Rank Size Rule

8 Key excerpts on "Rank Size Rule"

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  Population Geography

  Pergamon Oxford Geographies

  • John I. Clarke, W. B. Fisher(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  8. Rank-size of towns of England and Wales with more than 5 0 , 0 0 0 inhabitants, 1961. 54 Population Geography A number of scholars have noted the empirical relationships, others have made mathematical contributions, but it was Zipf who most publicized the rank-size rule, attempting to explain the relationships in a general theory of human behaviour. His explanation involved two forces, those of diversification and unification. The former tends to split a population into small autarchic communities as a result of economical location near raw materials. Diversifi-cation tendencies minimize the problems of transportation of raw materials to factories, while unification tendencies minimize difiiculties of transportation of manufactured goods. When both forces are operating, optimum popu-lation distribution with reference to these forces is said to occur. The rule has not gained universal acceptance. Rosing has claimed that the rank-size rule is meaningless by itself, as it is only a special case of skewed distribution. Testing by chi-square the five largest cities of 132 countries, he obtained results which indicated that the rank-size rule was an inadequate description in all cases. But the rule must be seen in conjimction with central place theory. The German, Christaller, conceiving a town as a central place within a rural area, and, later, his compatriot Lösch, both assumed a functional size-class hierarchy of towns as a basis of analysis of their spatial arrangement, function and size. The basic assumptions are that a rural area supports a town which in turn serves the rural area, and that there are small towns for small areas and large towns for large areas. Christaller postulated an integrated system of towns according to size, which in theory would have to be evenly spaced. But whereas Christaller assumed that each order in the hierarchy contains a fixed number, of settlements for each settlement of the order above.

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  Soviet Urbanization

  • Olga Medvedkov(Author)
  • 2017(Publication Date)
  • Taylor & Francis

    (Publisher)

  Zipf’s approach to urban hierarchies is associated with a so called ‘Rank-Size Rule’. It predicts city sizes in a nation without rigid and distinct steps in grading. The reasoning suggested by Zipf is free from postulating any unique privileges for services in cities. If one sets aside numerous (and ingenious) attempts to portray the Rule as a clone, a derivative from Central Place Theories (Beckmann, 1958) or from trends observed for population density gradients (Okabe, 1979, 1987) the post-Zipf theory in Geography for the Rank-Size Rule is regretfully thin. It is unfair to the originality of Zip’s ideas, and there is more in the Rule than only an empirical and descriptive device for compressing data arrays.

  In this chapter a view is presented that independence of Zip’s tradition is a virtue. I am introducing a train of reasoning for strengthening the Rank-Size model in its sovereignty. One may find appealing its freedom from rigid assumptions. It permits more room for experiments in analysing why real-life data fit so well the Rank-Size model, the fact well testified by many authors (Alperovich, 1984; Assamy, 1986; Boventer, 1973; Cassetti et al., 1971; Medvedkov, Y., 1964; Models, 1970; Rashevsky, 1943, 1951; Richardson, 1978; Rosen and Resnik, 1980). Such features are just right for the job in mind.

  Consequently, there will be opportunities to return later, within this chapter, to the Rank-Size Rule.

  Why knowledge of urban hierarchy is important

  The material so far discussed, in this chapter, presents evidence that Geography pays much attention to urban hierarchy. One may legitimately ask, if a justification exists for bringing the same approach into this book.

  Why invest more into the same? What urgency is in learning the Soviet reality under the angle of urban hierarchy? Why may it benefit readers and give them a better understanding of social, cultural, economic and political problems in the Soviet Union?

  One answer to this is in indicating that knowledge comes with structuring data into information. A process of learning has steps to replace vague opinions by solid findings. For that purpose there are specialized tools of research. They rely on concepts (like that of hierarchy). It means acceptance of traditions in studies; one benefits from their potential.

