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214 Becoming Metric-Wise
when citation scores are not natural numbers then using this variant is, in
our opinion, the method of choice. It is defined as follows:
Let P(r) denote the magnitude value of the rth source and let P(x)denote
the piecewise linear interpolation of the sequence (r, P(r)), r5 1, 2, .. .,then
h r is the abscissa of the intersection of the function P(x) and the straight line
y 5 x (if this intersection exists). Its value is then given by:
ðh 1 1ÞPðhÞ 2 hPðh 1 1Þ
h int 5 (7.9)
1 2 Pðh 1 1Þ 1 PðhÞ
Clearly h int $ h with equality sign only if h 5 P(h).
Considering the previous examples we have:
For [7,4,3,1,0] h 5 3, P(3) 5 3; P(4) 5 1 and h int 5 (4 3 3 1)/
(1 2 1 1 3) 5 9/3 5 3. As long as the third publication receives exactly
3 citations h int stays equal to 3. For [6,5,4,3,2,1,0] we find
h int 5 (4 4 2 3 3)/(1 2 3 1 4) 5 7/2 5 3.5. For [10, 2, 2, 2] h int is clearly
2. For [10, 8, 1, 0] it is (3 8 2 2 1)/(1 2 1 1 8) 5 22/8 5 2.75. In most
cases, but not always, h int , h rat , see (Guns & Rousseau, 2009a).
It was mentioned that the interpolated h-index could also be used
when citation values are not necessarily natural numbers, which happens
e.g., when using fractional counting. It is then possible that the most-
cited citation of a set of publications has a value of 0.5, in which case the
line y 5 x does not intersect P(x). This explains the restriction in the
definition (the part “if the intersection exists”). A solution for this prob-
lem has been suggested in (Rousseau, 2014c).
7.4.3 The m-quotient
It has been mentioned that in its original version the h-index favors
scientists with a longer career. Hirsch (2005) provided a solution for this,
namely by dividing the standard h-index by the length of a scientist’s
career expressed in years. We note that there is an unspoken assumption
used in this proposal. This assumption is that in the “regular” case an
h-index grows linearly over the years. If this regular growth were curved
then either younger or older scientists would be favored by this correc-
tion. It has been observed that by and large the h-index of a scientist
grows linearly, but, of course, there are many exceptions. Surely, the
higher the h-index the more citations are needed (we remarked earlier
that this number can be as high as (2h 1 1)) to increase it by one unit. As
many articles in the h-core tend to be older ones, for which it is less