Relations between the shape of a size-frequency distribution and the shape of a rank-frequency distribution

  • Authors:

  • L. Egghe;L. Waltman

  • Affiliations:

  • Universiteit Hasselt, Campus Diepenbeek, Agoralaan, B-3590 Diepenbeek, Belgium;Centre for Science and Technology Studies, Leiden University, P.O. Box 905, 2300 AX Leiden, The Netherlands

  • Venue:
  • Information Processing and Management: an International Journal
  • Year:
  • 2011

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Abstract

We study the dependence of the shape of the rank-frequency distribution g on the shape of the size-frequency distribution f and vice versa. We show mathematically that g is convexly decreasing if and only if f is monotonically decreasing and that g has an S-shape (i.e., g is first convexly decreasing and then concavely decreasing) if and only if f is first increasing and then decreasing. To illustrate our mathematical results, we empirically analyze size- and rank-frequency distributions of the number of articles and of the impact factor of journals in various scientific fields. We find that most of the size-frequency distributions that we examine are first increasing and then decreasing. Most of the rank-frequency distributions that we examine have an S-shape. However, the concave part of the S-shape is sometimes very small.