Abstract
The h-index characterizes the publication achievement/impact of authors and is defined by the originator Jorge Hirsch as: ‘the number of papers with citation number ≥h’. The h-index has the inherent characteristic that authors with very different total citations can have the same h-index. In fact, no contributions to the h-index are made either by papers cited fewer times than h, or citations of an individual paper above h. Such citations are ‘excess’ citations not credited by the h-index. To address these deficiencies, we propose a simple, straightforward modification, the hb-index: 
, where h is the Hirsch h-index and e is the sum of all citations minus h2. Therefore, e is the excess citations not credited by the h-index.
Keywords: Publication impact, Author impact, Hirsch index, Modified Hirsch index, Bibliographic impact
Introduction
The h-index was first described by Hirsch1 in an article entitled: ‘An index to quantify an individual's scientific research output’. The h-index was originally intended to be a measure of an individual's achievement although it commonly now is used as a bibliographic measure of impact. Hirsch1 defined the h-index as ‘the number of papers with citation number ≥h’. The h-index has received acceptance and support,2–4 criticism,5 corrections, complements, and a single-number alternative has been devised.6. An extensive evaluation of the h index and its variants has been published.7 The h-index and the related area of journal impact has recently been the subject of articles published in journals of higher education.8,9 The h-index has been used recently to assess impact of disease10 and it has been scaled for different scientific fields that were shown to have significantly different average citations per paper.11 The h-index can be calculated using databases such as Thompson Scientific Web of Science12 or Google Scholar Citations.13 A book and associated software14 is useful generally for the subject of publication impact, specifically for analyzing academic citations, and for calculating the h-index.
Inherently, the h-index has the undesirable characteristic that individuals with very different numbers of citations can have the same h-index. Our goal in this report was to modify the h-index to express the differences resulting from increased total citations while maintaining the other attributes of the index that have been found useful in evaluating the publication impact of an individual including for promotion, tenure, asset allocation, and related administrative purposes.
Results
Every h-index can be generated by the minimum of h2 citations when exactly h papers receive exactly h citations. The effect of h2 citations justifies 
term in the hb-index. Also, any h-index theoretically could be generated by any total number of citations ≥h2. The greater than h2 citations are ‘excess’ citations and it appears reasonable and desirable to modify the h-index to credit these citations. This is especially important for young scientists who may have only a few papers and a comparatively low h-index. By analogy, it is desirable that a single-number index reveals the ‘ripening fruit’ whereas the h-index reveals only the ‘fully ripened’ fruit in one event as it creates a unitary increase in h on the special occasion that a citation occurs for a ‘nearly ripe’ paper.
Fig. 1 clarifies the h-index and reveals its limitation of not accounting for ‘excess’ citations. A specific publication record is shown for a hypothetical author who has published 90 papers cited 820 times. Citations are distributed as follows: 1 paper with 100; 9 papers each with 30; and groups of 10 papers each with 20, 10, 5, 4, 3, 2, 1, and 0 citations. By inspection, the h-index is 20 (number of papers with ≥h citations). However, the number of papers and their citation distributions are only one of many with the h-index of 20. Indeed, every h-index can be created from multiple distributions (except for the trivial case with one paper). Fig. 1 also illustrates that no citations of a paper that exceed 20 make any contribution to the h-index of 20. Likewise, no citation of lowly ranked papers (those below the twentieth paper in this case) makes any contribution to the h-index (by definition, exactly 20 papers cited 20 or more times, creates the h-index). The other citations are ‘excess’ because they do not contribute to the h-index.
Figure 1.
Graphic depiction for a hypothetical author with an h-index of 20 (see text for specifics).
There are other undesirable consequences of the h-index. For example, the first citation of any published paper of an individual creates an h-index of 1. However, the publication of any number of additional papers (however large the number) will not change the h-index until two papers become cited twice and (minimally) n papers must be cited n times for the h-index to become n. Thus, additional citations of ≤h for any number of papers cited h times will not raise the h-index. Also, a single paper, no matter how highly cited, cannot lead to an h-index that is larger than 1, and a body of papers, no matter how highly cited, cannot have an h-index larger than the number of papers in that body. Also, if a scientist publishes only one paper and that paper becomes cited any number of times, the h-index remains at 1. It is undesirable that an index behave in this manner.
Table 1 compares the h-index and the hb-index. Publication records were created with different citation distributions to clarify the concept of ‘excess citations’ and to make the math simple enough to evaluate by inspection. However, other citation distributions contain the same defects. Table 1 distributions A–F compare different authors with very different total citations but the same number of papers to isolate the effect of number of citations. For clarity, the excess citations for distributions A–H occur in papers with citations greater than h. It is obvious that publications with citations fewer than h have no effect on h.
Table 1.
