of the error term

is substantial and this is dimensioned so that

current local potential is most important but

that distant places from the current highest

levels of potential can always grow throughsuch errors being reinforced. Once potential

is computed in this way, the location of a

new agent i at co-ordinates Xit1 1,Yit1 1, is

based on identifying the maximum value of

equation (16). That is,

Xit1 1 , Yit1 15maxxy

{P(x, y, t1 1)} (17)

and thence the variable o(x, y, t) is set as aunit increment for the location Xi,Yi which

drives the function in equation (16).In Figure 7, we show the evolution of a

spatial system to t5 2000 using this model.We randomly locate some 6 initial resource

locations but, from then on, we use equations(16) and (17) to locate the agent population.

From this simulation, we see that several

clusters of population grow up, not adjacent

by any means to the initial resource loca-

tions. Although we do not show this, these

clusters do not grow uniformlythat is, they

do not emerge together. There is a complex

dynamics initiated by this process which is

reected in the interaction maps as well as

the polynucleations which result. The poten-

tial from equation (16) is also shown in

Figure 7 which reveals a diverse picture of

change in that there are locations with com-paratively high potential at time t5 2000

which have not yet had a chance to establish

their claim for the location of new agent

populations. It is the micro-diversity of this

potential surface that ensures competition be-

tween clusters of settlement within the sys-

tem.

We can now describe a much fuller ver-

sion of this model where we add the locationof new resources through time. Assuming

that our agents are populations and our re-

sources employment activities, we argue that

one new employment activity generates three

units of population and that the units of

employment activity are industrial and ser-

vice in their orientation. This is a long-stand-

ing distinction in urban systems and all it

means is that industrial activities are more

likely to be generated from other industrialactivities, while service activities are more

orientated to populations. Here, this implies

that the initial resource locations tend to

generate industrial resources more than

population or services. These relationships

are casually built into the model in terms of

the weighting of potential for new agents and

resources.

The potential function in equation (16)above can now be extended and generalised

with respect to the new activities and theirrelationships in the following way:

P(x, y, t1 1)5P(x, y, t)1 hP=2 P(x, y, t)1

Oi

oi(x, y, t)1 eP(x, y, t)

I(x, y, t1 1)5I(x, y, t)1 hI=2I(x, y, t)1

Oioi(x, y, t)1 eI(x, y, t)

S(x, y,t1 1)5 S(x, y, t)1 hS=2 S(x, y, t)1

Oi

oi(x, y, t)1 eS(x, y, t) (18)

The structure of the equations in (18) is



17/22

MICHAEL BATTY650

Figure 7. The emergence of polynucleated clusters of agents.

similar to the single potential for agents or

populations in equation (16) except that the

sum of objects oi(x, y, t) over i is meant toreect a sum of both new agents and new

resources. The parameters of diffusion and

the error terms are of course specic to thepotential involved. Finally, we should note

that the initial levels of potential are ran-

domly distributed. New activitiespopu-

lation P(xit, yit), industry I(xit, yit), and

services S(xit, yit)are then located in cells of

maximum potential from equation (18) ac-

cording to the previous logic in equation(17).

We have run this model up to t5 2000,and we show these simulations in Figure 8.

In this model, there are many outputs that we

can examine, but here we will concentrate

rst on the patterns of population that are

produced. It is quite clear that these patterns

do not bear much resemblance to the initial

distribution of resources. Moreover, it is also

clear that new spatial clusters emerge

throughout the process, thus ensuring that no

particular cluster becomes singly dominant.

This is, of course, the typical signature of a

polynucleated urban landscape. In Figure 8,we also show the pattern of transport routes

which indicates how the earlier clusters have

more intensive interaction than the later clus-

ters. In fact, as new resources begin to domi-

nate the initial distribution, the initial pattern

of resources is soon wiped out. As new

resources depend on new populations andvice versa, the structures that emerge are in

fact highly consistent with those that are seenin real city systems such as in the north

eastern seaboard of the US, in the Randstad

or in the north-west of England. We also

show here the potential for the service re-

sources after 100 time-periods of the simu-

lation, and thence after 2000. From an initial



18/22


Figure 8. The emergence of the polynucleated urban landscape.



19/22

MICHAEL BATTY652

pattern which is highly correlated with the

early population clusters, the potential gradu-ally diffuses. In this paper, we are not going to

dwell on the mechanics of tting hypothetical

simulation models such as these. Sufce it to

say that even with the very small number of

parameters that these models use, there are still

considerable variations in the outputs that can

be generated. However, in the case of the

potentials critical to the models growth

dynamics, it is likely that the errors rates need

to be higher than those we have used to reect

greater micro-diversity. However, even with

those used, the spatial structures which emerge

clearly represent strong tendencies towards

polynucleation.

6. The Simulated Evidence

It is a straightforward matter to compute

rank size relations from the urban develop-ment patterns produced by the full model as

shown in Figure 8. However, the model only

produces uniform densities in each cell, and asthe space lls up, rank size relations will

inevitably atten in their long tails. What we

have done is to take the scanned patterns

captured on 4003 400 grids and to aggregate

these to 203 20 grids from which counts of

the occupied cells are made. We have excluded

all those cells which have zero occupancy.

Although the total possible lled cells is

400 (which compares with the previous em-

pirical data for Great Britain and the aggregatetheoretical model), only a subset of these

have activity which rises from N5 88 at

t5100 to N5 282 at t5 2000. The rank size

relations are shown in Figure 9. These

are much less smooth than the previous

empirical and theoretical aggregates which is

due to the somewhat coarse nature of thespatial simulations. However, they reveal clear

evidence of scaling, similar to that shownearlier in Figures 1 3. The relations shown in

Figure 9 are based on the 7 patterns shown in

Figure 8 for t5 100, 200, 500, 900, 1000, 1900

and 2000.

