PROPOLIS

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Tuesday, 24 January 2006 Said Business School
CABDyN Seminar Cities and Complexity
Explaining the Dynamics of Urban
Scaling Michael Batty University College
London m.batty_at_ucl.ac.uk http//www.casa.ucl.ac.
uk/
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I will tell the story as I go along of small
cities no less than of great. Most of those which
were great once are small today and those which
in my own lifetime have grown to greatness, were
small enough in the old days From Herodotus
The Histories Quoted in the frontispiece by
Jane Jacobs (1969) The Economy of Cities, Vintage
Books, New York
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• Outline of the Talk
• What is Scaling? What is Rank Size?
• City Size/Rank-Size Dynamics
• Explanations Lognormality, Stochasticity,
  Hierarchy
• Volatility within Stability
• Reworking Zipf The US Urban System The
  Emergence of Cities
• The UK Urban System
• Rank Clocks
• Next Steps
• Acknowledgements
• Rui Carvalho, Richard Webber (CASA, UCL)
• Tom Wagner, John Nystuen, Sandy Arlinghaus (U
  Michigan)
• Yichun Xie (U Eastern Michigan) Naru Shiode
  (SUNY-Buffalo).

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• What is Scaling? What is Rank Size?
• Things that scale are things that look the same
  as we make the scale bigger or smaller these
  are fractals in fact and what this means is that
  we never have any characteristic scale on which
  to ground the phenomena
• So for example if we look at a graph of
  frequencies of an object occurring against its
  size, if it scales then this means that whatever
  portion of the curve we look at, it appears the
  same
• A curve based on a power law scales in that if we
  change the scale, then this simply magnifies or
  reduces the curve

