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An inquiry into the form and function of Zipfs Law Thomas Wagner with John Nystuen China Data Center University of Michigan Updated on 21 th February 2003.

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Presentation on theme: "An inquiry into the form and function of Zipfs Law Thomas Wagner with John Nystuen China Data Center University of Michigan Updated on 21 th February 2003."— Presentation transcript:

1 An inquiry into the form and function of Zipfs Law Thomas Wagner with John Nystuen China Data Center University of Michigan Updated on 21 th February 2003

2 In 1950: 29% of 2.5 billion people lived in cities and towns. By 2005: 50% of 6.5 billion people will live in urban areas. By 2025: 60% of 8.3 billion people will live in urban areas.

3 Urban areas form, grow, change, and sometimes disappear increasingly concentrate people, resources, and enterprises in small areas the world over are dynamic, highly productive, interactive open ecosystems (involving geography, energy, people, information, production) create complex hierarchal networks of proximity, infrastructure, migration, communication, and commodity exchange are not randomly distributed over the landscape

4 What determines urban patterns?

5 US Urbanization 1790-2000

6 What is Zipfs Law? Within a network of cities, the size of a citys population is inversely proportional to its rank on an ordered list: e.g. the largest city is #1, the second largest is #2 with ½ the population of #1, the third largest city is #3 with 1/3 the population of #1, etc. The populations of urban areas follow consistent power law relationships within very narrow limits with an exponent close to 1. This relationship applies to large urban systems around the world and over long periods of time.

7 Why do we want to talk about Zipfs Law? Its remarkably consistent Its frequently referenced Its causes are unknown Its variations are many Its applications are growing It leads to interesting questions

8 Who was George Kingsley Zipf (1902-1950)? Prof. & Lecturer, Harvard Univ., 1930-50. Proposed many linguistic and social power law relationships. Authored Human Behavior & The Principle of Least Effort (1949). Not well known in his time. Not the inventor of the rank-size rule.

9 Zipf footnote (p374): The first person to my knowledge to note the rectilinear distribution of communities in a country was Felix Auerbach in 1913 who, however, generalized incorrectly upon the value, p =q =1, and was quoted therein by A.J.Lotka. In 1931 R. Gilbrat reported the rectilinearity of the large communities of Europe with a value of p less than 1.

10 Some power law relationships noted by Zipf

11 Zipfs Law

12 from Human Behavior and the Principle of Least Effort (1949)

13 Zipfs urban unit definition Since our argument referred to the natural boundaries of communities, as opposed to their political boundaries, we shall use the data for the populations of Metropolitan Districts in the US in 1940, although… we shall find that politically bounded regions follow the same equation. (p 375)

14 Growth of US urban populations: 1790 - 1930

15 Changes in the urban areas of other countries, e.g. India

16 Zipfs Law (updated)

17 Size-rank of the 135 largest cities in the US (2000) is remarkably linear on a log-log scale (left)

18 Size-rank of the 135 largest US cities (2000) has a power law exponent close to 1

19 Zipfs Law doesnt explain why, but if true… All urban areas are nodes within a network of urban areas within a region Urban networks are scale-free –relations within subsets apply to the whole –transitioning from chaos to order

20 Questions about Zipfs Law Does it work? How consistent is it? What does it tell us about urban networks or their growth? Whats the right urban unit? region? –minimum size? maximum size? –do they change over time? Why population? How about other things? What are its applications?

21 Who else is involved? Vilfredo Pareto (1898) Felix Auerbach (1913) R. Gilbrat (1931) Herbert Simon (1955) Paul Krugman (1994, 1996) X. Gabaix (1999) Laszlo Barabasi (1999) ~ 300 other people – see Li @

22 Why does it work ? Stochastic growth (Gilbrat)? Non-random growth (Simon)? Urban hierarchy (Christaller & Losch)? Scale economies and diseconomies (Henderson, Dobkins & Ioannides)? Natural resources (Prigogine and Allen)? Other reasons?

23 applications of Gilbrats Law The probability distribution of the growth process does not depend on the initial size of a citys population or economy. All cities grow and shrink stocastically, e.g. with common means and variances that are independent of city size.

24 Herbert Simon says Probability of (a firms) growth is related to its size. the rich get richer e.g. Paretos law. Big firms have preferential growth time is required to converge to Zipfs Law

25 maybe Spatial Hierarchies (Christaller & Losch)? from G. William Skinners The City in Late Imperial China (1977)

26 Chinas different economic zones

27 Economist J.V Henderson says: Cities differ in size because: they specialize in the production of goods with different scale economies. Firms seek to avoid diseconomies of population sizes and qualities. The mismatch of population diseconomies and firm scale economies results in Zipfs Law.

28 what about resource availability? Cities are open systems for in-flows of people, water, and energy Big cities use people, water, and energy more efficiently than smaller cities, and far more than rural areas. Different efficiencies result in different patterns.