The scaling relations appear similar through

time, notwithstanding the gradual attening

of these curves. In fact, relations are near

pure rank size to begin with but, as thesimulation proceeds, they become progres-

sively like those observed in reality (Figures 1

and 2). To gauge this, we have tted the

rank size equation from (6) to the data at

these time-periods and Table 3 presents these

results. Strictly speaking, we should only

t these relations to the long tail of the

distributions in Figure 9, but it is still clear that

the slopes are higher than those of the

empirical and theoretical aggregate models.

However, although the slope increases slightly

at rst, it then decreases through time, without

any signicance in terms of increasing or

decreasing the concentration of settlementclusters.

To test the changes between time-

periodsa better test of the stability of the

systemwe have computed the percentage

shifts in population shares given by C 2000 and

the shifts in rank order K 2000 and k2000. We havecomputed these between t5 1900 and

t5 2000, although this period is probably too

small. This is thus only indicative of the sortsof shift that are taking place in this model as

well as in real cities and in the theoretically

aggregate model. The percentage shift C 2000 is

of the order of 8 per cent which is much smaller

than that given by the British data and the

aggregate model. This is reected too in the

average rank shift K 2000 which is 33 out of 282,

with a percentage rank shift k2000 of 12 per cent.

We have plotted the t5 1900 and t5 2000

logged distributions of pir(1900) and pir(2000)against their log of rank in Figure 10 where we

also show log(pir(2000)) against log(rir(1900)) to

show the shifts in rank. In fact, although the

shift in rank is similar to that for the aggregate

model given previously, the nature of the shift

is different with less numbers of aggregated

cells changing much more than in the previousanalyses.

Nevertheless, what this analysis doesshow is that the spatial disaggregate model

produces rank size relations which are con-

sistent with our earlier examples and that,

with some redimensioning of the temporal

process over which these changes are com-

puted, both models in this paper are producing



20/22


Figure 9. Rank size distributions from t5 100 to t5 2000 for the agent-based model.

Table 3. The rank size relation tted to the agent-based simulation data

Period t Correlation R2 Intercept Kt P*1t5 10Kt Slope at

100 0.962 2.508 322.474 2 1.033200 0.952 3.325 989.708 2 1.195500 0.887 3.324 2113.321 2 1.222900 0.888 3.590 3892.044 2 1.256

1000 0.869 3.643 4399.487 2 1.2561900 0.804 3.757 5721.635 2 1.1852000 0.799 3.798 6278.507 2 1.197

similar results which bear out the empirical

analysis.

7. Next Steps

Plausible though the models in this papermight appear, they are only one variety from

several new approaches which appear prom-ising in the analysis of the evolving spatial

structure of city systems. The notion that

cities exist at a critical threshold which they

maintain through growth and redistribution is

also one which is consistent with persistent

urban structures. Work along these lines has

shown that cities can grow and change rad-

ically in terms of their densities but, at the

same time, remain within strong limits (Bak,

1996). Batty and Xie (1999) explain how

cities continually readjust their form to a

supercritical level, lling their geometric

space in the same way they have done for

decades. For cities to make the transition toother spatial forms, there have to be radical

changes in technology and/or behaviour and

it would appear that, in contemporary times,

this might only come if people were radically

to change their patterns of movement. We

have not explored the extent to which these



21/22

MICHAEL BATTY654

Figure 10. Simulated rank size and population shifts between t51900 and t52000.

models keep within critical limits, although

casual observation of the forms that they

generate suggest that they do.

At one level, the evidence for persistent

urban structures that we have presented here,

and the kinds of polynucleated urban land-

scapes that emerge over centuries and mil-

lennia, are obvious. It is easier to build and

replace on what has gone before than to

strike out anew. In this very denite sense,

history matters. But, at another level, thiskind of persistence is harder to explain and

simulate than new activities. Weak positive

feedbacks are more difcult to pin down than

the stronger forces that continually impress

economies of scale. In fact, persistence may

be due more to interactions in the economy

than to building on and around what already

exists, in that individuals can adjust their

interaction patterns quite dramatically with-out changing the spatial structures in which

those patterns exist. These are ideas for the

future but all the elements for their develop-

ment are contained in the approach which we

have chosen to pursue here. It could also be

remarked that the approach here is too phys-

icalistalthough once very long-term dy-

namics are invoked, physical competition

between cities provides an implicit economicdynamics which generates plausible spatial

structures. What these dynamics also show is

that polynucleated spatial structures are more

likely to be the rule rather than the exception.

Moreover, the edge city phenomenon is but

one variety of node in a sea of development

which is continually restructuring and chang-

ing.Finally, this paper has presented a some-

what unusual exercise in analytical mod-

elling. In fact, we have presented two

different types of modelone aggregate

based on random proportionate growth where

the spatial characteristics of the system are

completely absent, and the second a spatially

disaggregate version which we argue con-

tains the same kind of weak growth mecha-nisms that characterise the aggregate model.

Both these models generate persistent struc-

tures which are similar to those found in real

city systems, while the spatial model sug-

gests that the usual form of such growth is

likely to be polynucleated. There is much



22/22


work to do on this approach in that our mix

of the empirical and theoretical at different

scales and levels enables us to capture the

kind of diversity that is characteristic of cit-

ies and that we have found hard to deal with

hitherto. In future work, we will take these

ideas further, linking interaction to locationthrough explicit feedbacks and dimensioning

the growth levels and time-paths of our

theoretical models more appropriately than

we have been able to do here.

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Polynucleated Urban Landscapes (Batty 2001)