A general framework for describing diversity within systems and similarity between systems with applications in informetrics

Abstract

Building on the ideas of Stirling (J R Soc Interface, 4(15), 707–719, 2007) and Rafols and Meyer (Scientometrics, 82(2), 263–287, 2010), we borrow models of genetic distance based on gene diversity and propose a general conceptual framework to investigate the diversity within and among systems and the similarity between systems. This framework can be used to reveal the relationship of systems weighted by the similarity of the corresponding categories. Application of the framework to scientometrics is explored to evaluate the balance of national disciplinary structures, and the homogeneity of disciplinary structures between countries.

Appendices

Appendix A

In this appendix we provide further necessary mathematical explanations.

Proposition 1

Using the notation introduced in the main text we show that: $\sum\nolimits_{i,j = 1}^{N} {p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} D_{ij} } = 1 - \sum\nolimits_{i,j = 1}^{N} {p_{{{\text{X}}{}_{i}}} p_{{{\text{X}}_{j} }} S_{ij} }$

Proof

$\begin{aligned} \mathop \sum\nolimits_{i,j = 1}^{N} {p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} D_{ij} } & = \mathop \sum\nolimits_{i,j = 1}^{N} {p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} (1 - S_{ij} )} = \mathop \sum\nolimits_{i = 1}^{N} {p_{{{\text{X}}_{i} }} } \left( {\sum\nolimits_{j = 1}^{N} {p_{{{\text{X}}_{j} }} } - \sum\nolimits_{j = 1}^{N} {p_{{{\text{X}}_{j} }} S_{ij} } } \right) \\ & = \mathop \sum \nolimits_{i = 1}^{N} p_{{{\text{X}}_{i} }} \left( {1 - \mathop \sum \nolimits_{j = 1}^{N} p_{{{\text{X}}_{i} }} S_{ij} } \right) = \mathop \sum \nolimits_{i = 1}^{N} p_{{{\text{X}}_{i} }} - \sum\nolimits_{i,j = 1}^{N} {p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} S_{ij} } = 1 - \sum\nolimits_{i,j = 1}^{N} {p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} S_{ij} } \\ \end{aligned}$

Proposition 2

Using the notation introduced in the main text we show that $L_{\text{X}} = 1 - \sum\nolimits_{i = 1}^{N} {p_{{{\text{X}}_{i} }}^{2} }$ can be derived from $SR_{\text{X}} = \sum\nolimits_{\begin{subarray}{l} i,j = 1 \\ i \ne j \end{subarray} }^{N} {p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} D_{ij} }$ by taking all D _ij  = 1 (i ≠ j).

Proof

As $\sum\nolimits_{i = 1}^{N} {p_{{{\text{X}}_{i} }} } = 1$ we have: $1 = \left( {\sum\nolimits_{i = 1}^{N} {p_{{{\text{X}}_{i} }} } } \right)^{2} = \sum\nolimits_{i = 1}^{N} {p_{{{\text{X}}_{i} }}^{2} } + \sum\nolimits_{i \ne j} {p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} }$. Hence $1 - \sum\nolimits_{i = 1}^{N} {p_{{{\text{X}}_{i} }}^{2} } = \sum\nolimits_{i \ne j} {p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} } .$ The left-hand side of this equality is L _X, the Simpson diversity measure, while the right-hand side is SR _X in the case that for i ≠ j all D _ij  = 1.

We would like to show that the expression used in Eq. 11, namely $\frac{{\mathop \sum \nolimits_{i,j = 1}^{N} p_{{{\text{X}}_{i} }} p_{{{\text{Y}}_{j} }} S_{ij} }}{{\sqrt {\left( {\mathop \sum \nolimits_{i,j = 1}^{N} p_{{{\text{X}}_{i} }} p_{{{\text{X}}_{j} }} S_{ij} } \right).\left( {\mathop \sum \nolimits_{i,j = 1}^{N} p_{{{\text{Y}}_{i} }} p_{{{\text{Y}}_{j} }} S_{ij} } \right)} }}$ is always smaller than or equal to 1. For this we recall the notion of a (real) inner product on a real vector space.

