A New Explanation of Some Leiden Ranking Graphs Using Exponential Functions
 Author: Egghe Leo
 Organization: Egghe Leo
 Publish: Journal of Information Science Theory and Practice Volume 1, Issue3, p6~11, 30 Sep 2013

ABSTRACT
A new explanation, using exponential functions, is given for the Sshaped functional relation between the mean citation score and the proportion of top 10% (and other percentages) publications for the 500 Leiden Ranking universities. With this new model we again obtain an explanation for the concave or convex relation between the proportion of top 100θ% publications, for different fractions of
θ .

KEYWORD
Leiden ranking , exponential function , mean citation rate , Sshaped

1. INTRODUCTION
For 500 universities (from 41 countries) from the Leiden Ranking 2011/2012 one observes in Waltman et al. (2012) the relation between the mean normalized citation score (MNCS) and the proportion of top 10% publications (PP_{top10%}).
Upon specifying a field, MNCS is the mean number of citations of the publications of a university in this field (normalized in several ways ？ see Waltman et al.). PP_{top10%} is the proportion (fraction) of the publications of a university in this field that, compared with other publications in this field, belong to the top 10% most frequently cited.
In Waltman et al. one finds an Sshaped relation between PP_{top10%} and MNCS: first convex then concave (see their Fig. 2). Allowing some other percentages, Waltman et al. find a convex relation between PP_{top10%} (as abscissa) and PP_{top5%} (as ordinate) and a concave relation between PP_{top10%} (as abscissa) and PP_{top20%} (as ordinate) ？ again see their Fig. 3.
In Egghe (2013) we explained all these regularities using the shifted Lotka function
where C > 0,
α >1,n ≥0, which was documented in Egghe and Rousseau (2012). Heref (n ) is the continuous version of the number of publications withn citations. Using (1) we studied the functional relation between the nonnormalized variant of MNCS, denoted MCS and PP_{top10%} . Puttingx =MCS andy =PP_{top10%} we proved in Egghe (2013) thatwhere
α is the exponent of Lotka in (1), where we also proved the Sshape, hereby explaining this empirical relationship in Waltman et al. (2012).For general fractions
θ we obtained in Egghe (2013)where
x =MCS andy = PP(θ )？PP_{top 100θ%} .From this model we proved, for any fractions
θ _{1},θ _{2}, the following functional relation between PP(θ _{1}) and PP (θ _{2}) :which is an explanation of the convex and concave graphs in Waltman et al.: concave if
θ _{2}>θ _{1} and convex ifθ _{2}<θ _{1}.In this paper the same problems are studied: explaining the Sshaped relationship between MCS and PP (
θ ) for anyθ and the convex or concave relationships between twoPP (θ _{1}) andPP (θ _{2}) as found in Waltman et al. Now, however, we do not use the shifted Lotka function (1) but the exponential function (a very classical function)where
C >0,α >1,n ≥0 where the functionf(n) has the same meaning as explained above. Withx = MCS andy =PP (θ ) we will prove (in the next section) thatwhich is a clearly different function when compared with (3). But also this regularity explains the one found (empirically) in Waltman et al. since (6) is also Sshaped.
Remarkably, in the third section, using (6) for two fractions
θ _{1} andθ _{2}, we will reprove (4); i.e., the same regularity between any twoPP (θ _{1}) andPP (θ _{2}) is found using exponential functions as when we used the shifted Lotka function (which are clearly different functions). But, at the end of the paper, we will also give two cases where (4) is not valid.The paper closes with a conclusion and open problems section.
2. EXPLANATION OF THE RELATION BETWEEN MCSAND PP(
θ )As indicated in the introduction we use the exponential function
denoting the continuous version of the number of publications with
n citations in a field whereC >0,α >1,n ≥0. Since the field is fixed, we have also thatC andα are fixed.For a university we use the exponential function
(
C’ > 0,α’ >1,n ≥0 ) denoting the continuous version of the number of publications of this university withn citations. Since we deal with several universities (e.g. 500 in the case of Waltman et al. (2012)) we have here thatC’ andα’ are variables.As noted by one referee, the fact that, per university, we use (8) with
C’ andα’ variables, does not necessarily lead to (7) for the entire field. Therefore, formula (7) should be considered as an assumption to be valid in practice:Denote by
