Evolutionary Biological Data Science Laboratory

h_m-index

While the normalized h_i-index accounts for the number of authors of a publication by normalizing the citation count for that publication, h_m (Schreiber 2008), also known as the fractional paper h and h_F-index (Egghe 2008), accounts for author number by normalizing the ranks. For this index, one counts the number of citations and ranks the publications as for the h-index, but rather than counting the rank of the i^th publication as i, it is instead calculated as the cumulative sum of 1/A_i. In formal terms, when calculating the h-index the rank of the i^th publication is

$$r_i=\sum\limits_{j=1}^{i}{1}=i.$$

For h_m we instead calculate the effective rank as

$$r_{\text{eff}}\left(i\right)=\sum\limits_{j=1}^{i}{\frac{1}{A_i}}.$$

h_m is determined as the largest value of r_eff(i) for which r_eff(i) ≤ C_i or

$$h_m=\underset{r_{\text{eff}}\left(i\right)}{\max}\left(r_{\text{eff}}\left(i\right) \leq C_i\right).$$

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
Authors (Ai)	3	3	2	3	1	4	4	4	1	4	2	1	1	4	2	1	1	1
Author Effort (Ei = 1/Ai)	0.33	0.33	0.50	0.33	1.00	0.25	0.25	0.25	1.00	0.25	0.50	1.00	1.00	0.25	0.50	1.00	1.00	1.00
Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Adjusted Rank (reff(i) = ΣEi)	0.33	0.67	1.17	1.50	2.50	2.75	3.00	3.25	4.25	4.50	5.00	6.00	7.00	7.25	7.75	8.75	9.75	10.75
							hm = 3.00

The largest adjusted rank where r_eff(i) ≤ C_i is 3.0000.

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