Definition

If V is a real vector space then $\langle . , .\rangle :V \times V \to \mathbb{R}$ is a real inner product if

• (1) $\forall x \in V: \langle x, x\rangle \ge 0$ with equality only if x = 0

• (2) $\forall x,y \in V: \langle x, y\rangle = y, x$

• (3) $\forall x,y,z \in V\,{\text{and}}\,c \in \mathbb{R}: \langle x + y, z\rangle = \langle x, z\rangle + \langle y,z \rangle\,{\text{and}}\,\langle cx, y\rangle = c \langle x, y \rangle$

If x, y belong to $\mathbb{R}^{n}$ then $x , y = \mathop \sum \nolimits_{i = 1}^{n} x_{i.} y_{i}$, where (x _i ) _i and (y _i ) _i are the components of x and y, is the standard inner product in $\mathbb{R}^{n}$. In an inner product space (a vector space equipped with an inner product) one can define a norm, denoted as $||\,.\,||,$ by $||x|| = \sqrt {\langle x , x} \rangle$. In such spaces the Cauchy–Schwarz–Bunyakowski (C–S–B) inequality is valid: $\left| {|\langle x,y\rangle |} \right| \le || x||.||y||.$ Rewriting C–S–B using the standard inner product in $\mathbb{R}^{n}$, yields: $\frac{{\mathop \sum \nolimits_{i = 1}^{n} x_{i} y_{i} }}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{n} x_{i.}^{2} } \sqrt {\mathop \sum \nolimits_{i = 1}^{n} y_{i.}^{2} } . }} \le 1$ if all components of x and y are non-negative (and x and y are not both equal to the zero vector). This inequality just shows that the cosine measure always gives values between −1 and +1; and in the case of vectors with non-negative components yields values between 0 and 1.

It can be shown that if M is an N × N symmetric, positive definite matrix, then the expression $\langle x |y\rangle = \sum\nolimits_{i = 1}^{n} {\sum\nolimits_{j = 1}^{n} {x_{i} m_{ij} y_{j} } }$ defines an inner product in $\mathbb{R}^{n}$ (of course in general different from the standard one). For symmetric matrices being positive definite is equivalent to having only strictly positive eigenvalues. The matrix S expressing the similarity between SCs is symmetric, and has moreover the following two properties: its diagonal contains only the value 1; and the values outside the diagonal are all numbers between 0 (included) and 1 (not included). Is such a matrix positive definite? Unfortunately the answer is no. Already in three dimensions it is possible to give counterexamples. The following matrix is symmetric and satisfies the extra requirements:

$$ M_{\text{H}} = \left( {\begin{array}{*{20}c} 1 & {0.8} & {0.9} \\ {0.8} & 1 & 0 \\ {0.9} & 0 & 1 \\ \end{array} } \right) $$

Yet, the eigenvalues of M _H are: 2.20416; 1 and −0.204159. This means that it is not possible to use an arbitrary matrix expressing the similarity between categories to form an inner product. Fortunately, the specific S-matrix we used IS positive definite (using Matlab we checked that all its eigenvalues are strictly positive). Hence we may define an inner product satisfying the Cauchy–Schwarz–Bunyakowski inequality:

$$ \left| {\frac{{\mathop \sum \nolimits_{i,j = 1}^{N} x_{i} y_{j} S_{ij} }}{{\sqrt {\left( {\mathop \sum \nolimits_{i,j = 1}^{N} x_{i} x_{j} S_{ij} } \right).\left( {\sum\nolimits_{i,j = 1}^{N} {y_{i} y_{j} S_{ij} } } \right)} }}} \right| \le 1 $$

This implies that values of the weighted cosine measure, as used in this contribution, are always situated between 0 and 1.

Next we would like to show that standard cosine values are never larger than the corresponding values for the weighted cosine measure. This turns out to be true for all cases that occurred here, so that we can indeed say that the standard cosine measure underestimates the similarity between countries. Yet we are not able to show this mathematically (for this particular S). We leave this as an open problem.

