Chinese Science Bulletin
© 2007 Science in China Press
Springer-Verlag
www.scichina.com www.springerlink.com Chinese Science Bulletin | March 2007 | vol. 52 | no. 6 | 855-863
ARTICLES SCIENTOMETRICS
The R- and AR-indices: Complementing the h-index
JIN BiHui
1†
, LIANG LiMing
2,3
, Ronald ROUSSEAU
3,4,5
& Leo EGGHE
3,5
1
National Science Library, Chinese Academy of Sciences, Beijing 100080, China;
2
Institute for Science, Technology and Society, Henan Normal University, Xinxiang 453002, China;
3
University of Antwerp (UA), IBW, B-2610 Wilrijk, Belgium;
4
KHBO, Industrial Sciences and Technology, B-8400, Oostende, Belgium;
5
Universiteit Hasselt (UHasselt), Agoralaan, B-3590 Diepenbeek, Belgium
Based on the foundation laid by the h-index we introduce and study the R- and AR-indices. These new
indices eliminate some of the disadvantages of the h-index, especially when they are used in combina-
tion with the h-index. The R-index measures the h-core’s citation intensity, while AR goes one step
further and takes the age of publications into account. This allows for an index that can actually in-
crease and decrease over time. We propose the pair (h, AR) as a meaningful indicator for research
evaluation. We further prove a relation characterizing the h-index in the power law model.
h-index, A-index, R-index, AR-index, g-index, performance evaluation, power law
1 The Hirsch index
The h-index, also known as the Hirsch index, was in-
troduced by Hirsch
[1]
as an indicator for lifetime
achievement. Considering a scientist’s list of publica-
tions, ranked according to the number of citations re-
ceived, the h-index is defined as the highest rank such
that the first h publications received each at least h cita-
tions. It became soon clear that the h-index can not only
be used for lifetime achievements, but also in the con-
text of many―― but not all―― other source-item rela-
tionships
[2,3]
. Consequently, the Hirsch index has been
calculated for journal citations
[2,4]
, topics
[5,6]
, library
loans per category
[7]
, and, pre-dating its actual intro-
duction, even cycling
[8]
. In this paper we will, however,
mainly use the terminology of publications and cita-
tions.
1.1 The Hirsch core
All publications ranked between rank 1 and rank h form
the Hirsch core. If there are several publications with the
same number of citations, then one may use two ap-
proaches to determine the Hirsch core. Either one in-
cludes all publications with h citations (hence the Hirsch
core may contain more than h elements), or one intro-
duces a secondary criterion for ranking. A good idea is
ranking articles with the same number of citations in
anti-chronological order so that more recent articles
have a larger probability to belong to the Hirsch core
than older ones. The Hirsch core can be considered as a
group of high-performance publications, with respect to
the scientist’s career. Hence the term ‘high-performance’
should be understood in a relative sense.
1.2 Advantages and disadvantages of the h-index
We recall some advantages and disadvantages of the
h-index that have been put forward in the literature
[1,9]
.
Advantages
●
It is a mathematically simple index.
●
It encourages a large amount of high quality (at
least highly
visible
) work.
●
The h-index can be applied to any level of aggre-
gation.
●
It combines two types of activity (in the original
setting this is citation impact and publications).
Received February 5, 2007; accepted February 26, 2007
doi: 10.1007/s11434-007-0145-9
†Corresponding author (email: jinbh@mail.las.ac.cn)
Supported by a Major State Basic Research Special Program China under grant (No.
2004CCC00400) and National Natural Science Foundation of China (Grant No.
70376019)
856 JIN BiHui et al. Chinese Science Bulletin | March 2007 | vol. 52 | no. 6 | 855-863
●
It is a robust indicator
[10]
. Increasing the number
of publications alone does not have an immediate effect
on this index.
●
Single peaks (top publications) have hardly any
influence on the h-index.
●
In principle, any document type can be included.
●
Publications that are hardly ever cited do not in-
fluence the h-index. In this way, the h-index discourages
publishing unimportant work.
