The Dynamics of Zipf John Nystuen Michael Batty Yichun Xie Tom Wagner 19 May 2003

Knowledge Gap Studies of urban areas are often aim at understanding individual cities or towns or sub- divisions of cities Studies of urban areas are often aim at understanding individual cities or towns or sub- divisions of cities Understanding “systems of cities” – how urban areas interconnect – may be increasingly important in a globalizing world, e.g. 9/11 attacks, SARS Understanding “systems of cities” – how urban areas interconnect – may be increasingly important in a globalizing world, e.g. 9/11 attacks, SARS Most analytical techniques don’t consider dynamic, non-linear behavior of urban processes Most analytical techniques don’t consider dynamic, non-linear behavior of urban processes Is there a role for Zipf’s Law? Is there a role for Zipf’s Law?

Optimal Size Cities Throughout history, many people have suggested the existence of an “optimal” city size – a population concentration that maximizes human productivity and quality of life (e.g. Aristotle, Karl Marx, Ebeneser Howard) Throughout history, many people have suggested the existence of an “optimal” city size – a population concentration that maximizes human productivity and quality of life (e.g. Aristotle, Karl Marx, Ebeneser Howard) Observation suggests that no such place exists or can exist. Observation suggests that no such place exists or can exist.

If an optimal size city existed, all cities would tend toward that mean. Instead we see log-normal rather than normal distributions of city sizes (peaked curve on the far left).

Urban Areas in the US “… 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) “… 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)

EMU Geographer Mark Jefferson noted: “Astonishing are the differences in the growth of American cities, and astonishing too, is the distinctness with which that growth responds to nature and the extent of each city’s sustenance space, its tributary space.” “Astonishing are the differences in the growth of American cities, and astonishing too, is the distinctness with which that growth responds to nature and the extent of each city’s sustenance space, its tributary space.” [How American Cities Grow, Bull of the Am Geographical Society, 1915] [How American Cities Grow, Bull of the Am Geographical Society, 1915]

George Kingsley Zipf ( ) noted the highly skewed distribution of populations of towns and cities across national landscapes – i.e. many small towns but few big cities; noted the highly skewed distribution of populations of towns and cities across national landscapes – i.e. many small towns but few big cities; (1) documented the skewed distribution as a “rank-size” rule: a power law with an exponent ~1: “Zipf’s Law”; (1) documented the skewed distribution as a “rank-size” rule: a power law with an exponent ~1: “Zipf’s Law”; (2) proposed that the skewed distribution resulted from social-economic process he called “Principle of Least Effort”; (2) proposed that the skewed distribution resulted from social-economic process he called “Principle of Least Effort”; started 50 year search by social scientists for a process that might explain the organization of “systems of cities” started 50 year search by social scientists for a process that might explain the organization of “systems of cities”

Zipf’s Law Has many forms Has many forms K = r X P K = r X P a K is the population of largest cityK is the population of largest city r is the rank (from the largest city)r is the rank (from the largest city) P is the city populationP is the city population a is a scaling factora is a scaling factor log K = log r – a log P log K = log r – a log P

An illustration of Zipf’s law

U.S. Distribution

Many social scientists have tried to explain the precision of Zipf’s Law across space and time Stochastic or Deterministic? Stochastic or Deterministic? Paul Krugman (1994): “…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.” [p44, Development, Geography, and Economic Theory] Paul Krugman (1994): “…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.” [p44, Development, Geography, and Economic Theory]

U.S. Census Data Civil ( ) Civil ( ) PMSAs, CMSAs, and SMSAs PMSAs, CMSAs, and SMSAs Minor Civil Divisions Minor Civil Divisions Urbanized Areas & Urban Clusters Urbanized Areas & Urban Clusters Places Places

Departures from Zipf’s distribution

Explanations for departures from an exponent of 1 Low exponents: Relatively even distribution of city sizes, reduced diversity within the distribution (many medium size cities, few large and small cities). Low exponents: Relatively even distribution of city sizes, reduced diversity within the distribution (many medium size cities, few large and small cities). High exponents: Increasing size diversity, large sample sizes High exponents: Increasing size diversity, large sample sizes

Urban Areas + Urban Clusters n = 3630 Ni = N1/ib Ni = N1/ib log N i = log N i = log N 1 – b log i log N 1 – b log iwhere N i is the population of i th cityN i is the population of i th city N 1 is the population of the largest cityN 1 is the population of the largest city b ~ 1b ~ 1

Evidence for Zipf’s Law

Zipf dynamics: Zipf’s Law is static but changes over time and space. Zipf’s Law is static but changes over time and space. Zipf: “Specialization of enterprise, conditioned by the various advantages offered by a non- homogeneous terrain, naturally presupposes an exchange of goods…” [p.6] Zipf: “Specialization of enterprise, conditioned by the various advantages offered by a non- homogeneous terrain, naturally presupposes an exchange of goods…” [p.6] Population migration promotes dynamic processes, e.g. “with a mobile population, more productive districts will be abandoned for more productive districts” -- Zipf. Population migration promotes dynamic processes, e.g. “with a mobile population, more productive districts will be abandoned for more productive districts” -- Zipf.

Zipf’s Law

Further research: How are urban systems organized in space and time? How are urban systems organized in space and time? What is an “urban system” and what are its vulnerabilities? What is an “urban system” and what are its vulnerabilities? What can we do to protect our urban system? What can we do to protect our urban system?

Urban Systems Old assumptions Old assumptions Cities emerge independently of other cities within rural landscapesCities emerge independently of other cities within rural landscapes Cities form vertical (Christaller) hierarchiesCities form vertical (Christaller) hierarchies Big cities threaten environmentsBig cities threaten environments New ideas New ideas Cities have many horizontal links that build networks and strengthen economiesCities have many horizontal links that build networks and strengthen economies Urban networks have unique stabilities and vulnerabilitiesUrban networks have unique stabilities and vulnerabilities Better organization follow understandingBetter organization follow understanding