1. The Dynamics of Zipf John Nystuen (UM) Michael Batty (UCL) Yichun Xie (EMU) Xinyue Ye (EMU) Tom Wagner (UM) 19 May 2003 Presented at the China Data Center University of Michigan

2. Knowledge Gap Studies of urbanization are often focused on individual cities or towns, or sub-divisions of cities and towns. Understanding “systems of cities” – how urban entities are distributed, connect, and interact -- may be increasingly important in a globalizing world, e.g. 9/11, SARS. Most analytical techniques don’t consider the dynamic, non-linear behavior of urban system processes.

3. Purpose of this Seminar 3 rd and final seminar of the Series Highlight the complementary analyses of the authors regarding a common interest in the dynamical aspects of Zipf’s law. Illustrate: –the nature of city-size distributions over time and space and the use of power-law approximations –the use of current and historical US census data to show city-size transitions and patterns –the use of China census data to show urban changes and the likely impacts of dramatic urban policies on city development patterns

4. Is there an ideal city size? Throughout history, many people have postulated the existence of an ideal size of city – one with a population and physical area that maximizes human productivity and the quality of life (e.g. Aristotle, Karl Marx, Ebenezer Howard) Observation suggests that no such ideal exists or can exist: all sizes flourish and occasionally die. Efforts to create news cites (e.g. in the Soviet Union, China) have been at a high cost and often fail.

5. If an optimal size city existed, all small towns would grow toward that mean but no larger. There would be no increasing economies of scale.

6. However, there may be an optimal city-size distribution within “systems of cities”. Large systems tend toward a log-normal distribution with a few very large cities and many many small cities.

7. National population maps usually show this uneven distribution of city sizes “… differences in the kind and degree of benevolence of soil-climate-contour are capable of inducing differences in the density of the population throughout the entire territory, but only if all persons pursue the advantages inherent in their locations.” George Kinsley Zipf (p. 6, National Unity & Disunity: The Nation as a Bio-Social Organism; 1941)

8. George Kingsley Zipf (1902-1950) documented the skewed distribution of city sizes for many countries as a power law with an exponent very close to -1 proposed that this skewed distribution resulted from a natural human process he called the “Principle of Least Effort” started a 50 year search by social scientists for an explanation for this very precise distribution which became known as “Zipf’s law”

9. Zipf’s Law Takes many forms K = r X P a –K is the population of largest city –r is the rank (from the largest city) –P is the city population –a is a scaling factor ~ -1 log K = log r – a log P

10. Linear (curving) and log-log (straight line) illustrations of Zipf’s power law RANKSIZE 110.0 25.0 33.3 42.5 52.0 61.7 71.4 81.3 91.1 101.0

11. Using census data, Zipf showed the remarkably constant straight line log-log distributions of US city-sizes between 1790-1930

12. Many social scientists have tried to explain the precision of Zipf’s Law across space and over time. None have been entirely successful. Paul Krugman: “…we have to say that the rank-size rule is a major embarrassment for economic theory: one of the strongest statistical relationships we know, lacking any clear basis in theory.” [p.44, Development, Geography, and Economic Theory, 1994]

13. Zipf dynamics: The rank-size rule is static but Zipf clearly recognized the dynamic nature of its underlying processes. Zipf stated: –“Specialization of enterprise, conditioned by the various advantages offered by a non-homogeneous terrain, naturally presupposes an exchange of goods…” –“with a mobile population, less productive districts will be abandoned for more productive districts”

14. Departures from the Zipf exponent of -1 (red curve on left graphs) indicate variations in city-size distributions and different urbanization processes.

15. Departures from a Zipf exponent of -1 Exponents between 0 and -1 (level slopes): even distributions of city sizes, relatively little urban diversity, characteristic of immature systems or perhaps managed efforts to promote inter-urban equity. Exponents greater than -1 (steep slope): diverse city-sizes, mature dynamic systems, large sample sizes.

