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There is More than a Power Law in Zipf

Cristelli, Matthieu, Batty, Michael, Pietronero, Luciano·2012·Scientific Reports
DOI 10.1038/srep00812PMC PMC3492871
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Figure 1 Zipf’s Law for National Gross Domestic Products 1900-2008. The Gross Domestic Products of nations appear to show a more and more a Zipfian behavior over the last one hundred years. We propose a fascinating interpretation of this evidence in terms of globalization. In fact we have said that a set is Zipfian if there exists an internal coherence among its elements. As the world has become more fully globalized, we observe that Zipf’s Law holds for an increasing number for countries. In fact in 2000 and 2008 we observe that not only the highest GDPs satisfy Zipf’s Law (red line) but also the top fifty economies and that the rank at which the deviation from a Zipf’s Law behavior starts increases in time, suggesting the idea that world economic system is getting more and more coherent, i.e. globalized. Globalization is making the world fully coherent/integrated with respect to the richness distribution among its units (i.e. countries) while this degree of integration has not yet been reached by the world’s national populations (see Fig. 3). Sources: Wikipedia: various pages on GDP http://en.wikipedia.org/wiki/List_of_countries_by_GDP_(nominal) and http://en.wikipedia.org/wiki/List_of_regions_by_past_GDP_(PPP).
Figure 1 Zipf’s Law for National Gross Domestic Products 1900-2008. The Gross Domestic Products of nations appear to show a more and more a Zipfian behavior over the last one hundred years. We propose a fascinating interpretation of this evidence in terms of globalization. In fact we have said that a set is Zipfian if there exists an internal coherence among its elements. As the world has become more fully globalized, we observe that Zipf’s Law holds for an increasing number for countries. In fact in 2000 and 2008 we observe that not only the highest GDPs satisfy Zipf’s Law (red line) but also the top fifty economies and that the rank at which the deviation from a Zipf’s Law behavior starts increases in time, suggesting the idea that world economic system is getting more and more coherent, i.e. globalized. Globalization is making the world fully coherent/integrated with respect to the richness distribution among its units (i.e. countries) while this degree of integration has not yet been reached by the world’s national populations (see Fig. 3). Sources: Wikipedia: various pages on GDP http://en.wikipedia.org/wiki/List_of_countries_by_GDP_(nominal) and http://en.wikipedia.org/wiki/List_of_regions_by_past_GDP_(PPP).CC BY-NC-ND · PubMed Central
Figure 2 Zipf’s Law for the Richest Billionaires in the United States. The richest 390 persons in the US are billionaires whose wealth we plot against their rank as the uppermost set of points (the first 195 richest being grey circles, the second 195 poorest being red circles). The second set is the sub-sample that we translate to the original ranks and plot as the set of red circle points below the diagonal straight line which is the pure Zipf plot associated with . The inset is a pure Zipf plot dimensioned to the entire set of 390 billionaires and the poorest sub-sample of 195. (Source: Forbes List http://www.forbes.com/).
Figure 2 Zipf’s Law for the Richest Billionaires in the United States. The richest 390 persons in the US are billionaires whose wealth we plot against their rank as the uppermost set of points (the first 195 richest being grey circles, the second 195 poorest being red circles). The second set is the sub-sample that we translate to the original ranks and plot as the set of red circle points below the diagonal straight line which is the pure Zipf plot associated with . The inset is a pure Zipf plot dimensioned to the entire set of 390 billionaires and the poorest sub-sample of 195. (Source: Forbes List http://www.forbes.com/).CC BY-NC-ND · PubMed Central
Figure 3 Rank Size of World Cities and Deviation from Zipf’s Law. The top 61 ‘world cities proper’ normalized as plotted in the grey squares are used to compute the modified rank size equation Eq.(3) – dashed red line – compared to the pure rank size equation which is the solid black line. Note the value of , below which values do not accord to Zipf’s Law in contrast to those above. The data is from the compilation of 65 separate databases (Sources: Wikipedia http://en.wikipedia.org/wiki/List_of_cities_proper_by_population).
Figure 3 Rank Size of World Cities and Deviation from Zipf’s Law. The top 61 ‘world cities proper’ normalized as plotted in the grey squares are used to compute the modified rank size equation Eq.(3) – dashed red line – compared to the pure rank size equation which is the solid black line. Note the value of , below which values do not accord to Zipf’s Law in contrast to those above. The data is from the compilation of 65 separate databases (Sources: Wikipedia http://en.wikipedia.org/wiki/List_of_cities_proper_by_population).CC BY-NC-ND · PubMed Central
Figure 4 Rank-Size Laws Illustrating Different Degrees of Coherence. Top Left: Rank-size for the largest world cities showing the absence of truly global cities which have developed in relation to all other cities. Therefore the world is not a coherent/fully integrated system and represents the wrong scale at which city size samples must be aggregated to obtain a Zipf’s Law. Instead Nigeria (green solid line) shows a nearly perfect Zipfian behavior. Nigeria which is separate from the rest of Africa, represents a city system which has developed more uniformly in a more integrated fashion, The Nigeria rank size law has been rescaled for clarity; Centre Left: Rank-size of the fifty largest European continental cities (i.e. the European part of Russia and UK are excluded). As in the case of the world cities, we observe absence of coherence at this geographical scale; Bottom Left: River formations are mainly due to geographical and morphological constraints on the Earth’s surface. Hence a Zipf’s Law is not expected and in fact the river rank size rule is well approximated by the curve (dashed black line), predicted by an independent sampling procedure without any screening effect; Top Right: For the frequency of words in the Corpus of Contemporary American English, a quasi-perfect Zipf’s Law is observed over the 2000 (and more) most used words. Linguistic systems are fully coherent with respect to our interpretation of Zipf’s Law. Centre Right: If we rank the Gross Domestic Product (GDP) of world countries, we observe a Zipfian behavior for the 30 richest. Bottom Right: As for words, a Zipf’s Law also appears in the frequency of usage of Python modules in a computer science project domain. Sources: Nigeria from the Mathematica database see Fig. 5; the top 61 country populations is from Wikipedia, see Fig. 2; the top 50 continental European city populations from http://www.citymayors.com/features/euro_cities.html; the top 161 river lengths from Wikipedia http://en.wikipedia.org/wiki/ List_of_rivers_by_length; top 2000 COCA words from http://corpus.byu.edu/coca/; GDP from Wikipedia http://en.wikipedia.org/wiki/List_of_countries_by_GDP_(nominal); the frequency of Python module usage from http://www.algorithm.co.il/blogs/math/python-module-usage-statistics/
