A citation-based indicator for interdisciplinarity has been missing hitherto among the set of available journal indicators. In this study, we investigate network indicators  1 Among the various journal indicators based on citations, such as impact factors, the immediacy index, cited half-life, etc., a specific indicator of interdisciplinarity has hitherto been lacking (Kajikawa et al., 2009; Porter et al., 2006 and Wagner et al., 2009; Zitt, 2005) . Journals obviously vary in terms of the range of disciplines they cover. In their early studies, Francis Narin and his colleagues already noted that Science and Nature did not fit into their hierarchical clustering scheme (Narin et al., 1972; Carpenter & Narin, 1973) . At lower levels of the journal hierarchy, however, journals may also fulfill inter-or transdisciplinary functions. For example, Limnology & Oceanology has been considered by Leydesdorff (1986) as a journal that overarched the domains of freshwater and marine biology, and therefore had a higher impact factor. Given the matrix of aggregated journal-journal citations as derived from the Journal Citation Reports (JCR) of the (Social) Science Citation Index, a clustering algorithm usually aims to partition the database in terms of similarities in the distributions-in this case, distributions of citations. Some journals reach across boundaries because they relate different subdisciplines into a single (disciplinary) framework. For example, the Journal of the American Chemical Society (JACS) accepts contributions to organic, inorganic, physical, analytical chemistry, etc. Other journals combine intellectual contributions based on methods or instruments used in different disciplines. Thus, a network is woven in which relatively distinguishable clusters can be considered as representations of specialties, while hierarchies across specialties can be shaped in different directions. Note 2 that these hierarchies may not only operate on individual journals, but on setsrepresenting specialties-in which one can construct a center (or "centroid"). A generalized measure of interdisciplinarity at the journal level therefore is far from obvious. Furthermore, interdisciplinarity may be a transient phenomenon. As a new specialty emerges, it may draw heavily on its mother disciplines/specialties, but as it matures a set of potentially new journals can be expected to cite one another increasingly, and thus to develop a type of closure that is typical of "disciplinarity" (Van den Besselaar & Leydesdorff, 1996) . Interdisciplinarity, however, may mean something different at the top of the journal hierarchy (as in the case of Science and Nature) than at the bottom, where one has to draw on different bodies of knowledge for the sake of the application (e.g., in engineering). Similarly, in the clinic one may be more inclined to integrate knowledge from different specialties at the bedside than a laboratory where the focus is on specialization and refinement. In the bibliometric tradition, therefore, "interdisciplinarity" has remained a difficult issue despite its high policy-relevance (Laudel & Origgi, 2006; Wagner et al., 2009 ). More recent additions to the database (that is, the JCR) such as eigenfactor and article influence scores (Bergstrom, 2007; Rosvall & Bergstrom, 2008) focus heavily on weighing journals in terms of their hierarchical status (cf. Bollen et al., 2006) . Most of these new measures rank journals like the PageRank in Google, using information from the entire network of journal-journal citation relations (Page et al., 1998; cf. Pinski & Narin, 1976) 3 However, none of these new indicators hitherto focused specifically on interdisciplinarity with perhaps the exception of studies of betweenness centrality (Bollen et al., 2009a and b; Leydesdorff, 2007 and 2009 ). Among the network indicators, betweenness centrality seems an obvious candidate for the measurement of interdisciplinarity (Freeman, 1977 (Freeman, , 1978. One of us experimented with betweenness centrality as an indicator of interdisciplinarity in aggregated journaljournal citation networks (Leydesdorff, 2007) . Large journals (e.g., Nature and Science), however, may show high betweenness centrality because of their degree centrality: a star in a network can be expected to contribute to a relatively large percentage of the possible pathways among other nodes (Bollen et al., 2009a, at p. 6). This can be corrected by normalizing the vectors (rows or columns), for example, by using the cosine measure (Salton & McGill, 1983; Ahlgren et al., 2003; Bonacich, personal communication, 22 May 2006). In local environments, betweenness centrality could then be shown to perform outstandingly as an indicator of "interdisciplinarity." However, the usefulness of this indicator at the level of the set has not yet been tested. Using rotated factor analysis, Bollen et al. (2009b, at pp. 4 ff.) found betweenness centrality positioned near the origin of a two-factor solution; this