Bilateral Co-authorship Indicators Based on Fractional Counting

2.1.1 Full counting for country C. When using full counting the production of country C (misusing the term production for the number of publications as given by a counting scheme) is set equal to the number of publications in which at least one author has at least one address in country C.

2.1.2 (Basic) fractional counting: if at least one of the authors of a publication has at least one address in country C then this publication contributes to the publication score of country C. Assuming that each author has exactly one address (or all addresses of this author are in the same country), then the contribution of country C in a publication with N authors is p_C = AU/N, where AU is the number of authors with all addresses in country C.

It may, however, happen that some authors have addresses in more than one country. Then p_C must be adapted. We proceed as follows: if AU authors with at least one address in country C contribute to this N-author publication, then the fractional contribution of country C to this publication is (1)pC=1N∑j=1AUajaj+bj,{p_C} = {1 \over N}\sum\nolimits_{j = 1}^{AU} {{{{a_j}} \over {{a_j} + {b_j}}}} , where a_j is the number of addresses of author j in country C, and b_j is the number of addresses of author j not situated in country C. It is clear that, allowing b_j = 0, formula (1) is always valid, whether or not author j has addresses outside country C or not.

If p_K represents the fractional contribution of a country K in a fixed article then Σ_K:countryp_K = 1.

Finally, the fractional production of country C in the set under investigation is the sum of all fractional scores of all publications involving country C.

We start from the idea that the most intense co-authorship pattern between two countries in one article occurs when the two countries contribute equally (the perfectly balanced case) and no other country contributes. We give this situation a score of 1. In the purely theoretical case that two countries never publish on their own, always publish co-authored publications, and this always in a perfectly balanced way, i.e. p_C1 = p_C2, (here and further on p-values are calculated as proposed in equation (1)) then their total co-authorship score is equal to their total number of publications. We denote the co-authorship score of two different countries C₁ and C₂ in one publication by cs(C₁,C₂).

Definition: co-authorship score of two different countries C₁ and C₂ in one publication.

The formula for cs is two times the harmonic mean of the fractional contributions of countries C₁ and C₂. We multiply the harmonic mean by two so that in the perfectly balanced case cs(C₁,C₂) receives a score of 1 (and not 0.5). This is just a practical agreement and not essential for comparing scores over time or for different countries. If p_C1 + p_C2 = 1, then cs (C₁, C₂) = 4p_C1p_C2.

Definition: co-authorship score, cs, of two different countries C₁ and C₂ for a given set of publications.

If PUB is the number of publications under consideration, then the co-authorship score cs (C₁,C₂) is defined as: (3)cs(C1,C2)=∑j=1PUBcsj(C1,C2)cs\left( {{C_1},{C_2}} \right) = \sum\nolimits_{j = 1}^{PUB} {c{s_j}\left( {{C_1},{C_2}} \right)}

We introduced the new cs-score without proper arguments. Next we show a list of good properties of cs. This list serves as argumentation for the introduction of this new cs-score. Indeed, we think that the properties shown in this list are necessary, or at least highly desirable for a proper cs-score. We first note that if a C₁–C₂ co-authored publication is added to the set under consideration then cs(C₁,C₂), equation (3), increases. So, besides an increase for the full counting score, also the fractional counting score increases.

The cs-value as defined in equation (2), i.e. for one publication, has the following properties.

P1) For all countries C₁, C₂ and for all publications: cs(C₁,C₂) = cs(C₂,C₁); cs is a symmetric measure. This symmetry property also holds for equation (3).

P2) For all countries C₁ and C₂ and for all publications: 0 ≤ cs(C₁,C₂) ≤ 1; cs is a non-negative measure, with upper bound equal to one.

P3) The co-authorship score cs(C₁,C₂) = 1 if and only if p_C1 = p_C2 = 0.5; this is the upper bound for the co-authorship score of two countries in one publication. This upper bound corresponds to the perfectly balanced case.

P4) If for countries C₁ and C₂: p_C1 ≤ p_C2, then for all countries C₃: cs(C₁,C₃) ≤ cs(C₂,C₃), which is a monotonicity property.

