*h*_{α}: An index to quantify an individual’s scientific leadership

## Abstract

The \(\alpha\) person is the dominant person in a group. We define the \(\alpha\)-author of a paper as the author of the paper with the highest *h*-index among all the coauthors, and an \(\alpha\)-paper of a scientist as a paper authored or coauthored by the scientist where he/she is the \(\alpha\)-author. For most but not all papers in the literature there is only one \(\alpha\)-author. We define the \(h_\alpha\) index of a scientist as the number of papers in the *h*-core of the scientist (i.e. the set of papers that contribute to the *h*-index of the scientist) where this scientist is the \(\alpha\)-author. We also define the \(h'_\alpha\) index of a scientist as the number of \(\alpha\)-papers of this scientist that have \(\ge\) \(h'_\alpha\) citations. \(h_\alpha\) and \(h'_\alpha\) contain similar information, while \(h'_\alpha\) is conceptually more appealing it is harder to obtain from existing databases, hence of less current practical interest. We propose that the \(h_\alpha\) and/or \(h'_\alpha\) indices, or other variants discussed in the paper, are useful complements to the *h*-index of a scientist to quantify his/her scientific achievement, that rectify an inherent drawback of the *h*-index, its inability to distinguish between authors with different coauthorships patterns. A high *h* index in conjunction with a high \(h_\alpha /h\) ratio is a hallmark of scientific leadership.

## Keywords

*h*-Index Coauthorship Scientific leadership

## Notes

### Acknowledgements

The author is grateful to a colleague for thoughtful comments.