Appendix B: closer look at each of the six profiles separately

To further explain the map of the six profiles, the proportions of 32 countries/regions in 14 disciplines and 3 major fields are shown in Table 12, 13, 14, and 15. Top disciplines of each country have been put in bold italics.

Table 12 The proportions of countries/regions in Profiles 1–3 in 14 disciplines and 3 major fields (in %)

Table 13 Proportions of countries in Profile 4 in 14 disciplines and 3 major fields(in %)

Table 14 The proportions of countries in Profile 5 in 14 disciplines and 3 major fields (in %)

Table 15 Proportions of countries in Profile 6 in 14 disciplines and 3 major fields (in %)

For Profile 2, which includes China, Singapore and Taiwan, the four Top disciplines of the three countries/regions are the same, but the ranks of the four Top disciplines are not the same. Actually, among these three China’s disciplinary structure is the most similar to that of Russia (Profile 1). China has Materials Sciences and Chemistry both much more prioritized than the other two. Materials Sciences is the dominant discipline of Singapore, and Computer Science ranks second. The disciplinary structure of Taiwan is very similar to Singapore’s, the only difference being Computer Sciences’ rank as its number one Top discipline.

The third profile group, including India, Iran and South Korea, exhibits more evenness than Profiles 1 and 2. Iran has Chemistry as the number one Top discipline, but the proportions of Computer Science, Materials Sciences and Biomedical Sciences are almost equal. India has Chemistry as the number one Top discipline, but with more focus on Biomedical Sciences. South Korea pays the most attention to Materials Sciences and Biomedical Sciences. Clinical Sciences is the fourth Top discipline of this country.

Profile 4 is a cluster including Japan, the Czech Republic, Spain, France, Poland, Mexico and Portugal, which can be divided into two parts according to the 3 major fields (see Table 13). One part, Sect. 1, consists of Japan, the Czech Republic, Spain and France. They pay more attention to the Life Sciences with a mean proportion of Life Sciences in the four countries at 47 % ± 2.4, and a mean proportion of Physical Sciences in the four countries at 39 % ± 1.2. The other section of Profile 4 is Poland, Mexico and Portugal, with a mean proportion of Life Sciences in the four countries at 38 % ± 1.4, and a mean proportion of Physical Sciences in the four countries at 43 % ± 2.6. So the disciplinary structure of the second part, Sect. 2, is more balanced than Sect. 1. The countries in Profile 4 all have Biomedical Sciences as their prime discipline, but the countries from Sect. 2 are more focused on Chemistry than those belonging to Sect. 1.

As mentioned earlier we notice that South Korea has the most balanced disciplinary structure among the Asian countries in our study. Japan’s too belongs to profile 4. Its disciplinary structure is more like Spain’s or France’s because they all have Clinical Sciences as their second Top discipline.

The 11 countries constituting Profile 5 are displayed in Table 12. For each country, the proportion of Biomedical Sciences and Clinical Sciences is even larger than 10 % and the proportion of Biomedical Sciences is larger than 20 %. Profile 5 has more polarized countries than Profile 4, with Biomedical Sciences and Clinical Sciences as dominant disciplines.

One notable feature of Profile 6 is its inclusion of Brazil. Brazil’s growing economy places it within Profile 6. It is similar to the USA, the UK, Denmark, Australia, Norway and the Netherlands, where Biomedical Sciences and Clinical Sciences are commonly the focus of activity, a typical characteristic of Profile 6. Yet, Brazil is still located in the margin of the MDS map. Agriculture is the fourth dominant discipline of Brazil, which is different from the other countries in Profile 6.

Within this profile the UK is very similar to the USA. Along with Brazil, Norway is another country showing different characteristics from the other countries in Profile 6 as ecology is its third priority discipline.

The common features of Profiles 5 and 6 can be found when comparing Tables 14 and 15. These two profiles have Life Sciences as their most dominant major fields, while their main difference lies in the fact that the disciplinary structure of Profile 5 is more balanced than Profile 6.