29 theres Gabaixs shocks Population growth comes from migration Need city-specific shocks (changes in taxes, pollution, floods, civil unrest, etc.) to generate common urban growth means and variances (Gilbrats law). Shocks overcome migration costs and cause populations to move to new cities based on opportunities that are independent of size. from Zipfs Law of Cities: An Explanation, Quarterly Journal of Economics, August 1999

30 thoughts from Barabasi* (regarding power laws, not urban networks) As long as we thought networks were random, we modeled them as static graphs. The scale-free model reflects our awakening to the reality that networks are dynamic systems that change constantly through the addition of new nodes and links. (p.106) Normally nature hates power laws…But all that changes if the system is forced to undergo a phase transition. Then power laws emerge – natures unmistakable sign that chaos is departing in favor of order. (p.77) * Linked (2002), Perseus Pub., Cambridge, MA

31 What about the US? Decadal Census data are available by –Civil divisions, census tracts, e.g. incorporated cities (detailed local data including most data before 1950) –Metropolitan Areas units (PMSAs, CMSAs), incorporates county-size areas. –Urbanized Areas (UAs) & Urban Clusters (UCs) newly created for the 2000 Census includes urban populations in Porto Rico and Guam 464 UAs incorporate 194,323,824 people on 71,693 sq. miles (~2%US) 3175 UCs incorporate 30,036,715 people on 20,672 sq. miles UAs and UCs have 79% of total US population

32 Standard Metropolitan Areas (MAs) cover large areas and include non-urban land

33 Urbanized Areas include only urban land uses (448 UAs for the lower 48 states shown here)

34 Comparison of an Urbanize Area (blue) map with a civil division/census tract map of Ann Arbor, Michigan

35 Comparison of Urbanized Area patterns for western US (left) & eastern US (right)

36 Zipfs exponent for 48 states Western ½ = -1.07, Eastern ½ = -1.13 Western cities = 50.3M people (26% of urban population) Eastern cities = 140.0M people (74% of urban population)

37 US Urban Patterns: Pacific Coast (upper-left), East & Gulf Coast (lower-left) Great Plains & Mountains (upper-right), Middle America (lower-right)

38 Regional Zipfs exponents: Pacific Coast = -1.26 Great Plains & Mountains = -1.19 East & Gulf Coast = -1.27 Middle and South = -1.07

39 interpreting Zipfs graph lines Distance from origin = total urban population Slope: an integrated scaling factor Curves (violate Zipfs Law) –concave –convex Tails (a problem for power laws) –Upper (a few big cities) Lower fat (many, many small cities)

40 What does the exponent mean? Slope of the line (amt of drop for 1 unit on the horizontal axis) e.g.1.0 = e.g. 2.0 = high = even distribution of urban populations among all cities low = urban population concentrated in certain size cities

41 What does the curve tell us? US Census 2000 data, n=448 Convex (see right): Big cities may be limited in size, population in many medium size cities Concave: Few medium size cities, big cities or small cities may account for large share of urban population

42 from Rosen and Resinik, 1980 (44 countries, 50 largest cities, 1970 census) sd= 0.196 32 countries Australia central city larger than metro area 30 countries had concave (upward) curves Large cities are growing faster than small cities Lack of a rigorous theoretical model *Size distribution of Cities, J of Urban Economics 8 (p165-186)

43 Should we care? Urban robustness and vulnerabilities Monitoring urban changes Monitoring global change Many possible futures

44 Urban robustness and vulnerabilities Do hub cities provide network robustness, e.g. resistance to disruption? Are urban networks vulnerable to particular types of shocks, e.g. –power-grid failures or fuel cutoffs? –abnormal weather events, e.g. droughts, excessive heat, excessive cold, floods, hurricanes? –earthquakes? –War?

45 Monitoring changes in urban networks What other things can we measures? –areas (size and patterns) –light emissions (energy use) –housing stocks (nos. of residences) –firms (size and location of production units) –impervious (built surfaces)

46 US population has a correlation with area at the R 2 = 0.9 level

47 Night lights observed by satellite provide urban patterns for US

48 Land use maps (left) or satellite-observed impervious surfaces (right) relate to urban population patterns. Detroit metropolitan area shown.

49 Conclusions: Zipfs law has an exponent of close to 1 for the 135 largest US cities, but greater than 1 (e.g. 1.13) for 448 cities over 50,000 population. Regional Zipf exponents ranging from -1.07 to -1.27, suggesting differences in regional patterns within a relatively mature urban network. The log rank-size line for the US is convex (downward curving) and suggests under-representation of large cities or over representation of medium size cities. Perhaps Zipfs law is not a law with a rigorous theory at all, but rather a rule expressing a natural (power law) tendency – more like the more well known normal distribution.

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