## References

- Alonso, S., Cabrerizo, F. J., Herrera-Viedma, E., & Herrera, F. (2009).
*h*-Index: A review focused in its variants, computation and standardization for different scientific fields.*Journal of Informetrics*,*3*, 273–289.CrossRefGoogle Scholar - Ancheyta, J. (2015). A correction of
*h*-index to account for the relative importance of authors in manuscripts.*International Journal of Oil Gas and Coal Technology*,*10*, 221–232.CrossRefGoogle Scholar - Ausloos, M. (2015). Assessing the true role of coauthors in the
*h*-index measure of an author scientific impact.*Physica A*,*422*, 136–142.MathSciNetCrossRefGoogle Scholar - Bornmann, L. (2014).
*h*-Index research in scientometrics: A summary.*Journal of Informetrics*,*8*, 478–485.CrossRefGoogle Scholar - Bornmann, L., & Daniel, H. D. (2007). What do we know about the
*h*index?*Journal of the American Socieyt for Information Science and Technology*,*58*, 1381–1385.CrossRefGoogle Scholar - Bornmann, L., & Daniel, H.-D. (2010). The citation speed index: A useful bibliometric indicator to add to the
*h*index.*Journal of Informetrics*,*4*, 444–446.CrossRefGoogle Scholar - Bornmann, L., Mutz, R., & Daniel, H. D. (2008). Are there better indices for evaluation purposes than the
*h*index? A comparison of nine different variants of the*h*index using data from biomedicine.*Journal of the American Society for Information Science and Technology*,*59*, 830–837.CrossRefGoogle Scholar - Crispo, E. (2015). A new index to use in conjunction with the
*h*-index to account for an author’s relative contribution to publications with high impact.*Journal of the Association for Information Science and Technology*,*66*, 2381–2383.CrossRefGoogle Scholar - Dorta-Gonzalez, P., & Dorta-Gonzalez, M. I. (2011). Central indexes to the citation distribution: A complement to the
*h*-index.*Scientometrics*,*88*, 729–745.CrossRefGoogle Scholar - Egghe, L. (2008). Mathematical theory of the
*h*- and \(g\)-index in case of fractional counting of authorship.*Journal of the American Society for Information Science and Technology*,*59*, 1608–1616.CrossRefGoogle Scholar - Egghe, L., & Rousseau, R. (2008). An
*h*-index weighted by citation impact.*Information Processing and Management*,*44*, 770–780.CrossRefGoogle Scholar - Galam, S. (2011). Tailor based allocations for multiple authorship: A fractional \(gh\)-index.
*Scientometrics*,*89*, 365–379.CrossRefGoogle Scholar - Gibb, B. C. (2012). Lies, damned lies and
*h*-indices.*Nature Chemistry*,*4*, 513–514.CrossRefGoogle Scholar - Guns, R., & Rousseau, R. (2009). Real and rational variants of the
*h*-index and the \(g\)-index.*Journal of Informetrics*,*3*, 64–71.CrossRefGoogle Scholar - Hirsch, J. E. (2005). An index to quantify an individual’s scientific research output.
*Proceedings of the National Academy of Science*,*102*, 16569–16572.CrossRefGoogle Scholar - Hirsch, J. E. (2010). An index to quantify an individuals scientific research output that takes into account the effect of multiple coauthorship.
*Scientometrics*,*85*, 741–754.CrossRefGoogle Scholar - Hu, X., Rosseau, R., & Chen, J. (2010). In those fields where multiple authorship is the rule, the
*h*-index should be supplemented by role-based*h*-indices.*Journal of Information Science*,*36*, 73–85.CrossRefGoogle Scholar - Jin, B., Liang, L. M., Rousseau, R., & Egghe, L. (2007). The R- and AR-indices: Complementing the
*h*-index.*Chinese Science Bulletin*,*52*, 863–863.Google Scholar - Lando, T., & Bertoli-Barsotti, L. (2014). New tools for complementing the
*h*-index: An empirical study.*Mathematical Methods in Economics*,*2014*, 566–571.Google Scholar - Liu, X. Z., & Fang, H. (2012). Modifying
*h*-index by allocating credit of multi-authored papers whose author names rank based on contribution.*Journal of Informetrics*,*6*, 557–565.CrossRefGoogle Scholar - Mazurek, J. (2018). A modification to Hirsch index allowing comparisons across different scientific fields.
*Current Science*,*114*, 2238–2239.Google Scholar - Perry, M., & Reny, P. J. (2016). How to count citations if you must.
*American Economic Review*,*106*, 2722–2741.CrossRefGoogle Scholar - Prathap, G. (2012). The Inconsistency of the
*h*-index.*Journal of the American Society for Information Science and Technology*,*63*, 1480–1481.CrossRefGoogle Scholar - Rousseau, R., & Ye, F. (2008). A proposal for a dynamic
*h*-type index.*Journal of the American Society for Information Science and Technology*,*59*, 1853–1855.CrossRefGoogle Scholar - Schreiber, M. (2008). To share the fame in a fair way, \(h_m\) modifies
*h*for multi-authored manuscripts.*New Journal of Physics*,*10*, 040201.CrossRefGoogle Scholar - Schreiber, M. (2008). A modification of the
*h*-index: The \(h_m\)-index accounts for multi-authored manuscripts.*Journal of Informetrics*,*2*, 211–216.CrossRefGoogle Scholar - Schreiber, M. (2009). A case study of the modified Hirsch index \(h_m\) accounting for multiple coauthors.
*Journal of the American Society for Information Science and Technology*,*60*, 1274–1282.CrossRefGoogle Scholar - Schreiber, M. (2018). A skeptical view on the Hirsch index and its predictive power.
*Physica Scripta*,*93*, 10201.CrossRefGoogle Scholar - Tscharntke, T., et al. (2007). Author sequence and credit for contributions in multiauthored publications.
*PLoS Biology*,*5*(1), e18.CrossRefGoogle Scholar - Van Eck, N. J., & Waltman, L. (2008). Generalizing the
*h*- and \(g\)-indices.*Journal of Informetrics*,*2*, 263–271.CrossRefGoogle Scholar - Van Raan, A. F. J. (2006). Comparison of the Hirsch-index with standard bibliometric indicators and with peer judgment for 147 chemistry research groups.
*Scientometrics*,*67*, 491–502.CrossRefGoogle Scholar - Vavrycuk, V. (2018). Fair ranking of researchers and research teams.
*PLoS ONE*,*13*(4), e0195509.CrossRefGoogle Scholar - Waltman, L., & Nees, J. (2012). The inconsistency of the
*h*-index.*Journal of the American Society for Information Science and Technology*,*63*, 406–415.CrossRefGoogle Scholar - Yaminfirooz, M., & Gholinia, H. (2015). Multiple
*h*-index: A new scientometric indicator.*Electronic Library*,*33*(547), 556.Google Scholar - Zhang, C.-T. (2009). The
*e*-index, complementing the*h*-index for excess citations.*PLoS ONE*,*4*, e5429.CrossRefGoogle Scholar