# Michael S. Rosenberg’s Laboratory

Computational Evolutionary Biology & Bioinformatics

E-mail: msr@asu.edu
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## hm-index

While the normalized hi-index accounts for the number of authors of a publication by normalizing the citation count for that publication, hm (Schreiber 2008), also known as the fractional paper h and hF-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 ith publication as i, it is instead calculated as the cumulative sum of 1/Ai. In formal terms, when calculating the h-index the rank of the ith publication is

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

For hm we instead calculate the effective rank as

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

hm is determined as the largest value of reff(i) for which reff(i) ≤ Ci 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) Authors (Ai) Author Effort (Ei = 1/Ai) Rank (i) Adjusted Rank (reff(i) = ΣEi) 42 36 14 11 9 9 3 2 2 2 1 1 1 0 0 0 3 3 3 2 4 1 4 4 1 1 4 2 1 4 2 1 0.33 0.33 0.33 0.50 0.25 1.00 0.25 0.25 1.00 1.00 0.25 0.50 1.00 0.25 0.50 1.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.33 0.67 1.00 1.50 1.75 2.75 3.00 3.25 4.25 5.25 5.50 6.00 7.00 7.25 7.75 8.75 hm = 3.00

The largest adjusted rank where reff(i) ≤ Ci is 3.0000.

Yearhm
19970.5833
19981.1667
19991.4167
20002.7500
20013.0000
20024.2500
20035.6706
20047.6706
20059.2615
200610.5949
200712.6782
200813.4282
200914.6782
201016.5191
201117.3941
201217.5767
201318.5767
201418.5767
201518.8267
201618.8267
201718.9695

## References

• Egghe, L. (2008) Mathematical theory of the h- and g-index in case of fractional counting of authorship. Journal of the American Society for Information Science and Technology 59(10):1608–1616.
• Schreiber, M. (2008) To share the fame in a fair way, hm modifies h for multi-authored manuscripts. New Journal of Physics 10:040201.