●
It has been shown that the h-index is closely cor-
related to total publication output
[1]
.
Disadvantages
●
The h-index, in its original setting
[1]
, puts new-
comers at a disadvantage since both publication output
and observed citation rates will be relatively low. In
other words, it is based on long-term observations.
●
The index allows scientists to rest on their laurels
since the number of citations received may increase
even if no new papers are published.
●
The h-index is only useful for comparing the bet-
ter scientists in a field. It does not discriminate among
average scientists.
●
This indicator can never decrease.
●
The h-index is only weakly sensitive to the num-
ber of citations received. Indeed, when a scientist’s
h-index is equal to h, then this scientist’s first h articles
received at least h times h, i.e. h
2
citations. This lower
bound is the only relation that logically exists between
publications and citations, when the h-index is known.
The two previously mentioned disadvantages may be
summarized by stating that the h-index lacks sensitivity
to performance changes.
Moreover, the h-index suffers from the same prob-
lems as all simple indicators that use citations.
●
Like most pure citation measures it is field-de-
pendent, and may be influenced by self-citations.
●
There is a problem finding reference standards.
●
There exist many more versatile indicators
[11]
.
●
It is rather difficult to collect all data necessary for
the determination of the h-index. Often a scientist’s
complete publication list is necessary in order to dis-
criminate between scientists with the same name and
initial. We refer to this problem as the precision prob-
lem.
It seems that in most applications colleagues have
used only Web of Science data. Such a practice is not
implied by the definition of the h-index, but when re-
stricting data to WoS data this punishes colleagues who
have highly cited articles in conference proceedings or
journals, including web journals, not covered by the
Web of Science (WoS).
Although (or because?) the h-index is a relatively
simple indicator it immediately attracted a lot of atten-
tion from the scientific community
[12
―
18]
.
1.3 Other h-type indices
In view of the advantages and disadvantages mentioned
above it is no surprise that colleagues proposed some
simple variations on the h-index idea
[19,20]
, elaborated
mathematical models
[3,21,22]
and proposed some new
‘Hirsch-type’ indices trying to overcome some of the
disadvantages. Among these we mention Egghe’s
g-index
[23,24]
, Kosmulski’s H
(2)
-index
[25]
and Jin’s
A-index
[26]
.
For the g-index as well as for the H
(2)
-index one
draws the same list as for the h-index. The g-index, on
the one hand, is defined as the highest rank such that the
cumulative sum of the number of citations received is
larger than or equal to the square of this rank. Clearly h
≤g. The H
(2)
-index, on the other hand, is k if k is the
highest rank such that the first k publications received
each at least k
2
citations. The main advantage of this
index is that it reduces the precision problem. We think
however that this index is not sensitive enough
[7]
and
will not consider it anymore in this article. The g-index
clearly overcomes the problem that the h-index does not
include an indicator for the internal changes of the
Hirsch core. Yet, it requires drawing a longer list than
necessary for the h-index, hence increasing the precision
problem.
1.4 The A-index and the new R-index
Jin’s A-index achieves the same goal as the g-index,
namely correcting for the fact that the original h-index
does not take the exact number of citations of articles
included in the h-core into account. This index is simply
defined as the average number of citations received by
the publications included in the Hirsch core. The name
of this index is derived from the fact that it is just an
average (A). Mathematically, this is,
A
=
1
1
.