suggests that betweenness centrality might form a separate (third) dimension in their array of 39 possible journal indicators. The occasion for returning to the research question of a journal indicator for "interdiscipinarity" was provided by the new interest in "interdisciplinarity" in bibliometrics (Laudel & Origgi, 2006; Wagner et al., 2009) and the availability of 4 another potential measure: diversity as defined by Stirling (2007; cf. Rao, 1982). Would it perhaps be possible to benchmark the various possible indicators of "interdisciplinarity" against each other? Using this new measure, Porter & Rafols (2009) and Rafols & Meyer (2010) , for example, suggested that this new measure would be useful to indicate interdisciplinarity at the article level. Might it also provide us with a useful indicator at the journal level? Stirling (2007, at p. 712) proposed to integrate the (in)equality in a vector with the network structure using the following formula for diversity D: This measure is also known in the literature as "quadratic entropy" (e.g., Izsáki & Papp, 1995) because unlike traditional measures of diversity such as the Shannon entropy and the Gini index, the probability distributions (p i and p j ) of the units of analysis (in our case, the citation distributions of the individual journals) are multiplied by the distance in the (citation) network among them (d ij ). The latter factor represents the journal ecology: a journal which is interdisciplinary within a specific domain (e.g., chemistry) may not reach beyond the confines of this domain. Other journals can be highly specialist, but yet combine citations from or to different domains. Thus, diversity has these two aspects which can also be studied separately as the properties of the citation distribution of each journal (e.g., the Gini index) versus using network indicators (e.g., betweenness 5 centrality). Stirling (2007) further suggested developing a heuristics by weighing the two components; for example, by adding exponents as follows: However, one then obtains a parameter space which is infinite (Ricotta & Szeidl, 2006) . Rao (1982, at pp. 6f.), however, uses the full matrix and even envisages that the main diagonal of the distance matrix may be unequal to zero depending on the research question. We used the full matrix; in our case the main diagonal values are always set to zero. Herfindahl-Hirschman index (in economics). The Gini coefficient is akin to this measure (Atkinson, 1970; Stirling, 2007, at p. 709) and has the advantage of having been widely used in bibliometrics (e.g., Burrell, 1991; Cole et al., 1978; Danell, 2000; Frame et al., 1977; Persson & Melin, 1996; Rousseau, 1992 Rousseau, , 2001 Stiftel et al., 2004; Zitt et al., 1999) . We chose to compare this measure at the level of the vector of each journal's citation distributions (cited and citing) with the Shannon entropy as another indicator used in bibliometric applications (e.g., Grupp, 1990; Leydesdorff, 1991) . The second part of Equation 1 ( ) represents a distance matrix in a network. We will use Euclidean distances and (1 -cosine) as two distance measures for reasons to be explained in the methods section below. Table 1 : Aggregated journal-journal citation matrix with citing and cited vectors for each journal, and six possible measures of "interdisciplinarity". This evaluation using these six measures will be performed examining both the cited and citing dimensions of the matrix. In the (later) discussion, we focus on the cited dimension 7 because in this dimension one also can compare with other journal indicators such as the impact factor and total cites (Bensman, 2007; Leydesdorff, 2009 ). Since there is as yet no established baseline for the interdisciplinarity of journals, we have to manoeuvre using correlation and factor analysis in order to understand whether two indicators measure something differently or capture to a large extent the same variation. Our initial hope was that the new Rao-Stirling diversity measure might provide us with such an obvious indicator of interdisciplinarity that one would be able to benchmark some of the other indicators against this dimension. A number of journals have thereafter no values other than on the main diagonal (that is, "journal self-citations"): 64 in the cited dimension and 20 citing. Although these journals cannot be provided with a value for the Rao-Stirling diversity indicator (since = 0), we have included the diagonal values in the computation of the Gini, the probabilistic entropy, and the centrality measures. Methods and materials Data processing Let us first turn to the vector-based measures. These are based on the frequency distributions of citations of each of the journals, either in the cited or citing directions. In the extreme case where a journal only cites or is cited by articles in the journal itself, the inequality in the citation distribution is maximal and the uncertainty minimal. 