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  Applications of the Expansion Method

  • Emilio Casetti, John Paul Jones III, Emilio Casetti, John Paul Jones III(Authors)
  • 1991(Publication Date)
  • Routledge

    (Publisher)

  The inequality measures referred to above were primarily intended as devices to determine overall systemic inequality. At most they could cope with the relation between overall systemic inequality and the systemic inequality of welldefined and pre-specified partitions of a system. The issue of continuous variation within the system is not and cannot be addressed by the use of these measures. It can, instead, be addressed by the use of the expanded rank-size functions discussed in the next sections of this paper. Specifically, the use of rank-size functions to measure inequality will be discussed and an application using the expansion method will be presented.

  THE RANK-SIZE APPROACH TO THE MEASUREMENT OF INEQUALITY

  The rank-size function has been widely employed to investigate the relationship between city size and rank. Rank-size models were developed to formalize the observation that, when all cities of a region are ranked in decreasing order of population size, the size of a city of a given rank is related to the size of the largest city in that region (Garner 1967). Zipf (1949) used rank-size relationships to examine a variety of issues, and concluded that the distribution of human activities, ranging from language, art, economic power, and social status, was subject to natural laws which he then adopted as a standard for defining an ideal distribution. In his work both frequency and size were used to represent the magnitude of a measurable quantity.1 In this paper ‘size’ is used as a general term for magnitude. The simplest representation of the rank-size relationship is:

  Figure 9.1 Zipf’s ideal rank-size distribution

  (9.1)

  where y is size, r is rank (largest size: rank 1), and a and b are parameters. In logarithmic form, equation (9.1) becomes

  (9.2)

  where a is the intercept and b is the slope of the rank-size curve. In the case of city size distribution, y is the population of a particular city and r is its rank according to population size. When the coefficient b equals –1, it is referred to by Zipf as an ideal distribution which reflects the orderliness of human activities, namely the rank-size rule. The coefficient b is the derivative of the logarithmic function (9.2). It evaluates the percentage rate of change in size associated with the percentage rate of change in rank. The rank-size rule suggested by Zipf prescribes that, if the rank of a unit increases (or decreases) by a certain percentage, its size also increases (or decreases) by the same percentage. On doubly logarithmic paper, the rank-size rule will appear as a straight line descending from left to right at an angle of 45°, indicating a slope of –1 (Figure 9.1

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  Time, Process and Structured Transformation in Archaeology

  • James McGlade, Sander E. van der Leeuw(Authors)
  • 2013(Publication Date)
  • Taylor & Francis

    (Publisher)

  Usually, the cities belonging to the same region or to the same country are strongly differentiated according to their size. There is strong evidence about a certain invariance in the statistical distribution of city sizes. The size distribution is highly skewed, with many small towns located close to each other, less and more distant medium-sized cities and only a few very large cities which are generally located far away from each other. The number of cities is roughly in an inverse geometrical relationship to their size. This distribution was identified (by Le Maître in 1682) and formalized (Auerbach 1913) a long time ago. It is usually known as the ‘rank-size rule’ as described by Zipf (1949, pp. 417–41): when plotted on logarithmic co-ordinates, the population size of a city is a linear function of its rank in the urban hierarchy. The cumulative frequency distribution of the sizes of cities is also similar to a Pareto function (Simon 1957, p. 148) or to a lognormal distribution (Gibrat 1931, p. 280). The slope of the Pareto distribution and the dispersion of the lognormal function are both indicators of the degree of concentration of urban populations (in other words, a measure of the inequalities in size among cities).

  Such types of city size distributions can be found everywhere in the world (Zipf 1949; Rosen and Resnick 1980; Parr 1985; Moriconi 1993). They seem to maintain the same shape for a very long time. Despite the numerous difficulties in evaluating the number of their inhabitants, it is thought highly probable by several historians that the hierarchical organization of French cities has existed at least since the Roman Empire. The existence of the hierarchy is not linked to demographic growth, since it has maintained itself even during the regressive periods of the Middle Ages.