h
j
j
cit
h
=
∑
(1)
In formula (1) the numbers of citations (cit
j
) are
ranked in decreasing order. Note that, as long as the
Hirsch core contains exactly h elements, the A-index is
unambiguously defined. The A-index, moreover, uses
JIN BiHui et al. Chinese Science Bulletin | March 2007 | vol. 52 | no. 6 | 855-863 857
ARTICLES SCIENTOMETRICS
the same data as the h-index so that the precision prob-
lem is exactly the same as for the original h-index, and
is not increased as in the case of the g-index. Clearly
h
≤A. Yet, the A-index suffers from another problem
illustrated by the following fictitious case. Assume that
scientist X
1
has published 20 articles, one cited 10 times
and all other ones just once. Scientist X
2
has published
30 articles, one cited 10 times and all other ones exactly
twice. Clearly, scientist X
2
is the better one. This is ex-
pressed by their h-indices which are 1 for X
1
and 2 for
X
2
. Yet their A-indices are 10 for X
1
and 6 for X
2
. The
better scientist is ‘punished’ for having a higher h-index,
as the A-index involves a division by h. This is, however,
only a small problem which can easily be solved by
simply taking the sum, or, the square root of the sum.
Taking the square root has the advantage of leading to
indicator values which are not very high and of the same
dimension as the A-index. As this new index is calcu-
lated using a (square) root we refer to it as the R-index.
As a mathematical formula the R-index is defined as
1
.
h
j
j
Rcit
=
=
∑
(2)
Clearly, R=
.
A
h . In general one may write R(X,Y),
where X denotes a particular scientist and Y the year for
which the R-index has been calculated. As this is of no
importance in our investigations we omit the symbols X
and Y. It is clear that h
≤R as each cit
j
is at least equal to
h. In the special case where each cit
j
is exactly equal to
h, R = h. This nice result is another advantage of using
the square root of the sum, and not the sum itself.
1.5 Further relations between h, A, R and g
We have already observed that h≤g, h≤A and that R =
.
A
h . Now we show one less obvious relation between
A and g, and hence between h, A, R and g.
Proposition 1. The following inequalities always
hold:
A
≥g≥h. (3)
Proof. The last inequality is already known. Now
A
=
1
h
j
j
cit
h
=
∑
≥
1
.
g
j
j
cit
g
=
∑
This inequality holds because the citations (cit
j
) are
ranked in decreasing order, hence the average number of
citations of the first m articles is a decreasing function of
m. As the g-index satisfies the relation
1
g
j
j
cit
=
∑
≥
2
g
or
1
g
j
j
cit
g
=
∑
≥g.
This proves the first inequality in line (3).
The following corollary, involving the four indices
under study follows immediately.
Corollary.
R = .
A
h ≥ .
g
h ≥ h . (4)
In practice the
R-index is correlated to the h-index
(see further) but, especially for individual scientists,
does add another view on scientist’s achievements.
1.6 Relations between h, A, R and g in the power law
model
In this section we show how the four indices: h, A, R and
g are related in the power law model. The power law
model
[27]
assumes that the number of sources producing
x items, e.g. authors’ articles receiving citations, is given
by the function
F:
:[1, [ ]0, ]: .
C
FCx
x
α
+∞ → → (5)
In eq. (5) C is a strictly positive constant, and
α
> 1.
Equivalently
[27]
, the corresponding rank-frequency func-
tion (number of citations received by the article ranked
r) is given by the function G:
]][ [
:0, 1, : ()
B
GT rGr
r
β
→+∞ → = (6)
with B,
β
> 0. The relation between the parameters α and
β
is
1
.
1
β
α
=
−
(7)
In the power law model the four Hirsch-type indices are
defined as follows:
h is the unique solution of r = G(r),
g is the unique solution of
2
0
() ,
r
rGsds=
∫
0
1
()
h
A
Grdr
h
=
∫
and
0
() .
h
RGrdr=
∫
Note that we do not claim that actual sources follow a
power law: we just apply this model as a first approxi-
mation of an observed frequency distribution. Assuming
further that
α
> 2, we prove the following proposition.
Proposition 2. Assuming a power law model as
858 JIN BiHui et al. Chinese Science Bulletin | March 2007 | vol. 52 | no. 6 | 855-863
described above with
α
> 2 or equivalently 0 <
β
< 1, we
have
1
2
A
h
α
α
−
⎛⎞
=
⎜⎟
−
⎝⎠
and
1
,
2
Rh
α
α
−
=
−
(8)
1/
1
2
A
g
α
α
α
−
⎛⎞
=
⎜⎟
−
⎝⎠
and
1/2
1
.