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Let me take a simple example surnames if we
rank the surnames from the most common to the
least, then what we get from the 1996 UK
electoral register is the following
1 SMITH 560122 2 JONES 431558 3 WILLIAMS 285836 4
BROWN 264869 5 TAYLOR 251567 6 DAVIES 216535 7 WIL
SON 192338 8 EVANS 173636 9 THOMAS 154557 10 JOHNS
ON 145459
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Now let us plot the graph of frequency versus
rank and then also transform this to a linear
scale for all 25630 names in the data
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1996 1881 SMITH 560122 SMITH 406573 JONES 43
1558 JONES 336447 WILLIAMS 285836 WILLIAMS 21260
2 BROWN 264869 BROWN 192061 TAYLOR 251567 TAYLOR
186584 DAVIES 216535 DAVIES 152450 WILSON 192338
WILSON 136222 EVANS 173636 EVANS 129757 THOMAS
154557 THOMAS 122449 JOHNSON 145459 ROBERTS 1116
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Changes in Rank from 1881 to 1996 in the British
Electoral Role
1996 1881 HUNT 83 HUNT 78 BATTY 1254 BATTY
957 STEADMAN 1835 STEADMAN 2377
The size of the population has increased from
around 26 to 40 million
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2. City-Size/Rank-Size Dynamics
The Strict Rank-Size Relation
The first popular demonstration of this relation
was by Zipf in papers published in the 1930s and
1940s
The Variable Rank-Size Relation
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Fixed or Variable Numbers of Cities and
Populations
Growth or decline pure scaling The number of
cities is expanding or contracting and all
populations expand or contract
mixed scaling Cities expanding or contracting,
populations expanding or contracting
The number of cities is expanding or contracting
and top populations are fixed.
The number of cities is fixed and all populations
are expanding or contracting
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Let us now look at a more conventional view of
frequency and rank size. If we examine the size
distribution of cities, we find they are not
normally distributed but lognorrnally
distributed, and can be approximated by a power
law
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On log log paper, the counter cumulative or rank
size distribution shows linearity over most of
its length and can be approximated note
approximated by a power law. Here is an example
for the distribution of over 20000 incorporated
places from 1970 to 2000 from the US Census
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This shows several things Remarkable macro
stability from 1970 to 2000 Classic lognormality
consistent with the most basic of growth
processes proportionate random growth with no
cities having greater growth rates that any other
A lack of economies of scale as cities get
bigger which is counter conventional
wisdom Remarkable linearity in the long or fat
or heavy tail which we can approximate with a
power law as follows if we chop off the data at,
say, 2500 population we will do this
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Parameter/Statistic 1970 1980 1990 2000
R Square 0.979 0.972 0.973 0.969
Intercept 16.790 16.891 17.090 17.360
Zipf-Exponent -0.986 -0.982 -0.995 -1.014
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Notice the slopes of these straight lines so
close to 1 This is Zipfs Law termed after
George Kingsley Zipf who first popularized it as
the rank-size rule Zipfs Law says that in a
set of well-defined objects like words (or cities
? Or incomes? Pareto), the size of any object is
inversely proportional to its rank and in the
strict Zipf case this is This is the strict
form because the power is -1 which gives it
somewhat mystical properties but a more general
form is the inverse power form
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3. Explanations Lognormality, Stochasticity
Hierarchy The last 10 years has seen many
attempts to explain scaling distributions such as
these using various simple stochastic processes.
Most do not take any account of the fact that
cities compete talk to each other. In essence,
the easiest is a model of proportionate effect or
growth first used for economic systems by Gibrat
in 1931 which leads to the lognormal
distribution There are many models based on this
from Simon (1955) to Solomon and Blank (2001)
all variants on this theme let me show you how
this works very quickly
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This is a good model to show the persistence of
settlements, it is consistent with what we know
about urban morphology in terms of fractal laws,
but it is not spatial. In fact to demonstrate
how this model works let me run a short
simulation based on independent events cities
on a 20 x 20 lattice using the Gibrat process
here it is it will produce a lognormal but the
volatility of the dynamics is suspect an
example of a simple phenomena simulated
beautifully by a simple model parsimony at its
best which is the hall mark of science, but
something that we know must be wrong !
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I will digress a little here. Recently there has
been a dramatic growth of network science due to
people like Watts, Barabasi, Newman and so on.
Essentially many of the results of scaling and
lognormality that appear in city size, income,
word distributions and so on, appear to hold for
interactions associated with networks. It is a
simple matter to generate a model of a growing
network where cities connect to each other
according to what Barabasi calls preferential
attachment ie the more the number of links in
a hub, the more likely they are to get links.
This is similar to Gibrats model of
proportionate growth. It is no surprise that we
get interactions distributed according to power
laws or rank size as we can show ??
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Three models a digression
Most models which generate lognormal or scaling
(power laws) in the long tail or heavy tail are
based on the law of proportionate effect. We will
identify 3 from many Gibrats Model Fixed
Numbers of Cities
Most models which generate lognormal or scaling
(power laws) in the long tail or heavy tail are
based on the law of proportionate effect. We will
identify 3 from many Gibrats Model Fixed
Numbers of Cities
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Gibrats Model with Lower Bound (the
Solomon-Gabaix-Sornette Threshold) Fixed Numbers
of Cities
Gibrats Model with Lower Bound Simons
Model Expanding (Contracting) Numbers of Cities
And there are the Barabasi models which add
network links to the proportionate effects. See
M. Batty (2006) Hierarchy in Cities and City