Figure 4 Rank-Size Laws Illustrating Different Degrees of Coherence. Top Left: Rank-size for the largest world cities showing the absence of truly global cities which have developed in relation to all other cities. Therefore the world is not a coherent/fully integrated system and represents the wrong scale at which city size samples must be aggregated to obtain a Zipf’s Law. Instead Nigeria (green solid line) shows a nearly perfect Zipfian behavior. Nigeria which is separate from the rest of Africa, represents a city system which has developed more uniformly in a more integrated fashion, The Nigeria rank size law has been rescaled for clarity; Centre Left: Rank-size of the fifty largest European continental cities (i.e. the European part of Russia and UK are excluded). As in the case of the world cities, we observe absence of coherence at this geographical scale; Bottom Left: River formations are mainly due to geographical and morphological constraints on the Earth’s surface. Hence a Zipf’s Law is not expected and in fact the river rank size rule is well approximated by the curve (dashed black line), predicted by an independent sampling procedure without any screening effect; Top Right: For the frequency of words in the Corpus of Contemporary American English, a quasi-perfect Zipf’s Law is observed over the 2000 (and more) most used words. Linguistic systems are fully coherent with respect to our interpretation of Zipf’s Law. Centre Right: If we rank the Gross Domestic Product (GDP) of world countries, we observe a Zipfian behavior for the 30 richest. Bottom Right: As for words, a Zipf’s Law also appears in the frequency of usage of Python modules in a computer science project domain. Sources: Nigeria from the Mathematica database see Fig. 5; the top 61 country populations is from Wikipedia, see Fig. 2; the top 50 continental European city populations from http://www.citymayors.com/features/euro_cities.html; the top 161 river lengths from Wikipedia http://en.wikipedia.org/wiki/ List_of_rivers_by_length; top 2000 COCA words from http://corpus.byu.edu/coca/; GDP from Wikipedia http://en.wikipedia.org/wiki/List_of_countries_by_GDP_(nominal); the frequency of Python module usage from http://www.algorithm.co.il/blogs/math/python-module-usage-statistics/CC BY-NC-ND · PubMed Central
Figure 5 Successive Conditioned Draws. When we extract an object (or a city size) such as x1, we remove a section (blue slice in panel 1) of the probability density around the drawn value. See the text for the details of how to compute the slice to remove (Eq.7). Then the density moves to the reduced distribution in panel 2 from which the next object is drawn in the same way with the slice associated with x2 being removed in panel 3.
Figure 5 Successive Conditioned Draws. When we extract an object (or a city size) such as x1, we remove a section (blue slice in panel 1) of the probability density around the drawn value. See the text for the details of how to compute the slice to remove (Eq.7). Then the density moves to the reduced distribution in panel 2 from which the next object is drawn in the same way with the slice associated with x2 being removed in panel 3.CC BY-NC-ND · PubMed Central
Figure 6 Real (Zipfian) and Sampled Theoretical Rank-Size Law. Left (a): Eight sets of samples of 250 cities each drawn using the random conditioning algorithm explained in the text, rescaled in order to have the same maximum value and compared with a pure Zipf’s Law (black solid line). In the inset, we report the average rank-size law (dashed orange line) of 200 simulated countries from which the eight reported in the main box are extracted. We compare this with the rank-size rule produced by independent random samplings (red dashed line) and with the average over the 16 countries of panel b) (maroon solid line). We observe that the conditioned sampling algorithm produces a striking result very close to a pure Zipf ‘s Law (blue solid line). Right (b): Rank-size rules for the cities in 16 world countries collapsed in order to have the same maximum values and comparable with Zipf’s Law for the same (black solid line) (Source: Wolfram Mathematica online database).
Figure 6 Real (Zipfian) and Sampled Theoretical Rank-Size Law. Left (a): Eight sets of samples of 250 cities each drawn using the random conditioning algorithm explained in the text, rescaled in order to have the same maximum value and compared with a pure Zipf’s Law (black solid line). In the inset, we report the average rank-size law (dashed orange line) of 200 simulated countries from which the eight reported in the main box are extracted. We compare this with the rank-size rule produced by independent random samplings (red dashed line) and with the average over the 16 countries of panel b) (maroon solid line). We observe that the conditioned sampling algorithm produces a striking result very close to a pure Zipf ‘s Law (blue solid line). Right (b): Rank-size rules for the cities in 16 world countries collapsed in order to have the same maximum values and comparable with Zipf’s Law for the same (black solid line) (Source: Wolfram Mathematica online database).CC BY-NC-ND · PubMed Central

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Research works printed but not linked: 20

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Authors
Cristelli, M., Batty, M., & Pietronero, L.
Venue
Scientific Reports
Publisher
Springer Science and Business Media LLC
Published
2012-11-08
Volume
2
Issue
1
ISSN
2045-2322
DOI
10.1038/srep00812
PMC
PMC3492871
Cited by
143
References
0
Language
en
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