Evolutionary Biological Data Science Laboratory

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 i^th 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)	57	26	16	12	11	10	4	3	2	1	1	1	1	0	0	0	0	0
Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
						h = 6
Cumulative Citations (ΣCi)	57	83	99	111	122	132	136	139	141	142	143	144	145	145	145	145	145	145
rw(i) = ΣCi / h	9.50	13.83	16.50	18.50	20.33	22.00	22.67	23.17	23.50	23.67	23.83	24.00	24.17	24.17	24.17	24.17	24.17	24.17
		r0 = 2

The largest rank where r_w(i) ≤ C_i is 2. The weighted h-index is the square-root of the sum of citations up to this rank, thus h_w = √83 = 9.1104

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