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BRIEF COMMUNICATION
A Thermodynamic Explanation for the Glänzel–Schubert
Model for the h-Index
Gangan Prathap
National Institute of Science Communication and Information Resources, New Delhi 110 012, India.
E-mail: [email protected]
Recently, it was shown that among existing theoretical
models for the h-index, the Glänzel–Schubert model provides the best fit for a chosen example involving the
research evaluation of universities. In this brief communication, we propose a thermodynamic explanation for the
success of the Glänzel–Schubert model of the h-index.
Recently, Ye (2010) showed that from the three main contenders among existing theoretical models for the h-index—
namely, Hirsch’s original approach, the Egghe–Rousseau
model, and the Glänzel–Schubert model—at the level of
universities, the Glänzel–Schubert model fits best. Ye also
provided a unified theoretical explanation for these three
models. In this brief note, we propose a simple thermodynamic explanation for the success of the Glänzel–Schubert
model.
The h-index (Hirsch 2005) and many of its variants purport
to be able to measure research performance as a composite of
a quality measure (usually taken as impact i expressed as the
ratio of citations C to papers published P) and the quantity
measure (i.e., P itself, or Q). The Glänzel–Schubert model
h ∼ P 1/3 (C/P)2/3
(1)
h ∼ C2/3 /P 1/3 = (C2 /P)1/3
(2)
can be rewritten as
The composite term C2 /P combines both size or quantity (C) and quality (C/P) of scientific effort and can be
interpreted as an energy-like term X = iC. A thermodynamic explanation or justification then can be constructed
proceeding from this identification. Consider that a person
Received November 11, 2010; revised January 25, 2011; accepted January
25, 2011
© 2011 ASIS&T • Published online 14 March 2011 in Wiley Online Library
(wileyonlinelibrary.com). DOI: 10.1002/asi.21508
has published a single paper in a publication window (i.e.,
period over which papers are published). If the total number
of papers is represented by P, then in this case, P = 1.Assume
this paper, over a fixed citation window (i.e., period over
which citations are counted) has collected c citations. The
total number of citations during this citation window is designated by C, and in this case, C = c. We now define the energy
e of the single paper as e = C2 /P = (c2 /1) = c2 . The basic or
elementary unit of effort or energy is defined as the energy
that a single paper gathering a single citation (measured over
the citation window) possesses. The term e = c2 then can be
considered to have c2 times the basic or elementary unit of
effort or energy. It is the knowledge energy in a paper as
measured over the citation window. We now present a few
structured exercises to expand on this idea.
Exercise 1
Let us now assume that a second person has two papers,
which have collected exactly c citations each. The total energy
(i.e., sum of the energies) is E = c2 + c2 = 2c2 . But what then
are the energies of the sum of the papers? We designate this
by X for reasons which will become clear as we proceed. If
we invoke the C2 /P definition, this leads to C = 2c, P = 2,
and X = 2c2 . Thus, in this thermodynamically “perfect” case,
E = X. The h-index for this case is 1 if c = 1, and 2 if c = 2.
If c > 3, the h-index remains at 2!
Exercise 2
We now introduce a thermodynamically “imperfect” case
where the two papers (P = 2) have collected c1 and c2 citations, respectively. The individual energies of the papers are
then e1 = (c1 )2 and e2 = (c2 )2 . The total energy (defined as
the sum of the individual energies) is E = (c1 )2 + (c2 )2 . Let
us now look at what we have termed the total exergy X. To
compute this, we need the sum of the citations, C = c1 + c2
JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY, 62(5):992–994, 2011
and the sum of the papers (P = 2). The “energy” of the sums
(i.e., the total exergy) is X = (c1 + c2 )2 /2. We see at once that
except in the case where c1 = c2 , the total energy will always
be greater than the total exergy. Let us define this discrepancy as the entropy, which we designate by the term S. Then,
S = E − X.
Exercise 3
We now consider yet another “perfect” case. An author has
n papers during the publication window, all of which have
collected c citations each during the citation window. Then,
the total number of papers published during the publication
window is P = n. The unit of energy of each paper is e = c2 .
The total energy (i.e., sum of the energies) is simply E = nc2 .
The “energy” of the sum of the papers (which we have agreed
to designate by the term X), called exergy, leads through the
C2 /P definition, with C = nc and P = n to X = nc2 . Again,
this is a thermodynamically “perfect” case with E = X, as
the entropy S is zero. This is not unexpected. Knowing the
number of citations of any one paper, that of all other papers
is predictable, being exactly the same. However, it is interesting to observe what happens to the evaluation of the h-index
and the p-index (Prathap 2010). Whatever the value of n, we
find that the p-index, defined as p = X1/3 yields p = n1/3 c2/3 .
