A New Explanation of Some Leiden Ranking Graphs Using Exponential Functions

A new explanation, using exponential functions, is given for the S-shaped 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.


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 (PPtop10%).
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.).PPtop10% 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 S-shaped relation between PPtop10% and MNCS: first convex then concave (see their Fig. 2).Allowing some other percentages, Waltman et al. find a convex relation between PPtop10% (as abscissa) and PPtop5% (as ordinate) and a concave relation between PPtop10% (as abscissa) and PPtop20% (as ordinate) -again see their Fig. 3.
In Egghe (2013) we explained all these regularities using the shifted Lotka function where  > 0,  >1,  ≥0, which was documented in Egghe and Rousseau (2012).Here () is the continuous version of the number of publications with  citations.Using (1) we studied the functional relation between the non-normalized variant of MNCS, denoted MCS and PPtop10% .Putting  =  and  = PPtop10% we proved in Egghe (2013) that where  is the exponent of Lotka in (1), where we also proved the S-shape, hereby explaining this empirical relationship in Waltman et al. (2012).
From this model we proved, for any fractions  1 ,  2 , the following functional relation between  ( 1 ) and  ( 2 ) : which is an explanation of the convex and concave graphs in Waltman et al.: In this paper the same problems are studied: explaining the S-shaped relationship between MCS and  () for any  and the convex or concave relationships between two  ( 1 ) and  ( 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  > 0,  >1,  ≥0 where the function () has the same meaning as explained above.With  =  and  =  () we will prove (in the next section) that which 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  1 and  2 , we will reprove (4); i.e., the same regularity between any two  ( 1 ) and  ( 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.

EXPLANATION OF THE RELATION BETWEEN MCS AND PP(𝝷)
As indicated in the introduction we use the exponential function denoting the continuous version of the number of publications with  citations in a field where  > 0,  >1,  ≥0.Since the field is fixed, we have also that  and  are fixed.
For a university we use the exponential function (＇> 0, ＇>1,  ≥0 ) denoting the continuous version of the number of publications of this university with citations.Since we deal with several universities (e.g.500 in the case of Waltman et al. (2012)) we have here that ＇and  ＇are variables.
As noted by one referee, the fact that, per university, we use (8) with ＇and ＇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  the total number of publications in the entire field and by ＇the total number of publications in a university in this field.We have, by definition of , using ( 10) and ( 13).
We first determine  0 defining the top 100% publications in the field (for any fraction ), by (7) : From ( 15) it follows that  - 0 =  and by ( 9) we have which is a positive number because 0 <  < 1.
Then the university proportion in these top 100% of the papers in the field is, by ( 8) (by ( 16)) (by ( 14)) or the function (6).This is an increasing function (since 0 <  < 1) for which and The number  is a fixed parameter (of the field).Fig. 1 is the graph of (18) for ln  = 0.8 from which the Sshape is clear, and it is close to the S-shape obtained in Waltman et al. (2012)

EXPLANATION OF THE RELATION BETWEEN ANY TWO VALUES OF PP(𝝷 1 ) AND PP(𝝷 2 )
For any two fractions  1 and  2 we have, by ( 18 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!
So ( 24) is valid when  and  are both shifted Lotka functions (proved in Egghe (2013)) and when  and  are both exponential functions (proved here).Now we present two cases where (24) is not valid.

Case I
We take  to be a shifted Lotka function and  to be an exponential function: by  the total number of citations in the entire field and by  ＇the total number of citations in a university in this field.We have, by definition of () seen using partial integration, by the fact that  >1 and the fact that