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 underrepresented and much misunderstood character of cities and city systems

2. City-Size/Rank-Size Dynamics Log population or Log P The Strict Rank-Size Relation Pr P1 r 1 K r 1 log Pr log P1 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 Pr K r log Pr log K1 log r Fixed or Variable Numbers of Cities and Populations log P Growth or decline: pure scaling P1 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 Gibrats Model: Fixed Numbers of Cities Pi (t 1) [1 g i (t )] Pi (t ), i 1,2, ..., n [1 g i (t )][1 g i (t 1)] ... [1 g i (0)] Pi (0), t [1 g i ( )] Pi (0) 0

Gibrats Model with Lower Bound (the Solomon-GabaixSornette Threshold) Fixed Numbers of Cities [1 g i (t )] Pi (t ), if Pi (t ) T Pi (t 1) T Gibrats Model with Lower Bound Simons Model Expanding (Contracting) Numbers of Cities Pi (t 1) [1 g i (t )] Pi (t ), i 1,2, ..., n Pi j (t 1) T , j i 1, i 2, ..., 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, 143168. 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 19 91 100 1911 80

1981 1921 1971 1931 60 40 20 0 1780 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 1941 1961 1951 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 Gibrats 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 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 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 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 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 Zipfs 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 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 For a sample of top cities we first show the dynamics of the Rank-Size Space 10000000 Log City Size New York City 1000000 Houston 100000 Boston Charleston Los Angeles Philadelphia 10000 Baltimore Richmond VA Norfolk

VA Chicago 1000 1 Log Rank 10 100 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 Log of Population 5.5 1991 5 1901 4.5 4 3.5 Log of Rank 3

0 0.5 1 1.5 2 2.5 3 Here is an example of the shift in size and ranks over the last 100 years 0 0.5 1 1.5 -1 Log of Population Shares -1.5 1901 -2

-2.5 1991 -3 -3.5 1991 Population based on 1901 Ranks -4 Log of Rank -4.5 2 2.5 3 This is what we get when we fit the rank size relation P r=P1 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 Correlation R2 1901 1911 1921 1931 1941 1951

1961 1971 1981 1991 0.879 0.880 0.887 0.892 0.865 0.869 0.830 0.815 0.816 0.791 Intercept Kt P*1t 10Kt Slope t 6.547 6.579 6.604 6.607 6.532 6.482 6.414 6.322 6.321 6.272 3526157.772

3801260.554 4025650.857 4046932.207 3410371.276 3034245.953 2595897.640 2101166.738 2095242.746 1872348.019 -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 1790 2000 1800 Richmond VA Norfolk VA 1990 1980 1810 1820 1830 1970 Boston 1960 Baltimore

Chicago 1950 1840 Time 1850 1940 1860 LA 1930 1870 Charleston 1920 1880 Houston 1910 1890 1900 Rank 1 20

40 60 80 100 The Log Rank Clock for the US data 1990 1980 2000 1790 1800 1810 Norfolk VA 1820 1830 Boston Baltimore 1970 1960 1840 Time NY Charleston

Chicago 1950 1850 Philly 1940 Houston 1860 LA 1870 Richmond VA 1930 1920 1880 1910 1900 (Log) Rank 1 1890 10 100

The Rank Clock for The UK data 1901 1991 1911 1981 1921 1971 1931 1961 1941 1951 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 250 1 200

14000000 10 200 180 2 160 140 12000000 120 100 80 10000000 9 150 3 60 40 20 8000000 0 100

6000000 8 4 4000000 50 7 2000000 0 1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 0

1901 5 6 1911 1921 1931 1941 1951 1961 1971 1981 1991 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 1 200 10 Population in Millions

50 40 180 2 160 140 120 Total Population in the Top 100 US Cities 100 80 9 3 60 40 20 0 8 4 30 7

20 6 Population NY City 10 0 1750 5 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 (SUNYBuffalo). 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/ 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