This is a formula that is first hinted at in Glänzel (2006),
and developed further in Glänzel (2008) and in Schubert and
Glänzel (2007). However, the h-index depends on whether
n is greater than or less than c. For all n ≤ c, h = n, and for
all n ≥ c, h = c. The h-index is initially limited by the number of papers published, and when n crosses this threshold
(n = h = c), h remains at the limiting value h = c. The p-index
does not suffer from such a limitation. Note the significance
of the curious conjunction at n = h = p = c. Here, we have
the thermodynamically “perfect” portfolio of p papers published having p citations per paper so that E = X = p3 . The
total energy is evenly distributed among all the states; that is,
p papers having p2 units of bibliometric energy each. This
also leads us to a geometric analogy of the whole process. Let
i be the impact given by C/P. The exergy is defined as C2 /P
(= iC = i2 P). Think of exergy as being represented by the
volume of a rectangular prism of sides i, i, and P. For a given
exergy X, p is the dimension of a cube that has zero entropy
dispersion of citations over papers, and has the same volume
(energy) as the rectangular prism. Note that for a single paper,
the exergy is the same as the energy (i.e., x = e).
Exercise 4
It is interesting now to consider a more general, thermodynamically “perfect” case where
X = Cα /P β
the publication window is P = n. The unit of energy of each
paper is e = cα . The total energy (i.e., sum of the energies)
is simply E = P cα . The “energy” of the sum of the papers
(which we have agreed to designate by the term X), called
exergy, leads through the Cα /P β definition, with C = nc and
P = n to X = P α cα /P β . As this is a thermodynamically “perfect” case with E = X, the entropy S is zero, leading to the
equality
Pcα = P α cα /P β ,
which leads to
α = β + 1.
The simplest thermodynamically consistent relationship
that can be proposed (Occam’s razor) is the case where α = 2
and β = 1. This, indeed, is the Glänzel–Schubert model for
the h-index.
This thermodynamic interpretation adds to the perspective
provided by the unification of the three theoretical models
for the study of h-indices (Ye, 2010). The Glänzel–Schubert
model, being based on the composite term C2 /P, which combines both size or quantity (C) and quality (C/P) of scientific
effort, leads easily to the energy-like term X = iC. This makes
it possible to complete a trinity of thermodynamic-like terms:
energy, exergy, and entropy. Unlike the classical statistical
thermodynamics entropy term which needs an additional
temperature term (i.e., requires the definition of a new entity
called temperature) to bring it into units of energy, here
we have a seemingly simple relationship that indicates that
entropy S = E − X, directly has the units of energy, and can
be interpreted as the unusable energy due to disorder in the
system. When there is perfect order (i.e., what we called
the thermodynamically “perfect” cases in Exercises 1, 3,
and 4 earlier), entropy becomes zero. E can be thought of
as the total internal energy of the system (disorder and all)
while X is the usable energy (akin to the free energy) that is
available to do work externally. Neither the Hirsch-type estimation (based only on C) nor the Egghe–Rousseau estimation
(based only on P) can lead to such interpretations.
There is promise that by introducing a thermodynamic
analogy to bibliometric research assessment, it is possible
to come up with more meaningful performance indicators.
Energy, exergy, and entropy terms are scalar quantities that
can be displayed as time series (variation over time) or in
event terms (variation as papers are published) and also in the
form of phase diagrams (energy–exergy–entropy representations). Exergy (related to h3 ) is possibly the most meaningful
single number scalar indicator of an entity’s performance
while entropy then becomes a measure of the unevenness
(disorder) of the research portfolio. The Glänzel–Schubert
model links exergy X to h, and this is why it appears to be
most successful (Ye, 2010).
(3)
References
An author has n papers during the publication window, all
of which have collected c citations each during the citation
window. Then, the total number of papers published during
Glänzel, W. (2006). On the h-index—A mathematical approach to a new
measure of publication activity and citation impact. Scientometrics, 67,
315–321.
JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY—May 2011
DOI: 10.1002/asi
993
Glänzel, W. (2008). On some new bibliometric applications of statistics related to the h-index. (OR 0801). Leuven, Belgium: Katholieke
Universiteit Leuven.
Hirsch, J.E. (2005). An index to quantify an individual’s scientific research
output. Proceedings of the National Academy of Sciences, USA, 102,
16569–16572.
Prathap, G. (2010). The 100 most prolific economists using the p-index.
Scientometrics, 84, 167–172.
994
Schubert, A., & Glänzel, W. (2007). A systematic analysis of Hirsch-type
indices for journals. Journal of Informetrics, 1, 179–184.
Ye, F.Y. (2011). A unification of three models for the h-Index. Journal
of American Society for Information Science and Technology, 62(1),
205–207.
JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY—May 2011
DOI: 10.1002/asi
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