Geodesic
Ranked analysis of the life cycle of polities
Dalʹnevostočnyj matematičeskij žurnal, Tome 17 (2017) no. 2, pp. 180-190.

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Many researchers, from Edward Gibbon to Arnold Toynbee, were interested in how large polities would emerge and collapse. Traditionally, the history of empires was considered both in temporal and spatial dynamics. This article focuses on the study of the external manifestations of polities' structural features which may be expressed by a limited set of mathematical curves described by the specified Zipf's law. An ideal Zipf's curve is characteristic of the classical empires with complex economies (China, Rome, and others). However, the curves of some empires have a distinctive feature – an ‘imperial tail’. The simpler the structure of large polities is, the closer is the line describing their livelihoods to a right line. 
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M. A. Guzev; N. N. Kradin; E. Y. Nikitina. Ranked analysis of the life cycle of polities. Dalʹnevostočnyj matematičeskij žurnal, Tome 17 (2017) no. 2, pp. 180-190. http://geodesic.mathdoc.fr/item/DVMG_2017_17_2_a5/

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