2
Rgh
α
α
α
−
⎛⎞
=
⎜⎟
−
⎝⎠
(9)
Proof. A is defined as the average number of cita-
tions received by publications belonging to the Hirsch
core. Hence
1
0
11
.
1
h
Bh
AdrB
hh
r
β
β
β
−
==
−
∫
As
1/( 1)
hB
β
+
= (by Theorem C)
[3]
, and by eq. (7) this
result implies that
1
11
11 11
.
122
h
A
Bh h
hh
β
ββ
αα
βαα
−
++−
−−
== =
−−−
This proves the first equality of line (8). It is further
shown by Egghe
[16]
that
1
1
.
2
g
h
α
α
α
α
−
−
⎛⎞
=
⎜⎟
−
⎝⎠
(10)
Eliminating h from eqs. (8) and (10) yields the first
equality of line (9). The corresponding relations for R
follow then easily from those for A.
Remark 1. As
α
> 2 eqs. (8) and (9) imply that A
and R are always larger than h. Moreover, A > g, while R
is larger than the geometric average of h and g
this fol-
lows from the fact that for
α
> 2
1/
1
2
α
α
α
−
⎛⎞
⎜⎟
−
⎝⎠
>1 . Note
that the power law model yields the same inequalities as
in the discrete case.
Remark 2. Eq. (8) or eq. (9) does not prove that h
and A, or h and R are linearly related. The reason is that
in the power law model
1/
hT
α
= , where T is the total
number of sources (here the total number of publica-
tions). Hence the factor
1
2
α
α
−
−
cannot be considered as a
constant.
Finally we prove a very remarkable relation, charac-
terizing the h-index in the power law model.
Characterization Theorem. Assuming a power
law model as described above with
α
> 2 and denoting
by
μ
the average production (here: total number of cita-
tions divided by the total number of articles in the au-
thor’s publication list) the following relations hold:
A
h
μ
=
and .Rh
μ
= (11)
Proof. Eqs. (11) follow immediately from equations
(8) and the fact that, in the power law model,
1
2
α
μ
α
−
=
−
(as shown on page 115 of ref. [27]).
This result shows that in the power law model h is the
unique number N such that the average number of cita-
tions of the first N publications is equal to the global
average multiplied by N. Uniqueness follows from the
fact that the average number of citations of the first N
publications is a decreasing function of N, while
μ
N is
an increasing function of N.
1.7 The h-, A-, R- and g-indices are highly correlated
in practice
Notwithstanding remark 2 above, we think that in most
practical cases the four Hirsch-type indices h, A, R and g
are linearly correlated. Indeed, they more or less use the
same, highly restricted, data set, and this with similar
objectives. In order to investigate this we study in this
section a number of practical cases.
Using Egghe’s data for Price awardees
[16]
we calcu-
lated the A- and the R-index of each of these colleagues.
We did the same for publications in the WoS of a num-
ber of physics, chemistry and biology subfields (1996
―
2005) and of the contribution of four large national re-
search institutes in the WoS (2001
―2005). Data were
obtained from the China in World Science Series
[28
―
30]
.
Details of the calculations can be found in the Appendi-
ces. Table 1 shows the observed Pearson correlation
coefficients (CCs).
Table 1 Correlation coefficients between R and g
Data set CC (R vs. g) CC (R/h vs. g/h)
Price awardees 0.998 0.995
Chemistry subfields 0.999 0.998
Biology subfields 0.999 0.997
Physics subfields 0.999 0.998
CAS physics subfields 0.999 0.995
Max Planck physics subfields 0.999 0.997
CNRS physics subfields 0.998 0.995
RAS physics subfields 0.991 0.959
These data speak for themselves: there is no doubt
that the R-index and the g-index are highly correlated in
practice. The same observation holds for the ratios R/h
and g/h. A similar remark (not shown) holds for A and g,
but with slightly smaller correlations. We further ob-
serve that the CC between R and g is always higher than
⎛
⎜
⎝
⎞
⎟
⎠
JIN BiHui et al. Chinese Science Bulletin | March 2007 | vol. 52 | no. 6 | 855-863 859
ARTICLES SCIENTOMETRICS
that between R and h or g and h. The latter two are very
similar (see appendices for details).