Systems, in D. Pumain (Editor) Hierarchy in
Natural and Social Sciences, Springer, Dordrecht,
Netherlands, 143-168.
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4. Volatility within Stability What is so
remarkable is the fact that we have such
volatility within such macro stability cities
are shifting their positions all the time as we
can see if we compare stable rank systems
population of US counties between 1940 and
2000 In essence the stochastic models generate
too much volatility and what we need is to add
more inertia and of course to add space Let me
show this volatility by showing how ranks shift
over this 60 year period
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Rank-size of Population of US Counties 1940 and
2000 with red plot showing 2000 populations but
at 1940 ranks
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5. Reworking Zipf The US Urban System The
Emergence of Cities I am now going to look at
the US, then the UK urban system. We have noted a
number of data sets but we will only deal here
with the top 100 towns or cities in terms of
population size from 1790 to 2000. There are in
fact 268 distinct cities that enter and leave the
top 100 between these dates but the data is
consistent in terms of definition. So we are
looking at the top of the rank-size hierarchy and
in fact it is not until 1840 that we actually get
100 towns defined in the US Census. As we will
see, the urban system is rapidly growing over
this 210 years.
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Now we are going to look at the dynamics from
1790 to 2001 in the classic way Zipf did and this
is an updating of Zipf. We have taken the top 100
places from Gibsons Census Bureau Statistics
which run from 1790 to 1990 and added to this the
2000 city populations We have performed log log
regressions to fit Zipfs Law to these We have
then looked at the way cities enter and leave the
top 100 giving a rudimentary picture of the
dynamics of the urban system We have visualized
this dynamics in the many different ways but
first we will show what Zipf did.
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There is a problem of knowing what units to use
to define cities and we could spend the rest of
the day talking on this. We have used what Zipf
used incorporated places in the US and to show
this volatility, we have examined the top 100
places from 1790 to 2000 But first we have
updated Zipf who looked at this material from
1790 to 1930 - here is his plot again
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In this way, we have reworked Zipfs data (from
1790 to 1930)
Year r-squared exponent
1790 0.975 0.876
1800 0.968 0.869
1810 0.989 0.909
1820 0.983 0.904
1830 0.990 0.899
1840 0.991 0.894
1850 0.989 0.917
1860 0.994 0.990
1870 0.992 0.978
1880 0.992 0.983
1890 0.992 0.951
1900 0.994 0.946
1910 0.991 0.912
1920 0.995 0.908
1930 0.995 0.903
1940 0.994 0.907
1950 0.990 0.900
1960 0.985 0.838
1970 0.980 0.808
1980 0.986 0.769
1990 0.987 0.744
2000 0.988 0.737
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For a sample of top cities we first show the
dynamics of the Rank-Size Space
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We have also worked out how fast cities stay in
the list we call these half lives We can
animate these
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6. The UK Urban System In the case of the US
urban system, we had an expanding space of cities
(except for the US county data which is a
mutually exclusive subdivision of the US
space) However for the UK, the definition of
cities is much more problematic. We do however
have a good data set based on 458 local
municipalities (for England, Scotland and Wales)
which has consistent boundaries from 1901 to
2001. So this, unlike the Zipf analysis, is for
a fixed set of spaces where insofar as cities
emerge or disappear, this is purely governed by
their size.
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Here is the data very similar stability at the
macro level to the US data for counties and
places but at the micro level.
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Here is an example of the shift in size and ranks
over the last 100 years
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This is what we get when we fit the rank size
relation PrP1 r - ? to the data. Rather similar
to the US data flattening of the slope of the
power law which probably implies decentralization
or diffusion of population dominating trends
towards centralization or concentration
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Now we show the changes in population for the top
ranked places from 1901 to 1991
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And now we show the changes in rank for these
places
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7. Rank Clocks I think one of the most
interesting innovations to examine these
micro-dynamics is the rank clock which can be
developed in various forms. Essentially we array
the time around the perimeter of a circular clock
and then plot the rank of any city or place along
each finger of the clock for the appropriate time
at which the city was so ranked. Instead of
plotting the rank, we could plot the population
by ordering the populations according to their
rank. For any time, the first ranked population
would define the first city, then adding the
second ranked population to the first would
determine the second city position and so on
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8. Next Steps The program to visualize many such
data sets Analysis of extinctions Many cities
and city systems The analysis for firms and
other scaling systems etc. etc.
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• Resources on these Kinds of Model
• http//www.casa.ucl.ac.uk/naru/portfolio/social.ht
  ml
• Arlinghaus, S. et al. (2003) Animated Time Lines
  Co-ordination of Spatial and Temporal
  Information, Solstice , 14 (1) at
  http//www.arlinghaus.net/image/solstice/sum03/
  and
• http//www.InstituteOfMathematicalGeography.org
• Batty, M. and Shiode, N. (2003) Population Growth
  Dynamics in Cities, Countries and Communication
  Systems, In P. Longley and M. Batty (eds.),
  Advanced Spatial Analysis, Redlands, CA ESRI
  Press (forthcoming). See http//www.casabook.com/
  
• Batty, M. (2003) Commentary The Geography of
  Scientific Citation, Environment and Planning A,
  35, 761-765 at
• http//www.envplan.com/epa/editorials/a3505com.pdf
  

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Academic Press, 1994 The MIT Press, 2005
http//www.casa.ucl.ac.uk/ http//www.casa.ucl.ac
.uk/citations/