16. List of U.S. Census data to illustrate Zipf’s law Incorporated towns (1790-2000) Standard Metropolitan Statistical Areas (1940-2000) Minor Civil Divisions (1950-2000) Urbanized Areas >50,000 & Urban Clusters >2,500 (1960-2000) Named Places (including unincorporated locations)

17. Primary MSAs (major central cities) Concave distribution, Zipf exponent = -1.2

18. U.S. Urban Areas + Urban Clusters Concave distribution Zipf exponent = -1.32

19. Distribution of named U.S. places for 1980 22500 Census recognized places (same as civil divisions for upper portion) Log-log distribution is linear in upper part, exponential in lower part

20. MCD Change 1980-90

21. Extent of Zipf’s Law Zipf’s Law is useful for illustrating distribution of cities in the upper or fat tail of its log-normal distribution. Krugman suggests it applies only to U.S. cities of 200,000 people but we consider that it extends to smaller cities as well. Here is data for the US urban system from 1970 to 2000 based on populations of 22,500 ‘places’ which shows that the Law extends over at least 3 orders of magnitude Using just the upper (fat) tail, it is be seen that the distribution is remarkably stable between 1970 and 2000

23. Parameter/Statistic1970198019902000 R Square0.9790.9720.9730.969 Intercept16.79016.89117.09017.360 Zipf-Exponent-0.986-0.982-0.995-1.014

24. Zipf Dynamics Reworked: The US Urban System: 1790 to 2000 We have taken the top 100 places from Gibson’s Census Bureau Statistics from 1790 to 1990 and added the 2000 city populations We performed log-log regressions to fit Zipf’s Law We then looked at the way cities enter and leave the top 100 giving a rudimentary picture of the dynamics of the urban system We may visualize these dynamics in many different ways

25. 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

26. 0 10 20 30 40 50 60 175018001850190019502000 Total Population in the Top 100 US Cities Population in Millions Population NY City

30. Here we look at the half lives of cities: the average number of years cities remain in the list of largest 100 cities Here is a plot

31. Applications of Zipf’s law to China China has the world’s largest urban population and one of the most dynamic of large urban systems. During the past half century, urban areas in China have undergone phenomenal changes that reflect both severe restrictions on large city growth and dramatically enforced decentralization policies. These Zipf calculations are, to some extent, constrained by limitations in data consistency and periodicity.

32. China census data Two categories of urban census data: Provincial- and Prefecture-level cities (n = 234 in 1996); County-level cities (n = 400 in 1996). Provincial- and Prefecture-level are highly urbanized and have a time- series since 1949. We used the total population living within the boundary of cities’ districts (shi qu) as the urban size. The total numbers of P&P cities: 1949: 56 cities; 1957: 60 cities; 1965: 63 cities; 1978: 95 cities; 1985:100 cities. Time series: 1949/1957/1965/1978/1985/1988/1992/1996. Starting with 1949, the largest 56 cities of each year were examined in rank-size space.

33. Map of all P&P cities in 1996 Map of the 56 largest cities in 1949 Map of the 56 largest cities in 1996

38. Zipf exponents for China’s 56 largest Prefecture and Provincial-level cities: Decreased from -1.02 in 1949 to a low of -0.52 in 1992 and then started to increase to -0.53 in 1996. Suggests the effects of stringent measures taken during the Maoist period, 1949-77, to limit urban migration, large city growth, and the concentration of coastal cities. Indicates independently functioning urban regions not well integrated into a single national system.

40. Thoughts about urban systems: Old assumptions –Cities emerge independently of other cities within rural landscapes –Cities form vertical (Christaller) hierarchies –Big cities threaten environments New ideas –Cities have many horizontal links that build networks and strengthen economies –Urban networks have unique stabilities and vulnerabilities –Better human environments may result from a better understanding of how “systems of cities” work

41. Questions for further research: How do urban systems organize themselves in space and time? What defines an “urban system” and what provides the basis for its stability and its vulnerabilities? What can we do to protect and enhance the quality of our urban systems and environments?