1.8 A preliminary conclusion
It seems that the g-index and R-index are highly corre-
lated while the latter has a computational advantage. Yet,
as a stand-alone index R may be overly sensitive to one
article receiving an extremely high number of cita-
tions
[31]
. In the extreme case one may encounter a scien-
tist with an h-index of 1 and an R-index of 10 (any high
number). This observation similarly applies to the
g-index (in particular when fictitious articles with zero
citations are added
[16]
). For this reason we suggest using
the R-index in conjunction with h. Consequently we
propose, as a preliminary conclusion, the pair (h, R) as a
good indicator for research evaluation. For practical
evaluation purposes applying time windows, e.g. a
5-year window, seems advisable. Moreover, the ratio R/h
might be an interesting indicator in its own right.
2 An age-dependent indicator: The
AR-index
In order to overcome the problem that the h-index may
never decrease and that scientists may, so to speak, ‘rest
on their laurels’ we propose the following adaptation of
the R-index
[32]
.
2.1 Definition: the AR-index
If a
j
denotes the age of article j we define the
age-dependent R-index, denoted by AR, by the following
equation:
1
h
j
j
j
cit
AR
a
=
=
∑
. (12)
If there are several publications with exactly h citations
then we include the most recent ones in the h-core. This
means that we include those with the more favorable
(cit/a) ratio.
Advantages of the AR-index are clear. Besides taking
the actual number of citations into account, it makes also
use of the age of the publications. In this way, the
h-index is complemented by an index that can actually
decrease. Such behavior is, in our opinion, a necessary
condition for a good research evaluation indicator. We
note that, moreover, the AR-index is based on the
h-index as it makes use of the h-core. For the AR-index
the inequality h
≤AR is not necessarily true anymore,
contrary to the corresponding relation involving the
R-index (see eq. (4)). We note that calculation of the
AR-index only requires the age of the publications in the
h-core, besides the data necessary for the calculation of
the h-index. This does not make the calculation of the
AR-index more difficult than that of the h-index. Note
that for source-item relations where age has no meaning
this indicator just does not apply. This is somewhat
similar to the h-index, which also does not apply for all
possible source-item relations.
Favorable points concerning the R-index also apply
here. Hence we propose the pair (h, AR) as a good indi-
cator for research evaluation.
2.2 An example
We calculated the AR-index over several years for the
articles written by B.C. Brookes after 1971 (WoS publi-
cation and citation data on January 1, 2007). Recall that
B.C. Brookes was a Price awardee in 1989. He died in
1991. Results are shown in Table 2.
Table 2 Evolution of B.C. Brookes’ AR-index
Year R-index AR-index
2002 18.60 3.93
2003 18.81 3.89
2004 18.97 3.84
2005 19.13 3.79
2006 19.34 3.76
2007 19.54 3.73
Brookes’ h-index over the whole period (2002―
2007) stays fixed at h = 12 (hence here h > AR). Be-
tween 2002 and 2007 his R-index increased by 5% while
the AR-index decreased by about 5%. A year written in
the first column of Table 2 stands for January 1 of that
year. The (average) age of an article on January 1 of year
Y is (k-0.5) if the article is published during the year Y-k.
Indeed, if an article is published during the year Y-3 then
it is, on January 1 of the year Y at least two years, and at
most three years old. On average it is 2.5 years old
[33]
.
This is how we calculated the average age of an article
in order to obtain the AR-index. Figure 1 illustrates the
decrease of Brookes’ AR-index over the latest years.
Figure 1 Decrease of Brookes’ AR-index.
