Zipfian and Lotkaian continuous concentration theory: Research Articles

  • Authors:

  • L. Egghe

  • Affiliations:

  • Limburgs Universitair Centrum (LUC), Universitaire Campus, B-3590 Diepenbeek, Belgium and Universiteit Antwerpen (UA), Campus Drie Eiken, Universiteitsplein 1, B-2610 Wilrijk, Belgium

  • Venue:
  • Journal of the American Society for Information Science and Technology
  • Year:
  • 2005

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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. © 2005 Wiley Periodicals, Inc.