, Volume 110, Issue 3, pp 1403–1413 | Cite as

Modified Benford’s law for two-exponent distributions

  • Hsiang-chi Tseng
  • Wei-neng Huang
  • Ding-wei HuangEmail author


Motivated by applications in scientometrics, we study the occurrence of first significant digits in Lavalette distribution and in double Pareto distribution. We obtain modifications of Benford’s law. When the exponents are small, significant deviations to Benford’s law are observed; when the exponents are large, the two distributions conform with Benford’s law. Both analytical and numerical results are presented. Scientometric data can fairly well be described by the modifications.


Benford’s law Zipf’s law Double Pareto distribution Lavalette distribution 


  1. Alves, A. D., Yanasse, H. H., & Soma, N. Y. (2014). Benford’s law and articles of scientific journals: Comparison of JCR and Scopus data. Scientometrics, 98, 173–184.CrossRefGoogle Scholar
  2. Alves, A. D., Yanasse, H. H., & Soma, N. Y. (2016). An analysis of bibliometric indicators to JCR according to Benford’s law. Scientometrics, 107, 1489–1499.CrossRefGoogle Scholar
  3. Benford, F. (1938). The law of anomalous numbers. Proceedings of the American Philosophical Society, 78, 551–572.zbMATHGoogle Scholar
  4. Berger, A., & Hill, T. P. (2011). Benford’s law strikes back: No simple explanation in sight for mathematical gem. The Mathematical Intelligencer, 33, 85–91.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Campanario, J. M. (2010). Distribution of ranks of articles and citations in journals. Journal of the American Society for Information Science and Technology, 61, 419–423.CrossRefGoogle Scholar
  6. Campanario, J. M., & Coslado, M. A. (2011). Benford’s law and citations, articles and impact factors of scientific journals. Scientometrics, 88, 421–432.CrossRefGoogle Scholar
  7. Egghe, L., & Guns, R. (2012). Applications of the generalized law of Benford to informetric data. Journal of the American Society for Information Science and Technology, 63, 1662–1665.CrossRefGoogle Scholar
  8. Hill, T. P. (1995). A statistical derivation of the significant-digit law. Statistical Science, 10, 354–363.MathSciNetzbMATHGoogle Scholar
  9. Hürlimann, W. (2009). Generalizing Benfords law using power laws: Application to integer sequences. International Journal of Mathematics and Mathematical Sciences, 2009. doi: 10.1155/2009/970284.
  10. Hürlimann, W. (2015). On the uniform random upper bound family of first significant digit distributions. Journal of Informetrics, 9, 349–358.CrossRefGoogle Scholar
  11. Lavalette, D. (1996). Facteur d’impact: Impartialité ou impuissance? Report INSERM U350, Institut Curie-Recherche, Bât. 112, Centre Universitaire, 91405 Orsay, France.Google Scholar
  12. Mansilla, R., Köppen, E., Cocho, G., & Miramontes, P. (2007). On the behavior of journal impact factor rank-order distribution. Journal of Informetrics, 1, 155–160.CrossRefGoogle Scholar
  13. Miller, S. J. (2015). Benford’s law: Theory and applications. Princeton, New Jersey: Princeton University Press.CrossRefzbMATHGoogle Scholar
  14. Newcomb, S. (1881). Note on the frequency of use of the different digits in nature numbers. American Journal of Mathematics, 4, 39–40.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Nigrini, M. (2012). Benford’s law: Applications for forensic accounting, auditing, and fraud detection. Hoboken, New Jersey: Wiley.CrossRefGoogle Scholar
  16. Pietronero, L., Tosatti, E., Tosatti, V., & Vespignani, A. (2001). Explaining the uneven distribution of numbers in nature: The laws of Benford and Zipf. Physica A, 293, 297–304.CrossRefzbMATHGoogle Scholar
  17. Pinkham, R. S. (1961). On the distribution of first significant digits. Annals of Mathematical Statistics, 32, 1223–1230.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Reed, W. J. (2001). The Pareto, Zipf and other power laws. Economics Letters, 74, 15–19.CrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  • Hsiang-chi Tseng
    • 1
  • Wei-neng Huang
    • 1
  • Ding-wei Huang
    • 1
    Email author
  1. 1.Department of PhysicsChung Yuan Christian UniversityChung-liTaiwan

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