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rational h-index

The rational h-index (Ruane and Tol 2008) (hΔ or hrat) is a continuous version of the h-index which not only measures the standard h-index but includes the fractional progress toward the next higher value of h. It is calculated as

$$h^\Delta = h + 1 -\frac{n}{2h+1},$$where \(n\) is the number of citations necessary to reach the next value of \(h\). The divisor, \(2h+1\), is the maximum number of possible citations needed to move from \(h\) to \(h+1\) (one additional citation for each of the \(h\) publications in the core plus \(h+1\) citations for a publication outside of the core with no citations). Practically speaking, \(n\) is the number of publications in the core with exactly \(h\) citations (thus needing one more to allow a move to \(h+1\)) plus \(h+1-C_{h+1}\) (the number of citations the \(h+1\)th ranked publication needs to reach \(h+1\) citations).

Example

Publications are ordered by number of citations, from highest to lowest.

Citations (Ci)592616111110433211100000
Rank (i)123456789101112131415161718
h = 6

The h-index is 6. To reach an h of 7, we would need to add 0 citations to the publications within the core and 3 citations to the 7th publication (for a total of 3). The maximum number of citations that could be required is 13 (if all 6 core publications had exactly 6 citations and the 7th publication had zero), thus the rational h is 6 + 1 - 3/13 = 6.7692.

History

YearhΔ
19971.3333
19983.4286
19993.7143
20005.8182
20016.7692
20028.7647
200310.9524
200412.9600
200515.8710
200616.9697
200719.9231
200821.9767
200924.9796
201025.9804
201128.9825
201232.9231
201333.9552
201434.9710
201535.9718
201635.9859
201737.7867
201837.8267
201937.8667
202038.9740
202139.9873
202241.9277
202341.9759
202442.9765
202544.9213

References