3 Maximum inequality corresponds to a Gini of unity and minimum uncertainty is equal to a Shannon entropy of zero. The journal is then extremely mono-disciplinary. In the opposite case, one would expect lower values for the Gini coefficient and higher entropy values. 3 If one drew a citation, one would know for certain which journal is involved. The Gini coefficient is a well established measure of inequality or unevenness in a distribution; it can be formulated as follows (e.g., Buchan, 2002): with n being the number of elements in the population and x i being the number of citations of element i in the ranking. The Gini ranges between zero for a completely even distribution and (n -1)/n for a completely uneven distribution, approaching one for large populations. For comparisons among smaller populations of varying size, this requires a normalization that brings Gini coefficients for all populations to the same maximum unity, i.e., one. The formula for this normalised Gini coefficient is: The uncertainty contained in a distribution can be formalized using Shannon's (1948) formula for probabilistic entropy: The maximum information is the same and thus a constant for all vectors, namely log 2 (8,207) = 13.00 bits. (When the two-base of the logarithm is used, uncertainty is expressed in bits of information.) Entropy can therefore be used as an indicator without the need for further normalization. However, the entropy is aggregated for each vector from the n cells with only non-zero values (n ≤ N; N = 8,207); n is a variable and log(n) can be considered as a local-journal-specific-maximum of the entropy. In a later section, the observed entropy values will also be assessed as a percentage of this local maximum of the entropy. Betweenness centrality is a network indicator and will be computed on both the (asymmetrical) citation matrix in both directions and the cosine-normalized matrices. After cosine-normalization the matrix is symmetrical, but one can normalize both over the cited and citing axis of the asymmetrical citation matrix. Thus, four evaluations are possible. Betweenness centrality is defined as follows: In order to compute the (Freeman) betweenness centrality, the matrix is first binarized.  In other words, betweenness centrality in the cosine-normalized vector space is after binarization for analytical reasons equal to betweenness centrality in the co-occurrence matrices (cited or citing). This inference is only valid when using no threshold values for the cosine. In this study, however, we did not set a threshold because a threshold introduces another parameter into the model and would thus complicate the comparison. The advantage of using co-occurrence-in our case, co-citation-matrices instead of cosine-normalized matrices is computational given the large size of the matrix: the (1-4 In the meantime, algorithms for weighting the line values are available (Brandes, 2001 (Brandes, , 2008 , but the implementation in Visone (at http://www.visone.info) cannot (yet) handle data matrices of this size (Brandes, personal communication, 25 February 2010). 5 Using SPSS (v. 15) for the construction of cosine matrices and Ucinet (v. 6.267) for the affiliations matrices (both cited and citing), we found correlations between the betweenness values of r = 0.987 (cited) and r = 0.990 (citing). These values should be precisely 1.0. We cannot explain these differences other than as an effect of potentially different roundings of decimals. 12 mode) co-citation matrix can be obtained from the asymmetrical (2-mode) citation matrix by multiplication with the transposed of the latter matrix. This can be done both in the cited (AA T ) and the citing dimension (A T A) of the asymmetrical citation matrix. We shall thus compare betweenness centrality as an indicator of interdisciplinarity by using both the asymmetrical citation matrix (in both the cited and citing directions) and the two symmetrical co-citation matrices (that is, using the numerators of Equation 7 for distinguishing between zeros and ones). Euclidean distances seem a most natural candidate for the distance matrix used for and third, Euclidian distances are more familiar, parsimonious, and intuitively accessible than most other distance measures. However, in our case, two journals with precisely the same distributions but with different sizes would be counted as distant from each other. For this reason, one first needs to normalize the data for size. In order to prevent this size effect, one could consider using z-scores (Bornmann & Daniel, 2009 ). However, Ahlgren et al. (2003) have shown that the Pearson correlation can be heavily affected by the large number of zeros in citation matrices; they therefore proposed the cosine as a non-parametric similarity criterion. Analogously to the Pearson 13 correlation, z-scores are based on averages and equally affected by this effect of the zeros on the mean. The cosine, however, is a similarity measure and not a distance. Yet, (1 -cosine) represents