T the total number of publications in the entire field and byT’ the total number of publications in a university in this field. We have, by definition off(n) (since
α >1) and similarlyDenote by
A the total number of citations in the entire field and byA’ the total number of citations in a university in this field. We have, by definition off(n) which is easily seen using partial integration, by the fact that
α >1 and the fact thatSimilarly we have
By definition of MCS, being the average number of citations per publication of a university, we have
, using (10) and (13).
We first determine
n _{0} defining the top 100θ % publications in the field (for any fractionθ ), by (7) :From (15) it follows that
and by (9) we have
which is a positive number because 0<
θ <1.Then the university proportion in these top 100
θ % of the papers in the field is, by (8)(by (16))
(by (14)) or the function (6). This is an increasing function (since 0<
θ <1) for which (x = MCS )and
The number
α is a fixed parameter (of the field). Fig. 1 is the graph of (18) for lnα =0.8 from which the Sshape is clear, and it is close to the Sshape obtained in Waltman et al. (2012). So this represents a new explanation of this regularity.3. EXPLANATION OF THE RELATION BETWEEN ANY TWO VALUES OF PP(
θ _{1}) AND PP(θ _{2})For any two fractions
θ _{1} andθ _{2} we have, by (18)Hence
from which it follows that for any two fractions
θ _{1} andθ _{2}hence
which is (4).
As already remarked in the introduction, this relation is the same as the one found in Egghe (2013) where the shifted Lotka function was used, a remarkable fact!
We have derived that (since 0<
θ _{1},θ _{2}< 1), ifθ _{2} >θ _{1} , then (by (24)),PP (θ _{2}) is a concave function ofPP (θ _{1}) and that, ifθ _{2} <θ _{1}, then (by (24)),PP (θ _{2}) is a convex function ofPP (θ _{1}) ？ see the graphs in Egghe (2013) which explain the corresponding graphs in Waltman et al. (2012).So (24) is valid when
f andφ are both shifted Lotka functions (proved in Egghe (2013)) and whenf andφ are both exponential functions (proved here). Now we present two cases where (24) is not valid.> Case I
We take
f to be a shifted Lotka function andφ to be an exponential function:(
C’ > 0,α’ >2,n ≥0 )(
C’ > 0,α’ >1,n ≥0 ) Following the method of the previous section we find, for every fractionθ From the method in this section we find, for any two fractions
θ _{1} andθ _{2},which is, clearly, not the function (24).
> Case II
We take
f to be an exponential function andθ to be a shifted Lotka function:(
C’ > 0,α >1,n ≥0 )(
C’ > 0,α’ >2,n ≥0 ) Following the methods of the previous section we find, for every fractionθ From the method in this section we find, for any two fraction
θ _{1} andθ _{2} ,which is, clearly, not the function (24).
4. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH
Experimental regularities in Waltman et al. (2012) are proved mathematically in this paper. Using the exponential function we proved the Sshaped functional relation between the mean citation rate and the proportion of top 100
θ % publications. We obtained a different function than in Egghe (2013) where a shifted Lotka function was used, but we obtained an Sshape in both cases.With this new model we could reprove the function (obtained in Egghe (2013))
for the relation between two
PP (θ )values. It is very remarkable that we obtain exactly the same function as in Egghe (2013) although different starting functions were used (shifted Lotka in Egghe (2013) and exponential here). We also showed that (33) explains the corresponding empirical regularities in Waltman et al. (2012).The importance of this paper is that the assumption of a simple exponential function (5) leads to an explanation of several regularities in Waltman et al.
We state as an open problem: can the Sshape in Waltman et al. for the relation between the mean citation rate and the proportion of top 100
θ % publications be proved using other starting functions (other than the shifted Lotka function and other than the exponential function)?Also the following is an open problem: characterise the functions
F(n) andφ (n ) (previous section) for which (33) is valid. From Egghe (2013) and this paper, this class of functions must include the shifted Lotka function and the exponential function.

3. Waltman L., CaleroMedina C., Kosten J., Noyons E.C.M., Tijssen R. J.W., van Eck N.J., van Leeuwen T.N., van Raan A. F.J., Visser M.S., Wouters P. (2012) The Leiden Ranking 2011/2012: Data collection, indicators, and interpretation. [Journal of the American Society for Information Science and Technology] Vol.63 P.24192432

[Fig. 1] Graph of (18) for ln a = 0.8