P5) The cs-value of countries C₁ and C₂ in one publication does not depend on how many (one, two or more) other countries and which other countries contribute to a joint publication. This is a form of anonymity property. Below we provide some comments on this property.

P6) Continuity

We can write cs(C₁,C₂) as 4pC1pC2pC1+pC2{{4{p_{{C_1}}}{p_{{C_2}}}} \over {{p_{{C_1}}} + {p_{{C_2}}}}} . Denoting the fractional contribution of other countries (i.e. not country C₁ or country C₂) by p_O, (O for “other”) we have that always p_C1 + p_C2 + p_O = 1. Hence, we can rewrite cs(C₁,C₂) as 4pC1(1−pC1−pO)1−pO{{4{p_{{C_1}}}\left( {1 - {p_{{C_1}}} - {p_O}} \right)} \over {1 - {p_O}}} . This shows that cs(C₁,C₂) can be written as a function of two independent variables, either p_C1 and p_C2 or p_C1 and p_O (and of course also as a function of p_C2 and p_O) where these two variables are always between zero and one and their sum is smaller than or equal to one. Considering these variables as real variables, we see that cs is a continuous function of two variables. Roughly speaking this means that a small change in any of these variables leads to a small change in cs(C₁,C₂).

P7) Next we show that the more uneven the contribution of the two countries, the smaller their cs-value. Conversely, when the two countries have an equal contribution, then their cs-value reaches a maximum, depending on p_O.

Assume that p_O (< 1) is fixed, then cs(C₁,C₂) increases for 0<pC1≤1−pO2{0 \lt p_{{C_1}}} \le {{1 - {p_O}} \over 2} and then decreases for 1−pO2< pC1<1−pO{{1 - {p_O}} \over 2} < {p_{{C_1}}} < 1 - {p_O} . Indeed: cs(C1,C2)=4pC1pC2pC1+pC2=4pC1(1−pC1−pO)1−pOcs\left( {{C_1},{C_2}} \right) = {{4{p_{{C_1}}}{p_{{C_2}}}} \over {{p_{{C_1}}} + {p_{{C_2}}}}} = {{4{p_{{C_1}}}\left( {1 - {p_{{C_1}}} - {p_O}} \right)} \over {1 - {p_O}}} . This is clearly a parabola in the variable p_C1 with top in the point with coordinates (1−pO2,1−pO)\left( {{{1 - {p_O}} \over 2},1 - {p_O}} \right) . This top is reached when the contribution of the two countries is equal. In the special case that p_O = 0, this top is situated in the point (0.5, 1).

P8) In this last property we show that when the relative contributions of the two countries are given, then their cs-value depends on p_O: the larger p_O the smaller their cs-value.

In the previous property we kept p_O ≠ 0 fixed. Now we keep P=pC1pC2≤1P = {{{p_{{C_1}}}} \over {{p_{{C_2}}}}} \le 1 fixed and show that the larger p_O, the smaller cs(C₁,C₂), which is again a property one may expect to hold for a proper co-authorship measure for two countries.

Proof: P=pC1pC2P = {{{p_{{C_1}}}} \over {{p_{{C_2}}}}} implies that P*p_C2 = p_C2 and hence cs(C1,C2)=4pC1pC2pC1+pC1=4P(pC2)2pC2(1+P)=4P*pC21+P{\rm{cs}}\left( {{{\rm{C}}_1},{{\rm{C}}_2}} \right) = {{4{p_{{C_1}}}{p_{{C_2}}}} \over {{p_{{C_1}}} + {p_{{C_1}}}}} = {{4P{{\left( {{p_{{C_2}}}} \right)}^2}} \over {{p_{{C_2}}}\left( {1 + P} \right)}} = {{4P*{p_{{C_2}}}} \over {1 + P}} . As p_C1 + p_C2 +p_O = 1 we have here: (1+P)* p_C2 + p_O = 1, or pC2=1−pO1+P{p_{{C_2}}} = {{1 - {p_O}} \over {1 + P}} . Consequently, cs(C1,C2)=4P(1−pO)(1+P)2{\rm{cs}}\left( {{{\rm{C}}_1},{{\rm{C}}_2}} \right) = {{4P\left( {1 - {p_O}} \right)} \over {{{\left( {1 + P} \right)}^2}}} . This clearly shows that if P is fixed, cs decreases in p_O.

