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Zipfian and Lotkaian continuous concentration theory

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  • L. Egghe

Abstract

In this article concentration (i.e., inequality) aspects of the functions of Zipf and of Lotka are studied. Since both functions are power laws (i.e., they are mathematically the same) it suffices to develop one concentration theory for power laws and apply it twice for the different interpretations of the laws of Zipf and Lotka. After a brief repetition of the functional relationships between Zipf's law and Lotka's law, we prove that Price's law of concentration is equivalent with Zipf's law. A major part of this article is devoted to the development of continuous concentration theory, based on Lorenz curves. The Lorenz curve for power functions is calculated and, based on this, some important concentration measures such as the ones of Gini, Theil, and the variation coefficient. Using Lorenz curves, it is shown that the concentration of a power law increases with its exponent and this result is interpreted in terms of the functions of Zipf and Lotka.

Suggested Citation

  • L. Egghe, 2005. "Zipfian and Lotkaian continuous concentration theory," Journal of the American Society for Information Science and Technology, Association for Information Science & Technology, vol. 56(9), pages 935-945, July.
    • Handle: RePEc:bla:jamist:v:56:y:2005:i:9:p:935-945
    DOI: 10.1002/asi.20186
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    Cited by:

    1. Lucio Bertoli-Barsotti & Marek Gagolewski & Grzegorz Siudem & Barbara .Zoga{l}a-Siudem, 2023. "Gini-stable Lorenz curves and their relation to the generalised Pareto distribution," Papers 2304.07480, arXiv.org, revised Jan 2024.
    2. Bertoli-Barsotti, Lucio & Gagolewski, Marek & Siudem, Grzegorz & Żogała-Siudem, Barbara, 2024. "Gini-stable Lorenz curves and their relation to the generalised Pareto distribution," Journal of Informetrics, Elsevier, vol. 18(2).
    3. Sarabia, José María, 2008. "A general definition of the Leimkuhler curve," Journal of Informetrics, Elsevier, vol. 2(2), pages 156-163.
    4. Egghe, L., 2010. "Conjugate partitions in informetrics: Lorenz curves, h-type indices, Ferrers graphs and Durfee squares in a discrete and continuous setting," Journal of Informetrics, Elsevier, vol. 4(3), pages 320-330.
    5. Bucci, Gabriella A. & Tenorio, Rafael, 2010. "Group diversity and salience: A natural experiment from a television game show," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 39(2), pages 306-315, April.
    6. Balakrishnan, N. & Sarabia, José María & Kolev, Nikolai, 2010. "A simple relation between the Leimkuhler curve and the mean residual life," Journal of Informetrics, Elsevier, vol. 4(4), pages 602-607.
    7. Chi, Pei-Shan, 2016. "Differing disciplinary citation concentration patterns of book and journal literature?," Journal of Informetrics, Elsevier, vol. 10(3), pages 814-829.
    8. Sarabia, José María & Prieto, Faustino & Trueba, Carmen, 2012. "Modeling the probabilistic distribution of the impact factor," Journal of Informetrics, Elsevier, vol. 6(1), pages 66-79.
    9. Bar-Ilan, Judit, 2008. "Informetrics at the beginning of the 21st century—A review," Journal of Informetrics, Elsevier, vol. 2(1), pages 1-52.
    10. Sarabia, José María & Gómez-Déniz, Emilio & Sarabia, María & Prieto, Faustino, 2010. "A general method for generating parametric Lorenz and Leimkuhler curves," Journal of Informetrics, Elsevier, vol. 4(4), pages 524-539.

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