a dissimilarity, and thus can be considered as a relevant measure of distance. Alternatively, using Euclidean distances one can prevent the noted effect of differences in size in otherwise similar vectors (rows or columns) by using the proportions of citations, that is, the relative instead of absolute frequency distributions. (Conveniently, the relative frequency distribution is by definition equal to the probability distribution that we need for computing the other part of the Rao-Stirling diversity.) The noted effect of size is neutralized by this normalization. In sum, we will perform the computation of RaoStirling diversity using the four possibilities of these normalized Euclidean distances and (1 -cosine) versus cited and citing. Throughout the paper, we use Spearman's rank-order correlations because our primary objective is an indication of interdisciplinarity as a variable attribute among journals. Some of the comparisons among specific indicators (e.g., before and after normalization) allow for finer-grained comparisons using the Pearson correlation, but in order to prevent confusion we use the Spearman correlation consistently. Let us first consider Gini coefficients and Shannon entropy as two vector-based journal indicators, both in the cited and the citing directions of the citation matrix. While the Gini coefficient indicates unevenness, Shannon entropy provides an indicator of evenness. In other words, the Gini coefficient can be considered as an indicator of specificity and therefore disciplinarity, whereas the entropy (H) increases both when more cells of the vector are affected and with greater spread among the different categories. Thus, this indicator can be expected to capture both the aspects of a wider range and a more equally distributed set of citing or cited journals, respectively. Therefore, the entropy measure can be expected to be sensitive to size, whereas the Gini coefficient is normalized in the denominator of Equation 4.  The negative signs of the rank-order correlations between the two indicators (Table 2) show the opposite directionality. Not surprisingly, there is no strong correlation between rankings in the cited and citing dimensions: journals that build on diverse knowledge bases (citing patterns) do not necessarily have diverse audiences (cited patterns). Table 2 shows that correlations between the two indicators in the cited dimension (ρ = -0.803) are higher than in the citing dimension (ρ = -0.658). 6 This is understandable, since the citing side represents the research front and therefore introduces variability, while the archive of science is cited and thus can be expected to be more stable (Leydesdorff, 1993) . One can expect these indicators to show such differences in functionality between cited and citing. Table 3 shows the top 20 journals in terms of Gini coefficients (after correction for the ascending and descending orders) and entropy, both cited and citing. As expected, the entropy measure is affected by size. The listed journals are recognizable as intuitively the most interdisciplinary among the journals (in both the cited and citing dimensions; columns b and d, respectively). The perception of the importance of journals is also heavily influenced by their size (Bensman, 2007) , so this intuitive recognition is not incidental (cf. Bollen et al., 2009a , at p. 6). 6 All correlations are highly significant because of the large number of cases (N = 8,207 18 Again, we find higher correlations on the cited side than citing (Table 4) . By listing the top 20 journals in each ranking (Table 5) , we are able to discern more details. The first two columns (a and b) of Table 5 show the predicted effect of size. Without normalization Science, Nature, and PNAS are indicated as the major interdisciplinary journals. On the citing side (third column), however, these three journals follow at the 8,207 ** Correlation is significant at the 0.01 level (2-tailed). * Correlation is significant at the 0.05 level (2-tailed). Table 6 : Rank-order correlations (Spearman's ρ) among diversity measures based on (1 -cosine) and Euclidean distances; cited and citing (N = 8,207). Not surprisingly, there is no strong correlation between rankings in the cited and citing dimensions. However, the negative (albeit not significant) correlation between the (1 -cosine)-based and distance-based diversities is unexpected. The resulting diversity is thus heavily dependent on the choice of the distance measure. Let us try to assess qualitatively which of the two measures would make more sense by listing the top 20 journals in these four cases (Table 7) . (
REF investigated network indicators (betweenness centrality), unevenness indicators (Shannon entropy, the Gini coefficient) and Rao-Stirling measures to understand interdisciplinarity and popularity of journals.
Indicators of the Interdisciplinarity of Journals: Diversity, Centrality, and Citations
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