  For the development of a dynamic theory of settlement systems, it would be important to know whether other kinds of distributions have been observed. This is a question which archaeology might help answer: do we, for instance, have examples of a normal distribution of settlement sizes? Such a distribution would suggest that there is an optimal size for human settlement, depending upon the availability of resources at a given distance and/or upon the internal management capacities of the group. The occurrence of a normal distribution of city sizes would imply that under such circumstances a permanent settlement is not viable below or above a given threshold of size.

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  Sustainable Transport for Chinese Cities

  • Roger L. Mackett, Anthony D. May, Masanobu Kii, Haixiao Pan, Roger L. Mackett, Anthony D. May, Masanobu Kii, Haixiao Pan, Stephen Ison, Jon Shaw(Authors)
  • 2013(Publication Date)
  • Emerald Group Publishing Limited

    (Publisher)

  These estimates have been based on a grid system in which the global land area is divided into a mesh, and the population distribution is estimated by downscaling the given regional or national urban populations. However, there are no established methods for this downscaling, and each study assumed an ad-hoc allocation function for the population. For instance, Gru¨ bler et al. tried to downscale the future regional population consistent with the Special Report on Emissions Scenarios (SRES) using a 0.5 1 0.5 1 global land grid system. In this study, grid population grows at an increasing rate as the population of the previous period increases. This reflects the population MASANOBU KII AND KENJI DOI 18 concentration in large cities, but these projections do not necessarily reproduce the well-known experimental city size distribution, that is the rank-size rule or Zipf’s law ( Zipf, 1949 ). The rank-size rule is an empirical law for city size distribution; it follows a Pareto distribution. In an early study by Simon (1955) , the rank-size rule was shown to be derived from a simple random growth model, where the growth probability of a city is proportional to its share of the total population. The model theoretically generates a class of city size distribution that includes the rank-size rule. However, it only provides the theoretical grounds for an aggregated city size distribution and requires Monte-Carlo experiments to estimate the growth of each city; this would lead to sets of results that are quite different from each other, even though their aggregated distributions are stable. Recently, complex network theories have provided other approaches, leading to a Pareto distribution in network node connectivity ( Albert & Barabasi, 2002; Barabasi & Albert, 1999 ). The continuum approach in the scale-free network theory provides a dynamic equation for the connectivity of each node, which leads to the Pareto distribution asymptotically.

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  Geocomputation

  A Practical Primer

  • Chris Brunsdon, Alex Singleton, Chris Brunsdon, Alex Singleton(Authors)
  • 2015(Publication Date)
  • SAGE Publications Ltd

    (Publisher)

  We will then extend our analysis of the US urban system using data for the metropolitan statistical areas (MSAs) from 1969 to 2008, and follow this with an examination of scaling in high buildings (skyscrapers) in New York City. High buildings are distinguished by the fact that newer buildings tend to be higher than old, while a building rarely declines in height due to the fact that high buildings tend to be demolished if they are changed at all. This poses a rather different dynamics that produces somewhat different patterns through time. We then examine the change in scaling associated with hubs in a network whose sizes are based on the number of travellers moving through these locations at different times of the working day. Our illustrations of dynamics will all be related to changes in ranks, exploiting the idea of how ranks change over time, where time is displayed as a clock not organised on the 12-hour cycle per se but calibrated to the time periods over which the dynamics is considered (Batty, 2006). In conclusion, we will argue that the real puzzle is to unpack the way spatial competition, which organises these patterns into strongly regular size distributions, gives rise to a continual shuffling and mix of cities as they move up and down the size distribution. Power Laws Explained and Rank Clocks Defined As Krugman (1996a) noted, the size distribution of US cities follows the simplest power law where the size of a city P varies in inverse proportion to its rank r as. Expressed in terms of frequencies as a probability distribution, the frequency of the occurrence of a city of size P is, which is the derivative of the previous rank-size expression. This kind of manipulation is quite simple, but it is worth noting that much confusion arises with power laws and rank size because different authors use one or other of these equations and the discussion as to the actual value of the power can become obtuse

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  Handbook of Regional and Urban Economics

  Cities and Geography

  • V. Henderson, J.F. Thisse(Authors)
  • 2004(Publication Date)
  • North Holland

    (Publisher)

  Physical Review Letters . 1997;79:523–526.

