g-index
The best known and most widely studied alternative to the h-index is known as the g-index (Egghe 2006a, b, c). The g-index is designed to give more credit for publications cited in excess of the h threshold. The primary difference between the formal definitions of the h- and g-indices is that g is based on cumulative citation counts rather than individual citation counts. Formally, the g-index is the largest value for which g publications have jointly received at least g2 citations.
$$g=\underset{i}{\max}\left(i^2\leq \sum\limits_{j=1}^{i}{C_j}\right)=\underset{i}{\max}\left(i\leq\frac{\sum\limits_{j=1}^{i}{C_j}}{i} \right)$$Although not usually formulated this way, the above also shows an alternative interpretation of the g-index, which makes it's meaning and relationship to h much clearer: the g-index is the largest value for which the top g publications average g citations, while h is the largest value for which the top h publications have a minimum of h citations.
Stricly speaking, it is possible for the number of citations in the g-core to exceed the square of the total number of publications (CP > P2), or using the alternate definition, for the average number of citations per publication to exceed the number of publications. Under this scenario, the threshold curve and the citation curve do not actually cross. Some authors have suggested adding phantom publications with zero citations until the curves cross (essentially, making g equal to the square-root of CP); a more conservative approach, illustrated here, is to set the maximum possible value of g equal to the number of publications.
Example
Publications are ordered by number of citations, from highest to lowest.
| Citations (Ci) | 47 | 26 | 19 | 15 | 11 | 10 | 4 | 3 | 2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cumulative Citations (ΣCi) | 47 | 73 | 92 | 107 | 118 | 128 | 132 | 135 | 137 | 138 | 139 | 140 | 141 | 141 | 141 | 141 | 141 | 141 |
| Rank Squared (i2) | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 | 256 | 289 | 324 |
| g = 11 | ||||||||||||||||||
| Rank (i) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
| Mean Citations (ΣCi / i) | 47.0 | 36.5 | 30.7 | 26.8 | 23.6 | 21.3 | 18.9 | 16.9 | 15.2 | 13.8 | 12.6 | 11.7 | 10.8 | 10.1 | 9.4 | 8.8 | 8.3 | 7.8 |
The largest rank where i2 ≤ ΣCi (or i ≤ mean Ci) is 11.
History
| Year | g |
|---|---|
| 1997 | 1 |
| 1998 | 3 |
| 1999 | 6 |
| 2000 | 8 |
| 2001 | 11 |
| 2002 | 15 |
| 2003 | 19 |
| 2004 | 25 |
| 2005 | 30 |
| 2006 | 36 |
| 2007 | 39 |
| 2008 | 41 |
| 2009 | 45 |
| 2010 | 51 |
| 2011 | 55 |
| 2012 | 56 |
| 2013 | 62 |
| 2014 | 64 |
| 2015 | 67 |
| 2016 | 69 |
| 2017 | 70 |
| 2018 | 75 |
| 2019 | 76 |
| 2020 | 77 |
| 2021 | 79 |
| 2022 | 84 |
| 2023 | 85 |
| 2024 | 86 |
References
- Egghe, L. (2006) An improvement of the h-index: The g-index. ISSI Newsletter 2(1):8–9.
- Egghe, L. (2006) How to improve the h-index: Letter. The Scientist 20(3):14.
- Egghe, L. (2006) Theory and practice of the g-index. Scientometrics 69(1):131–152.