Centre for Advanced Spatial Analysis and the Bartlett School
Emergence and ExtinctionEmergence and Extinctionin Cities & City Systemsin Cities & City Systems
Michael BattyMichael Batty University College University College LondonLondon
“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 TalkOutline of the Talk 1. Preamble: Emergence, Extinction, Growth,
Change 2. City-Size/Rank-Size Dynamics3. The Simplest Models: Baseline Explanations4. Visualizing Dynamics: A Demonstration5. The US Urban System6. The UK Urban System7. Rank Clocks8. Next Steps
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The basic idea
Log of rank
Log of size
1.1. Preamble: Emergence, Extinction, Preamble: Emergence, Extinction, Growth, Change 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
2.2. City-Size/Rank-Size DynamicsCity-Size/Rank-Size DynamicsLo
g po
pula
tion
or L
og P
Log rank or Log r
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rKrPPrrPPr logloglog 1
rKPrrKPr logloglog 1
The Strict Rank-Size Relation
The Variable Rank-Size Relation
The first popular demonstration of this relation was by Zipf in papers published in the 1930s and 1940s
log P
log r
P1
Growth or decline: pure scaling
The number of cities is expanding or contracting and all populations expand or contract
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
Fixed or Variable Numbers of Cities and Populations
3. The Simplest Models: Baseline 3. The Simplest Models: Baseline ExplanationsExplanations
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
nitPtgtP iii ...,,2,1),()](1[)1(
),0()]0(1[...)]1(1)][(1[ iiii Pgtgtg
t
ii Pg0
)0()](1[
Gibrat’s Model with Lower Bound (the Solomon-Gabaix-Sornette Threshold) Fixed Numbers of Cities
T
TtPiftPtgtP iii
i
)(),()](1[)1(
nitPtgtP iii ...,,2,1),()](1[)1(
Gibrat’s Model with Lower Bound – Simon’s ModelExpanding (Contracting) Numbers of Cities
]1,,0[,...,,2,1,)1( zzifiijTtP jji
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 4. Visualizing Dynamics: A DemonstrationDemonstration
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
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log frequency
log size
5. The US Urban System5. 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 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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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
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 reworkedZipf’s data (from 1790 to 1930)
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Year r-squared exponent1790 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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1 Log Rank 10 100
Chicago
Houston
Los Angeles
RichmondVA
NorfolkVA
Boston
Baltimore
Charleston
NewYorkCity
Philadelphia
Log CitySize
For a sample of top cities we first show the dynamics of the Rank-Size Space
We have also worked out how fast cities stay in the list & we callthese ‘half lives’
We can animatethese
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6. The UK Urban System6. 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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Log of Rank
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of P
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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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1991
Log of Rank
1991 Population based on 1901 Ranks
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of P
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Here is an example of the shift in size and ranks over the last 100 years
Year t Correlation R2 Intercept Kt tKtP 101* Slope t
1901 0.879 6.547 3526157.772 -0.8171911 0.880 6.579 3801260.554 -0.8101921 0.887 6.604 4025650.857 -0.8121931 0.892 6.607 4046932.207 -0.8021941 0.865 6.532 3410371.276 -0.7401951 0.869 6.482 3034245.953 -0.7001961 0.830 6.414 2595897.640 -0.6511971 0.815 6.322 2101166.738 -0.6011981 0.816 6.321 2095242.746 -0.6011991 0.791 6.272 1872348.019 -0.577
This is what we get when we fit the rank size relation Pr=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
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Now we show the changes in population for the top ranked places from 1901 to 1991
And now we show the changes in rank for these places
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7. Rank Clocks7. 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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Rank 1 20 40 60 80 100
Chicago
Houston
LA
RichmondVA
NorfolkVA
Boston Baltimore
Charleston
The Rank Clock for the US data
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(Log) Rank 1 10 100
Chicago
HoustonLA
Richmond VA
NorfolkVA
Boston Baltimore
CharlestonNY
Philly
The Log Rank Clockfor the US data
CamdenHackneyIslingtonLambethNewhamSouthwarkTower HamletsWandsworthWestminsterBarnetBrentBromleyCroydonEalingManchesterSalfordWiganLiverpoolSeftonWirralDoncasterSheffieldNewcastle SunderlandBirminghamCoventryDudleySandwellKirkleesLeedsWakefieldBristolEdinburghGlasgow
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The Rank Clock forThe UK data
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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.
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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
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Total Population in the Top 100 US Cities
Pop
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illio
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Population NY City
8. Next Steps8. Next Steps
The program to visualize many such data setsAnalysis of extinctionsMany cities and city systemsThe analysis for firms and other scaling systemsetc. etc………….
AcknowledgementsAcknowledgements
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 ModelResources 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/ andhttp://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
