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weighted h-index

Similar to the R-index, the weighted h-index (Egghe and Rousseau 2008) is designed to give more weight to publications within the core as they gain citations. The primary difference is that for this metric the core is defined differently. Publications are still ranked by citation count, but instead of using the raw rank, one uses a weighted rank of

$$r_w\left(i\right)=\frac{\sum\limits_{j=1}^{i}{C_j}}{h},$$

that is, the weighted rank of the ith publication is the cumulative sum of citations for the top i publications, divided by the standard h-index. With these weighted ranks, one finds the last publication in the weighted core, r0, as the largest value of i where \(r_w\left(i\right)\leq C_i\) (the last publication for which the weighted rank of that publication is less than or equal to the number of citations for that publication):

$$r_0=\underset{i}{\max}\left(r_w\left(i\right)\leq C_i\right).$$

The weighted index is then calculated as$$h_w=\sqrt{\sum\limits_{i=1}^{r_0}{C_i}},$$the square-root of the sum of citations for the weighted core.

Example

Publications are ordered by number of citations, from highest to lowest.

Citations (Ci)592616111110433211100000
Rank (i)123456789101112131415161718
h = 6
Cumulative Citations (ΣCi)5985101112123133137140143145146147148148148148148148
rw(i) = ΣCi / h9.8314.1716.8318.6720.5022.1722.8323.3323.8324.1724.3324.5024.6724.6724.6724.6724.6724.67
r0 = 2

The largest rank where rw(i) ≤ Ci is 2. The weighted h-index is the square-root of the sum of citations up to this rank, thus hw = √85 = 9.2195

History

Yearhw
19971.0000
19982.6458
19995.6569
20007.0000
20019.2195
200213.1149
200315.7162
200420.5183
200525.5539
200630.4302
200735.7211
200839.4081
200943.8862
201048.9285
201155.0636
201258.9830
201364.9230
201467.8454
201573.4983
201677.2399
201780.3928
201883.3247
201986.1510
202088.8594
202192.6930
202295.4620
202397.7548
2024100.0150
2025101.0346

References