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/