  Zipf, G.K. Human Behavior and the Principle of Least Effort . Oxford: Addison-Wesley; 1949.

  Uncited References

  Bairoch, P., Batou, J., Chevre, P., The population of European cities 800–1850: Data bank and short summary of results, Centre of International Economic History Series 2. Chicago: Geneva University; 1988.

  Fujita, M., Krugman, P., Venables, A.J. The Spatial Economy . New York: MIT Press; 1999.

  1 It is important to point out that there is no universally accepted definition of a city for statistical purposes. In the U.S. context, research has been conducted with both city proper data and data for metropolitan statistical areas. Differences in data availability worldwide may make international comparisons tricky. Rosen and Resnick (1980) show that the Pareto exponent of city size distributions tend to be closer to 1 when agglomerations are more carefully constructed, i.e., are closer to “true” agglomerations rather than administratively defined “cities”. We return to this below.

  2 Talking about steady-state distributions requires a normalization of this type.

  3 See Gabaix (1999a, Proposition 4) for a precise statement of the rank–size rule, and the goodness of fit one can expects from it. The rank–size rule is a good approximation for cities of high rank, but not for the largest cities. For instance, the rank–size rule says that ratio of the largest city to second largest city is 2. But Zipf’s law implies that this ratio is widely variable, indeed has a smallest 95% confidence interval equal to [1, 20]. This comes from the Rényi theorem described in Section 2.2.2 , which says that P (S (1) /S (2) > x ) = 1/x for x > 1. So [1, 1/.05] is the smallest 95% confidence interval for S (1) /S (2) .

  4 The Statistical Abstract of the U.S. lists all the agglomerations with size above 250,000 inhabitants. The exponent ζ is sensitive to the choice of the cutoff size above which one selects the cities. For a lower cutoff, the exponent ζ is typically lower. We come back to this issue, and a possible explanation, in Section 2.4

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  Great Urbanization Of China, The

  • Ding Lu(Author)
  • 2011(Publication Date)
  • World Scientific

    (Publisher)

  For the United States in particular, Rosen and Resnick (1980) gave an estimate of 1.000 for 1970 metropolitan population data, 1.184 for city proper data, and 1.310 for data on the 50 largest cities in the United States. Parr (1985) used US city proper data and reported an estimated exponent ranging from 1.066 to 1.147 during 1895–1981. Fourth, there is no theoretical reason to expect the Pareto law to hold with precision for city-size distribution, although a number of studies have theoretically derived a hierarchy in the urban system (e.g. Henderson, 1988; Fujita et al. , 1999). 1 As these studies have emphasized, there is 14 S. Song and K.H. Zhang 1 Gabaix (1999) theoretically proves that city-size distribution converges to a power law if all cities follow some proportional growth process (i.e. cities’ growths have the same mean and same variance). He calls this power law Zipf’s law and argues against using the Pareto law for city-size distribution. a tension between external economies associated with the geographic concentration of industry within a city, on one hand, and diseconomies such as congestion costs associated with large cities, on the other. The net effect of this tension determines the optimal size of a given city. A hierar-chy of city systems emerges because external economies tend to be specific to particular industries and hence vary greatly across industries, but diseconomies tend to depend on the overall size of a city. However, none of previous studies has proved that the hierarchy should follow the Pareto law or the rank-size rule. Fujita et al. (1999, p. 219) state that “the regularity of the urban size distribution poses a real puzzle, one that nei-ther our approach nor the most plausible alternative approach to city sizes seems to answer”. They further say that “we hope that future research will resolve this puzzle” (p. 225).

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