# Michael S. Rosenberg’s Laboratory

Computational Evolutionary Biology & Bioinformatics

E-mail: msr@asu.edu
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## tapered h-index

While the rational h-index gives a fractional value to those citations necessary to reach the next value of h, the tapered h-index (Anderson et al. 2008) is designed to give every citation for every publication some fractional value. The best way to understand this index is to first consider the contribution of every citation to the h-index. To have an h-index of 1, an author needs a single paper with a single citation. That citation has a weight (or score) of 1, because it accounts for the entire h value of 1. To move to an h-index of 2, the author needs three additional citations: one additional citation for the original publication and two citations for a second publication. As h has increased by one, each of these three citations is contributing a weight (or score) of 1/3 to the total h-index. This is most easily illustrated by a Ferrers graph of ranked publications versus citations which shows the specific contribution of every citation to a specific value of h

Citation
12 34 56
Ranked
publication
111/3 1/51/71/91/11
21/31/3 1/51/7
31/51/5 1/51/7
41/71/7
51/9

The largest filled-in square in the upper left corner (the Durfee square) has a length equal to h; the contents of the square also sum to h. Using this logic, one can determine the credit each citation would give to a larger value of h, regardless of whether that h has been reached. Consider this graph with respect to the rational h-index. In the above example, h is 3. If one just considers the citations necessary to reach an h of 4, we can see that 5 of the 7 necessary citations are already present. Each of these has a weight of 1/7 (since 7 total citations are necessary); adding these to h we get the rational h-index, $$h^\Delta=3.71$$. The tapered h-index is simply taking this same concept but expanding it to include all citations for all publications.

The tapered h-index for a specific publication is the sum of all of its scores and the total score of the index is the sum across all publications. In simple formulaic terms, the score hT(i) for the ith ranked publication is calculated as

$$h_{T\left(i\right)}=\left |\begin{matrix} \frac{C_i}{2i-1} & \text{if } C_i \leq i \\ \frac{i}{2i-1}+\sum\limits_{j=i+1}^{C_i}{\frac{1}{2j-1}} & \text{if } C_i > i \end{matrix}\right. ,$$

and the total tapered h-index is the sum of these scores for all publications,

$$h_T=\sum\limits_{i=1}^{P}{h_{T\left(i\right)}}.$$This index is consistent with the concept of the h-index, while also giving every citation some small influence on the score.

YearhT
19971.3333
19983.7437
19994.8642
20007.4899
20019.6576
200212.3077
200315.0441
200418.5548
200521.1995
200623.7368
200726.8198
200829.0324
200932.1712
201035.5884
201138.2048
201240.0496
201343.0222
201444.6786
201546.8178
201647.5942
201748.3933

## References

• Anderson, T.R., R.K.S. Hankin, and P.D. Killworth (2008) Beyond the Durfee square: Enhancing the h-index to score total publication output. Scientometrics 76(3):577–588.