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 CenterUniversity of Michigan

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.

Purpose of this Seminar• 3rd 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

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.

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.

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.

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)

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”

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

Linear (curving) and log-log (straight line) illustrations of Zipf’s power law

Power Law

y = 10x-1

R2 = 1

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 2 4 6 8 10 12

RANK

SIZ

E

Power Law

y = 10x-1

R2 = 1

1.0

10.0

1 10

RANK

SIZ

E

RANK SIZE

1 10.02 5.03 3.34 2.55 2.06 1.77 1.48 1.39 1.1

10 1.0

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

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]

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”

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

Linear Distribution

y = 15.979x-0.8347

R2 = 0.6968

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12

RANK

SIZE

Concave

y = 7.3543x-0.8419

R2 = 0.9941

1

10

1 10

RANK

SIZE

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.

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)

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

U.S. Urban Areas + Urban ClustersConcave distributionZipf exponent = -1.32

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

MCD Change 1980-90

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

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

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

In this way, we have reworkedIn this way, we have reworkedZipf’s data (from 1790 to 1930) Zipf’s data (from 1790 to 1930)

3.5

4

4.5

5

5.5

6

6.5

7

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

0

10

20

30

40

50

60

1750 1800 1850 1900 1950 2000

Total Population in the Top 100 US Cities

Pop

ulat

ion

in M

illio

ns

Population NY City

1000

10000

100000

1000000

10000000

1 Log Rank 10 100

Chicago

Houston

Los Angeles

RichmondVA

NorfolkVA

Boston

Baltimore

Charleston

NewYorkCity

Philadelphia

Log CitySize

189019001910

1790 1800

1810

1820

1830

1840 Time

1850

1860

1870

18801920

1930

1940

1950

1960

1970

1980

1990

2000

Rank 1 20 40 60 80 100

Chicago

Houston

LA

RichmondVA

NorfolkVA

Boston Baltimore

Charleston

189019001910

1790 1800

1810

1820

1830

1840 Time

1850

1860

1870

18801920

1930

1940

1950

1960

1970

1980

1990

2000

(Log) Rank 1 10 100

Chicago

HoustonLA

Richmond VA

NorfolkVA

Boston Baltimore

Charleston

NY

Philly

Here we look Here we look at the half at the half lives of cities: lives of cities: the average the average number of number of years cities years cities remain in the remain in the list of largest list of largest 100 cities100 cities

Here is a plotHere is a plot

0

20

40

60

80

100

1780 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

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.

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.

Map of Map of all P&P all P&P cities in cities in 19961996

Map of the 56 largest Map of the 56 largest cities in 1949cities in 1949

Map of the 56 largest Map of the 56 largest cities in 1996cities in 1996

Top 100 Chinese cities, 1949-1996

1

10

100

1000

1 10 100

rank

pop(

10,0

00 p

erso

ns)

1949 1957 1965 1978 1985 1988 1992 1996

Top 100 county-level cities in 1996

y = 222.41x-0.192

R2 = 0.94691

10

100

1000

1 10 100

rank

po

p (

10

,00

0 p

ers

on

s)

Top 100

Power (Top 100)

Total 400 county-level cities in 1996

y = 846.76x-0.5544

R2 = 0.61571

10

100

1000

1 10 100 1000

rank

po

p (

10,0

00 p

erso

ns)

total 400

Power (total 400 )

Top 100 Cities based on urban district population, 1996

y = 1051.1x-0.5425

R2 = 0.9838

1

10

100

1000

10000

1 10 100

rank

pop

1996 urban district population

Power (1996 urban district population)

Top 56 cities in China, 1949-1996

Year 1949, y = 706.36x-1.0713

R2 = 0.9526

Year 1957, y = 1048x-0.9883

R2 = 0.9377

Year 1992, y = 896.64x-0.5252

R2 = 0.9857

Year 1996, y = 1035.1x-0.5385

R2 = 0.981

Year 1978, y = 825.72x-0.6493

R2 = 0.9726

Year 1965, y = 1321.2x-0.9619

R2 = 0.9385

Year 1985, y = 1016.6x-0.629

R2 = 0.9797

1

10

100

1000

10000

1 10 100

rank

pop(1

0,000

perso

ns)

1949

1957

1965

1978

1985

1988

1992

1996

Power (1949)

Power (1957)

Power (1992)

Power (1996)

Power (1978)

Power (1965)

Power (1985)

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.

rank-time clock, absolute

0

10

30

50

60

70

80

90

1949

1957

1965

1978

1985

1988

1992

1996

ShanghaiNanjin

Suzhou

Guanzhou

Anshan

Wurumuqi

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

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?