Centre for Advanced Spatial Analysis and the Bartlett School Emergence and Extinction in Cities & City Systems Michael Batty University College London

“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 Log of rank Log of size

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 2. City-Size/Rank-Size Dynamics Log population or Log P Log rank or Log r 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 P1P1 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 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

Gibrat’s Model with Lower Bound (the Solomon-Gabaix- Sornette Threshold) Fixed Numbers of Cities Gibrat’s Model with Lower Bound – Simon’s 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,

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

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 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

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 reworked Zipf’s data (from 1790 to 1930) Yearr-squaredexponent

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 call these ‘half lives’ We can animate these

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 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.

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

Log of Rank 1991 Population based on 1901 Ranks Log of Population Shares Here is an example of the shift in size and ranks over the last 100 years

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

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

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

Time Rank Chicago Houston LA Richmond VA Norfolk VA Boston Baltimore Charleston The Rank Clock for the US data

Time (Log) Rank Chicago Houston LA Richmond VA Norfolk VA Boston Baltimore Charleston NY Philly The Log Rank Clock for the US data

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 The Rank Clock for The UK data

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.

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 Total Population in the Top 100 US Cities Population in Millions Population NY City

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 Arlinghaus, S. et al. (2003) Animated Time Lines: Co-ordination of Spatial and Temporal Information, Solstice, 14 (1) at 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 Batty, M. (2003) Commentary: The Geography of Scientific Citation, Environment and Planning A, 35, at