We note that if P = 1 (C₁ and C₂ have an equal contribution) cs(C₁,C₂) = 1-p_O.

We propose the following indicator as a co-authorship intensity indicator, denoted as CI(C₁,C₂): (4)CI(C1,C2)=cs(C1,C2)FCCI(C1,C2)CI\left( {{C_1},{C_2}} \right) = {{cs\left( {{C_1},{C_2}} \right)} \over {FCCI\left( {{C_1},{C_2}} \right)}}

On the one hand, two countries may co-author often (FCCI is high), but this collaboration usually involves many other countries or is highly asymmetric (most authors belong to one country). Then the CI-index is small. When, on the other hand, scientists of two countries work together it is mostly without third party and in a balanced way. Then the intensity index is—relatively—high. We write “relatively” as it is obvious that in real situations CI will rarely be close to one.

These data are presented to illustrate numerical values resulting from the definition of this new indicator.

The last row of Table 1 illustrates the perfectly balanced case: the two countries contribute equally and no other country contributes, while the first row of Table 2 is an illustration of the case with a three-country collaborative publication, where the target two countries contribute equally (as are some other rows).

As we do not have a program yet to do the complete data gathering we chose a small real-world example, doing all data gathering and calculations by hand and with the help of an Excel file. Data were collected from the Web of Science (WoS) as available at the KU Leuven, Belgium. We included the Proceedings but not the Book citation indexes. A search was performed on August 2, 2020 for CU=(Netherland* OR Holland) and CU=Belgium, within the two subject categories (SC) Mathematics and Applied Mathematics. This search was performed for PY=2016–2017 and for PY=2018–2019. Our aim is to find the cs- and CI-values for the pair Belgium-the Netherlands over these fields.

Fractional counting was performed as in formula (1). The number of authors (N) determines the basic fractions (each author receives a fraction 1/N for their country). If an author has several addresses then this fraction is further subdivided according to the number of addresses. If, for example, an author has five addresses, two in the Netherlands, one in Belgium, one in the United States and one in Germany, then their contribution to the Netherlands is 2/(5N), to Belgium it is 1/(5N) and to other countries it is 2/(5N). We used addresses as given in the WoS even if it was clear that they referred to the same physical space: clearly, the author played different roles (for different organizations) and it was considered important to mention this in the byline.

Results are given below, but numbers have only a limited importance. This example is just a small feasibility study. It provided us some experience with the practical difficulties in calculating cs-values. We note that a few articles were classified as Mathematics and also as Mathematics Applied: these are included twice. Moreover some articles published in 2020, but with an EA (Early Access Date) in 2019 were retrieved by PY=2019. This is according to the new way how PY= works in the WoS. These articles too are included.

In these fields, Belgium produced slightly more than the Netherlands. Joint work is increasing and is higher in applied mathematics than in (pure) mathematics. Yet, in all cases co-authorship between the two countries is low.

The average number of authors per publication is—not surprisingly—higher in applied mathematics than in mathematics, and seems to increase (based on two periods only). Relative contribution by other countries seems to be on the rise too. Mathematics provides an example where the absolute number of co-authored articles increases, but the cs-value decreased. Yet, the general rule is that the more joint publications, the higher the co-authorship score. For the two fields the average cs-value per publication decreased over the two periods. Quite a lot of researchers have several addresses, often in different countries. As mathematics is a field where generally co-authorship is low, this shows already that data collection for fractional scores is not a sinecure.

Table 5 shows the calculation and resulting values for the co-authorship intensity of the two countries, in two fields and over two periods. For mathematics as well as for applied mathematics the CI-values decrease. As we have no experience with this new indicator we cannot discuss the meaning of the obtained values, yet the decreasing trend might not be surprising. Our indicators consist of two parts: a part directly depending on the bilateral relation between the two countries under study and a part related to the contribution of other countries. As international collaboration has increased over the years—and certainly within the European Union—a decrease of cs- and CI-values is to be expected.