The Dynamics of Zipf The Dynamics of Zipf John NystuenJohn Nystuen

Michael BattyMichael Batty

Yichun XieYichun Xie

Tom WagnerTom Wagner

19 May 200319 May 2003

Knowledge GapKnowledge Gap Studies of urban areas are often aim at Studies of urban areas are often aim at

understanding individual cities or towns or sub-understanding individual cities or towns or sub-divisions of citiesdivisions of cities

Understanding “systems of cities” – how urban Understanding “systems of cities” – how urban areas interconnect – may be increasingly areas interconnect – may be increasingly important in a globalizing world, e.g. 9/11 important in a globalizing world, e.g. 9/11 attacks, SARSattacks, SARS

Most analytical techniques don’t consider Most analytical techniques don’t consider dynamic, non-linear behavior of urban processesdynamic, non-linear behavior of urban processes

Is there a role for Zipf’s Law?Is there a role for Zipf’s Law?

Optimal Size CitiesOptimal Size Cities

Throughout history, many people have Throughout history, many people have suggested the existence of an “optimal” suggested the existence of an “optimal” city size – a population concentration that city size – a population concentration that maximizes human productivity and quality maximizes human productivity and quality of life (e.g. Aristotle, Karl Marx, Ebeneser of life (e.g. Aristotle, Karl Marx, Ebeneser Howard)Howard)

Observation suggests that no such place Observation suggests that no such place exists or can exist.exists or can exist.

If an optimal size city existed, all cities would tend If an optimal size city existed, all cities would tend toward that mean. Instead we see log-normal toward that mean. Instead we see log-normal rather than normal distributions of city sizes rather than normal distributions of city sizes

(peaked curve on the far left).(peaked curve on the far left).

Urban Areas in the USUrban Areas in the US

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

EMU Geographer EMU Geographer Mark Jefferson noted:Mark Jefferson noted:

““Astonishing are the differences in the Astonishing are the differences in the growth of American cities, and astonishing growth of American cities, and astonishing too, is the distinctness with which that too, is the distinctness with which that growth responds to nature and the extent growth responds to nature and the extent of each city’s sustenance space, its of each city’s sustenance space, its tributary space.” tributary space.”

[How American Cities Grow, Bull of the Am [How American Cities Grow, Bull of the Am Geographical Society, 1915]Geographical Society, 1915]

George Kingsley ZipfGeorge Kingsley Zipf(1902-1950)(1902-1950)

noted the highly skewed distribution of noted the highly skewed distribution of populations of towns and cities across national populations of towns and cities across national landscapes – i.e. many small towns but few big landscapes – i.e. many small towns but few big cities;cities;

(1) documented the skewed distribution as a (1) documented the skewed distribution as a “rank-size” rule: a power law with an exponent “rank-size” rule: a power law with an exponent ~1: “Zipf’s Law”;~1: “Zipf’s Law”;

(2) proposed that the skewed distribution resulted (2) proposed that the skewed distribution resulted from social-economic process he called “Principle from social-economic process he called “Principle of Least Effort”; of Least Effort”;

started 50 year search by social scientists for a started 50 year search by social scientists for a process that might explain the organization of process that might explain the organization of “systems of cities”“systems of cities”

Zipf’s LawZipf’s Law

Has many formsHas many forms K = r K = r XX P P a

• K K is the population of largest cityis the population of largest city• r r is the rank (from the largest city)is the rank (from the largest city)• P P is the city populationis the city population• a a is a scaling factoris a scaling factor

log K = log r – a log Plog K = log r – a log P

An illustration of Zipf’s lawAn illustration of Zipf’s 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 SIZE1 10.02 5.03 3.34 2.55 2.06 1.77 1.48 1.39 1.1

10 1.0

U.S. DistributionU.S. Distribution1790-19301790-1930

Many social scientists have tried to explain Many social scientists have tried to explain the precision of Zipf’s Law across space and the precision of Zipf’s Law across space and

timetime

Stochastic or Deterministic?Stochastic or Deterministic?

Paul Krugman (1994): “…we have to say that the Paul Krugman (1994): “…we have to say that the rank-size rule is a major embarrassment for rank-size rule is a major embarrassment for economic theory: one of the strongest statistical economic theory: one of the strongest statistical relationships we know, lacking any clear basis in relationships we know, lacking any clear basis in theory.” [p44, Development, Geography, and theory.” [p44, Development, Geography, and Economic Theory]Economic Theory]

U.S. Census DataU.S. Census Data

Civil (1790-2000)Civil (1790-2000) PMSAs, CMSAs, and SMSAs PMSAs, CMSAs, and SMSAs Minor Civil DivisionsMinor Civil Divisions Urbanized Areas & Urban ClustersUrbanized Areas & Urban Clusters

PlacesPlaces

Departures from Zipf’s distributionDepartures from Zipf’s distribution

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

Explanations for departures Explanations for departures from an exponent of 1from an exponent of 1

Low exponents: Relatively even Low exponents: Relatively even distribution of city sizes, reduced distribution of city sizes, reduced diversity within the distribution diversity within the distribution (many medium size cities, few large (many medium size cities, few large and small cities).and small cities).

High exponents: Increasing size High exponents: Increasing size diversity, large sample sizesdiversity, large sample sizes

Urban Areas + Urban ClustersUrban Areas + Urban Clustersn = 3630n = 3630

Ni = N1/ibNi = N1/ib log Nlog Ni i = =

log Nlog N11 – b log i – b log i

where where

• NNi i is the population is the population of iof ith th citycity

• NN11 is the population is the population of the largest cityof the largest city

• b ~ 1b ~ 1

Evidence for Zipf’s LawEvidence for Zipf’s Law

Zipf dynamics:Zipf dynamics: Zipf’s Law is static but changes over time and Zipf’s Law is static but changes over time and

space.space.

Zipf: “Specialization of enterprise, conditioned by Zipf: “Specialization of enterprise, conditioned by the various advantages offered by a non-the various advantages offered by a non-homogeneous terrain, naturally presupposes an homogeneous terrain, naturally presupposes an exchange of goods…” [p.6]exchange of goods…” [p.6]

Population migration promotes dynamic Population migration promotes dynamic processes, e.g. “with a mobile population, more processes, e.g. “with a mobile population, more productive districts will be abandoned for more productive districts will be abandoned for more productive districts” -- Zipf. productive districts” -- Zipf.

Zipf’s LawZipf’s Law

Further research:Further research:

How are urban systems organized in How are urban systems organized in space and time?space and time?

What is an “urban system” and what What is an “urban system” and what are its vulnerabilities?are its vulnerabilities?

What can we do to protect our urban What can we do to protect our urban system?system?

Urban SystemsUrban Systems

Old assumptionsOld assumptions• Cities emerge independently of other cities Cities emerge independently of other cities

within rural landscapeswithin rural landscapes• Cities form vertical (Christaller) hierarchiesCities form vertical (Christaller) hierarchies• Big cities threaten environmentsBig cities threaten environments

New ideasNew ideas• Cities have many horizontal links that build Cities have many horizontal links that build

networks and strengthen economiesnetworks and strengthen economies• Urban networks have unique stabilities and Urban networks have unique stabilities and

vulnerabilitiesvulnerabilities• Better organization follow understandingBetter organization follow understanding