1. Centre for Advanced Spatial Analysis 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@ucl.ac.uk http://www.casa.ucl.ac.uk/

2. Centre for Advanced Spatial Analysis “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

3. Centre for Advanced Spatial Analysis Outline of the Talk 1.What is Scaling? What is Rank Size? 2.City Size/Rank-Size Dynamics 3.Explanations – Lognormality, Stochasticity, Hierarchy 4.Volatility within Stability 5.Reworking Zipf: The US Urban System: The Emergence of Cities 6.The UK Urban System 7.Rank Clocks 8.Next StepsAcknowledgements: Rui Carvalho, Richard Webber (CASA, UCL); Tom Wagner, John Nystuen, Sandy Arlinghaus (U Michigan); Yichun Xie (U Eastern Michigan); Naru Shiode (SUNY-Buffalo).

4. Centre for Advanced Spatial Analysis 1.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

5. Centre for Advanced Spatial Analysis 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: 1SMITH560122 2JONES431558 3WILLIAMS285836 4BROWN264869 5TAYLOR251567 6DAVIES216535 7WILSON192338 8EVANS173636 9THOMAS154557 10JOHNSON145459

6. Centre for Advanced Spatial Analysis 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

7. Centre for Advanced Spatial Analysis 19961881 SMITH560122SMITH406573 JONES431558JONES336447 WILLIAMS285836WILLIAMS212602 BROWN264869BROWN192061 TAYLOR251567TAYLOR186584 DAVIES216535DAVIES152450 WILSON192338WILSON136222 EVANS173636EVANS129757 THOMAS154557THOMAS122449 JOHNSON145459ROBERTS111602

8. Centre for Advanced Spatial Analysis 19961881 HUNT83HUNT78 BATTY1254BATTY957 STEADMAN1835STEADMAN2377 Changes in Rank from 1881 to 1996 in the British Electoral Role The size of the population has increased from around 26 to 40 million

9. Centre for Advanced Spatial Analysis 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

10. Centre for Advanced Spatial Analysis 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

11. Centre for Advanced Spatial Analysis 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

12. Centre for Advanced Spatial Analysis 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

13. Centre for Advanced Spatial Analysis 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

14. Centre for Advanced Spatial Analysis Parameter/Statistic1970198019902000 R Square0.9790.9720.9730.969 Intercept16.79016.89117.09017.360 Zipf-Exponent-0.986-0.982-0.995-1.014

15. Centre for Advanced Spatial Analysis Notice the slopes of these straight lines – so close to 1 This is Zipf’s Law – termed after George Kingsley Zipf who first popularized it as the rank-size rule Zipf’s 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

16. Centre for Advanced Spatial Analysis 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

17. Centre for Advanced Spatial Analysis 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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19. 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 Gibrat’s 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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22. 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 Gibrat’s 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 Gibrat’s Model: Fixed Numbers of Cities

23. Centre for Advanced Spatial Analysis 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, 143-168.

24. Centre for Advanced Spatial Analysis 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

25. Centre for Advanced Spatial Analysis Rank-size of Population of US Counties 1940 and 2000 with red plot showing 2000 populations but at 1940 ranks

26. Centre for Advanced Spatial Analysis 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.

27. Centre for Advanced Spatial Analysis 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 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 but first we will show what Zipf did.

28. Centre for Advanced Spatial Analysis 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

29. Centre for Advanced Spatial Analysis In this way, we have reworked Zipf’s data (from 1790 to 1930) Yearr-squaredexponent 17900.9750.876 18000.9680.869 18100.9890.909 18200.9830.904 18300.9900.899 18400.9910.894 18500.9890.917 18600.9940.990 18700.9920.978 18800.9920.983 18900.9920.951 19000.9940.946 19100.9910.912 19200.9950.908 19300.9950.903 19400.9940.907 19500.9900.900 19600.9850.838 19700.9800.808 19800.9860.769 19900.9870.744 20000.9880.737

30. Centre for Advanced Spatial Analysis For a sample of top cities we first show the dynamics of the Rank-Size Space

31. Centre for Advanced Spatial Analysis We have also worked out how fast cities stay in the list & we call these ‘half lives’ We can animate these

32. Centre for Advanced Spatial Analysis 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.

33. Centre for Advanced Spatial Analysis 1991 1901 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….

34. Centre for Advanced Spatial Analysis 1901 1991 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

35. Centre for Advanced Spatial Analysis 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

36. Centre for Advanced Spatial Analysis Now we show the changes in population for the top ranked places from 1901 to 1991

37. Centre for Advanced Spatial Analysis And now we show the changes in rank for these places

38. Centre for Advanced Spatial Analysis 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

39. Centre for Advanced Spatial Analysis 1890 1900 1910 1790 1800 1810 1820 1830 1840 Time 1850 1860 1870 1880 1920 1930 1940 1950 1960 1970 1980 1990 2000 Rank 1 20 40 60 80 100 Chicago Houston LA Richmond VA Norfolk VA Boston Baltimore Charleston The Rank Clock for the US data

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

41. Centre for Advanced Spatial Analysis 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 1901 1911 1921 1931 1941 1951 1961 1971 1981 199 1 The Rank Clock for The UK data

42. Centre for Advanced Spatial Analysis 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………….

43. Centre for Advanced Spatial Analysis 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.arlinghaus.net/image/solstice/sum03/ 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/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

44. Centre for Advanced Spatial Analysis Academic Press, 1994 The MIT Press, 2005 http://www.casa.ucl.ac.uk/ http://www.casa.ucl.ac.uk/citations/