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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)472619151110432111100000
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
19982.8000
19993.7143
20005.7273
20016.7692
20028.8235
200311.8696
200412.9600
200515.8710
200617.8857
200719.9487
200821.9535
200925.8824
201026.9245
201129.9322
201232.9538
201333.9254
201434.9710
201535.9577
201635.9859
201737.7867
201837.8133
201937.8667
202038.9481
202140.9506
202241.9639
202342.9529
202442.9529

References