# Michael S. Rosenberg’s Laboratory

Computational Evolutionary Biology & Bioinformatics

E-mail: msr@asu.edu
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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) Rank (i) Cumulative Citations (ΣCi) rw(i) = ΣCi / h 42 36 14 11 9 9 3 2 2 2 1 1 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 h = 6 42 78 92 103 112 121 124 126 128 130 131 132 133 133 133 133 7.00 13.00 15.33 17.17 18.67 20.17 20.67 21.00 21.33 21.67 21.83 22.00 22.17 22.17 22.17 22.17 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 = √78 = 8.8318

Yearhw
19971.0000
19982.4495
19994.3589
20006.0828
20018.8318
200211.6619
200315.1658
200418.1384
200523.1517
200629.1376
200732.7567
200837.0135
200940.3361
201043.6234
201146.4758
201249.1732
201353.2165
201456.0268
201559.3212
201662.7057
201764.5368

## References

• Egghe, L., and R. Rousseau (2008) An h-index weighted by citation impact. Information Processing and Management 44:770–780.