Centre for Advanced Spatial Analysis and the Bartlett School

Michael Batty University College London

[email protected]

www.casa.ucl.ac.uk

Emergence and Extinction in Cities & City Systems

“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

Outline of the Talk

1. Preamble: Emergence, Extinction, Growth, Change 2. City-Size/Rank-Size Dynamics 3. The Simplest Models: Baseline Explanations 4. Visualizing Dynamics: A Demonstration 5. The US Urban System 6. The UK Urban System 7. Rank Clocks 8. Next Steps

The basic idea

20 16

Log of size

12 8 4 0 0 3

Log of rank

6 9

1. Preamble: Emergence, Extinction, Growth, Change

What is emergence? And what is extinction?

Emergence can be of two forms – the addition of new objects or cities in this case, or the rapid, unexpected growth of existing cities Extinction can mean the disappearance of cities or it might be the rapid decline of cities These are part of growth and change, the much under represented and much misunderstood character of cities and city systems

2. City-Size/Rank-Size Dynamics

The Strict Rank-Size Relation P r

 log

P r P

1

r

 1  log 

P

1

K r

 1  log

r The first popular demonstration of this relation was by Zipf in papers published in the 1930s and 1940s Log rank or Log r The Variable Rank-Size Relation P r



K r

  log

P r

 log

K

1   log

r

P 1 Fixed or Variable Numbers of Cities and Populations log P Growth or decline: pure scaling The number of cities is expanding or contracting and all populations expand or contract log r 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 mixed scaling: Cities expanding or contracting, populations expanding or contracting

3. The Simplest Models: Baseline Explanations

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 Gibrat’s Model: Fixed Numbers of Cities P i

(

t

 1 )  [ 1 

g i

(

t

)]

P i

(

t

),

i

 1 , 2 , ...,

n

 [ 1 

g i

(

t

)][ 1 

g i

(

t

 1 )] ...

[ 1 

g i

( 0 )]

P i

( 0 ),  

t

  0 [ 1 

g i

(  )]

P i

( 0 )

Gibrat’s Model with Lower Bound (the Solomon-Gabaix Sornette Threshold) Fixed Numbers of Cities P i

(

t

 1 )    [

T

1 

g i

(

t

)]

P i

(

t

),

if P i

(

t

) 

T Gibrat’s Model with Lower Bound – Simon’s Model Expanding (Contracting) Numbers of Cities P i

(

t

 1 )  [ 1 

g i

(

t

)]

P i

(

t

),

P i



j

(

t

 1 ) 

T

,

i

 1 , 2 , ...,

j



i

 1 ,

i

 2 , ...,

n if



j



z

, [ 0 ,

z

, 1 ]

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.

4. Visualizing Dynamics: A Demonstration

I am working on a comprehensive program which will essentially combine all the techniques that I introduce below. The visual evidence of space-time change must be notated by P, r, and t.

I haven't finished the program but I can say that we will introduce the following

• • • • •

Rank-size and related distributions, Change in rank over time, population over time Change in rank and populations over time, Half lives of population change, rank-clocks, Frequencies of extinctions/declines in rank

20 16 12 8 4 0 0 3 6 9

1901 100 80 60 40 20 0 1780 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 1981 1971 19 91 1961 1951 1911 1941 1921 1931 log frequency log size

5. The US Urban System

I am now going to look at the US, then the UK urban system. There are several data sets for each but for the US, we will begin with the 20000 incorporated places for which we have populations from 1970 to 2000 This data – in fact all our ranges of data – do not show power laws per se but show lognormal distributions which can be approximated by scaling laws in their long tail.

In fact, there is some controversy over whether or not the dynamics implied by Gibrat’s Law leads to power law distributions in the steady state. Nevertheless …

This picture 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

Parameter/Statistic R Square Intercept Zipf-Exponent 1970

0.979

16.790

-0.986

1980

0.972

16.891

-0.982

1990

0.973

17.090

-0.995

2000

0.969

17.360

-1.014

Now let us look at the rank-size of population of US Counties 1940 and 2000 with red plot showing 2000 populations but at 1940 ranks

20 16 12 8 4 0 0 3 6 9

Now we are going to look at the dynamics from 1790 to 2001 in the classic way Zipf did. This is an updating of Zipf.

We have taken the top 100 places from Gibson’s 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 Zipf’s 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 we implied earlier and we show these as follows but first we will show what Zipf did.