Corrective effects of the hb-index
| Citation distributions* | Total citations | Excess citations (e)† | Publications | h-index | hb-index |
|---|---|---|---|---|---|
| A | 100 | 0 | 10 | 10 | 10 |
| B | 190 | 90 | 10 | 10 | 19.48 |
| C | 200 | 100 | 10 | 10 | 20.00 |
| D | 230 | 130 | 10 | 10 | 21.40 |
| E | 370 | 270 | 10 | 10 | 26.43 |
| F | 550 | 450 | 10 | 10 | 31.21 |
| G | 1050 | 650 | 20 | 20 | 45.50 |
| H‡ | 7150 | 5550 | 260 | 40 | 114.50 |
*Distribution A has the minimum number of publications and citations that yield the h-index of 10. The other citation distributions were made specific for clarity and devised to give a wide range of total citations, but are not unique to our argument. Distribution B is assigned 190 citations with 1 paper cited 100 times and 9 papers cited 10 times each. Distribution C has 1 paper cited 50 times, 2 papers cited 40 times each, and 7 papers cited 10 times each. Distribution D has 1 paper cited 50 times, 3 papers each cited 30 times, 3 papers each cited 20 times, and 3 papers each cited 10 times. Distribution E has 1 paper cited 100 times, 3 papers each cited 50 times, 3 papers each cited 30 times, and 3 papers each cited 10 times. Distribution F has 1 paper with each of the following number of citations: 100, 90, 80, 70, 60, 50, 40, 30, 20, and 10. Distribution G has 1 paper cited 200 times, 4 papers each cited 100 times, 5 papers each cited 50 times and 10 papers each cited 20 times. Distribution H is 1 paper cited 500 times, 1 paper cited 300 times, 3 papers each cited 150 times, 10 papers each cited 100 times, 25 papers each cited 40 times, 60 papers each cited 30 times, 70 papers each cited 20 times, 50 papers each cited 10 times, and 40 papers each cited 5 times.
†The sum of those citations of individual papers that are greater than or less than the value of h. Excess citations have the value of e as defined by 
. Therefore, e is the excess citations not credited by the h-index.
‡The citations are distributed to show the effects on an author with a high number of publications and citations that include some publications with citations less than h.
Discussion
It is apparent (Table 1) that many citation distributions can have the same h-index and number of publications but have unique hb-indices. The first row in Table 1 has the fewest citations (h2) for an h-index of 10 and it has zero ‘excess’ (e) citations. All citations contribute and the h-index and the hb-index are identical (the hb-index is 
). The second row in Table 1 also has an h-index of 10. However, the hb-index is 
. Thus, the hb-index is increased, relative to h, by the contribution of the 90 citations that otherwise do not contribute. Table 1 also shows that six authors, each with identical h-indices of 10 and exactly 10 papers but unique distributions of citations (A–F), have hb-indices that range from 10 to 31.21. Fundamentally, the h-index can never be larger than the total number of publications, but it can range downward to zero. The h-index precisely equals the total number of papers that are cited ≥h times. More specifically, as shown by citation distributions A–G (Table 1), once each paper has been cited a minimum of 10 times, the h-index becomes 10. Regardless of additional citations, the h-index remains at 10 until another paper is published and each of the now eleven papers is cited a least 11 times. These additional ‘excess’ citations are revealed and credited by the hb-index as the function 
where e is the value of the ‘excess’ citations. Incorporating 
provides a smooth increase in the hb-index.
To further compare the h-index with the hb-index, consider an additional case (not in Table 1) with citations distributed identically to distribution B (Table 1) except that all 90 excess citations are distributed among papers cited fewer times than the h index (10 in this case). This necessarily raises the total number of papers beyond 10 but these citations contribute nothing to the h-index. To evaluate the effects on the hb-index for this case, citations must be specifically assigned. Assume 10 papers with 10 citations each (to generate h = 10) and one paper each with the following number of citations: 9, 8, 7, 6, 5, 4, 3, 2, and 1. Now there are 19 papers with 145 total citations. The h-index remains 10 but the hb-index is now 
which equals 16.71 and credits the 9 papers that received a total of 45 citations. As a further example, if there had been 10 papers each cited 10 times and one paper cited an ‘excess’ 9 times, the h-index would, of course, be 10 but the hb-index would be 
. Likewise, for 1 ‘excess’ citation, the hb-index would be 
. The consequences for other cases are easily calculated.
It can be argued that the failure of the h-index to credit papers with citations below h is of low concern. However, researchers early in their careers are more likely to have such excess citations from papers not yet highly cited. Indeed, the hb-index recognizes this and gives appropriate credit based on the comparatively low (but significant for discriminating among authors) contribution of 
by lowly cited papers. However, when there are papers that receive many citations in excess of h (Table 1 distributions B–H; see footnote for specifics), these papers are appropriately highly credited by the hb-index which rises to comparatively large values because 
is comparatively large.
Conclusion
The h-index is improved by calculating the hb-index which is the h-index plus a value 
with e defined as the excess citations (the citations in excess of the minimum total number (h2) that can define the h-index). The hb-index credits citations that otherwise have no weight, differentiates among authors who have the same h-index, and is especially useful for distinguishing among publication records of young researchers with modest h-indices but significant papers that are ‘ripening’ but not yet highly cited.
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