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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.

History

YearhT
19971.3333
19984.3738
19995.4085
20008.3648
200110.2133
200213.1511
200316.9065
200419.9143
200523.3876
200625.7451
200728.5960
200831.1015
200934.1268
201037.4354
201140.8543
201242.9804
201345.5526
201447.1168
201549.0064
201650.0268
201751.2033
201853.0216
201953.8615
202055.7672
202157.5715
202259.6963
202360.5617
202461.9176
202563.7914

References