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

In this way, we have reworked Zipf’s data (from 1790 to 1930)

7 6.5

6 5.5

5 4.5

4 3.5

0 0.5

1 1.5

2 Year 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 r-squared 0.975

0.968

0.989

0.983

0.990

0.991

0.989

0.994

0.992

0.992

0.992

0.994

0.991

0.995

0.995

0.994

0.990

0.985

0.980

0.986

0.987

0.988

exponent 0.876

0.869

0.909

0.904

0.899

0.894

0.917

0.990

0.978

0.983

0.951

0.946

0.912

0.908

0.903

0.907

0.900

0.838

0.808

0.769

0.744

0.737

For a sample of top cities we first show the dynamics of the Rank-Size Space

10000000 Log City Size 1000000 New York City Houston 100000 Boston Charleston Los Angeles Philadelphia 10000 1000 Baltimore 1 Log Rank 10 Norfolk VA Richmond VA Chicago 100

We have also worked out how fast cities stay in the list & we call these ‘half lives’

100 80 60

We can animate these

40 20 0 1780 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

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.

Here is the data – very similar stability at the macro level to the US data for counties and places but at the micro level….

6.5

6 5.5

5 4.5

4 3.5

3 0 Log of Rank 0.5

1 1901 1.5

2 1991 2.5

3

Here is an example of the shift in size and ranks over the last 100 years

2.5

3 -1 0 -1.5

-2 -2.5

-3 -3.5

1991 0.5

on 1901 Ranks 1 1991 Population based 1.5

1901 2 -4 -4.5

Log of Rank

This is what we get when we fit the rank size relation P r =P 1 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

Year

t

1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 Correlation R 2 0.879

0.880

0.887

0.892

0.865

0.869

0.830

0.815

0.816

0.791

Intercept

K t

6.547

6.579

6.604

6.607

6.532

6.482

6.414

6.322

6.321

6.272

P

* 1

t

 10

K t

3526157.772

3801260.554

4025650.857

4046932.207

3410371.276

3034245.953

2595897.640

2101166.738

2095242.746

1872348.019

Slope 

t

-0.817

-0.810

-0.812

-0.802

-0.740

-0.700

-0.651

-0.601

-0.601

-0.577

Now we show the changes in population for the top ranked places from 1901 to 1991

1400000 1200000 1000000 800000 600000 400000 200000 0 1901 1911 1921 1931 1941 1951 1961 1971 1981 1991

And now we show the changes in rank for these places

0 20 40 60 80 100 120 140 1901 1911 1921 1931 1941 1951 1961 1971 1981 1991

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

The Rank Clock for the US data

1990 1980 1970 1960 2000 1790 Richmond VA Norfolk VA 1800 1810 1820 1830 Boston Baltimore Chicago 1840 Time 1950 1850 1940 LA 1860 1930 Charleston 1920 1910 Houston 1900 1890 1880 1870 Rank 1 20 40 60 80 100

The Log Rank Clock for the US data 1990 1980 2000 Norfolk VA 1970 1790 Boston Baltimore 1800 1810 1820 1830 1960 NY 1950 Philly 1940 1930 1920 Houston LA Richmond VA 1910 1900 Charleston 1890 1880 1870 1840 Time Chicago 1850 1860 (Log) Rank 1 10 100

The Rank Clock for The UK data 1991 1901 1981 1971 1961 1951 1911 1941 1921 1931 Camden Hackney Islington Lambeth Newham Southwark Tower Hamlets Wandsworth Westminster Barnet Brent Bromley Croydon Ealing Manchester Salford Wigan Liverpool Sefton Wirral Doncaster Sheffield Newcastle Sunderland Birmingham Coventry Dudley Sandwell Kirklees Leeds Wakefield Bristol Edinburgh Glasgow

Let me make a very slight digression on the population rank clock. Basically for the UK system, it is little different because the UK does not grow much in terms of the top 20 or so places.

16000000 14000000 12000000 250 200 10000000 8000000 150 6000000 4000000 2000000 0 1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 100 50 0 1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 9 8 10 7 200 180 160 140 120 100 80 60 40 20 0 1 6 2 5 3 4

But for the US system for the top few places the population changes very dramatically during the 210 year period and thus the population rank clock would be very different, more like a spiral. I have not had time to plot this yet but it would be like this in shape

60 50 40 Total Population in the Top 100 US Cities 9 10 200 180 160 140 120 100 80 60 40 20 0 1 2 3 8 4 30 7 5 20 6 10 Population NY City 0 1750 1800 1850 1900 1950 2000

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………….

Acknowledgements

Rui Carvalho, Richard Webber (CASA, UCL); Denise Pumain, U Paris 1 (Sorbonne) Tom Wagner, John Nystuen, Sandy Arlinghaus (U Michigan); Yichun Xie (U Eastern Michigan), Naru Shiode (SUNY Buffalo).

Resources on these Kinds of Model

http://www.casa.ucl.ac.uk/naru/portfolio/social.html

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/ http://www.InstituteOfMathematicalGeography.org

and 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