Characterizing structure of cross-disciplinary impact of global disciplines: A perspective of the Hierarchy of Science

As one of the largest scholarly databases, Microsoft Academic Graph (MAG) has a wide coverage of disciplines and publications. We used the version of MAG 2021.11, which is the latest version. MAG has a fine-grained discipline structure (i.e., “Fields of Study” or Topics) with a hierarchy from level 0 to level 5. The fields of study were generated by hierarchical topic modeling (Shen, Ma, & Wang, 2018). We only focused on the top two layers as the disciplines to be analyzed. Level 0 has 19 disciplines, including Art, Biology, Business, Chemistry, Computer Science, Economics, Engineering, Environmental Science, Geography, Geology, History, Materials Science, Mathematics, Medicine, Philosophy, Physics, Political Science, Psychology, and Sociology. A total of 292 disciplines are included in Level 1 of the hierarchy (Wang et al., 2020).

We collected 83,748,262 papers of the journal type from MAG. To ensure the number of the publications that meet the requirements of follow-up analysis, we selected publications published from 1971 to 2020 with at least one citation, and deleted the publications without discipline labels. The final dataset includes 82,646,847 publications and 4,774,260,411 citations. Figure 2(a) shows the number of annual journal publications with an increasing trend over the years. Figure 2(b) presents the number of publications in each discipline. Materials, Medicine, and Computer science have the most publications in MAG, while Environmental science, History, and Economics have the fewest. It should be well noted that one publication could belong to multiple disciplines.

Figure 2.

One well-acknowledged theory is that the system of model science is composed of a diversity of disciplines, which have their distinct research paradigms and communities. Based on the partitions of disciplines, the scientific impact of a publication in a discipline could be divided into intra-disciplinary impact on the publications in the same discipline and cross-disciplinary impact on the publications out of the discipline. Herein, we define cross-disciplinary impact of disciplines as the impact of knowledge from one discipline on other disciplines in addition to that on its original discipline. This term is also noted as transdisciplinary impact in Huang et al. (2022). Then, we refer to the three-dimensional framework of variety-balance-disparity (Leydesdorff & Rafols, 2010), which has been adopted to measure the interdisciplinarity of a field. The structure of cross-disciplinary impact of a discipline to all other disciplines could also be characterized with the three dimensions, including variety, balance, and disparity. Figure 3 illustrates the structure of a discipline’s cross-disciplinary impact.

Figure 3.

The original definition of variety is the number of categories into which system elements are apportioned (Stirling, 2007). Many scholars have redefined different calculation formulas for variety, for example, Leydesdorff (2018) defined variety as the number of valued categories divided by the total number of categories. Drawing on his definition and combining it with the calculation method proposed by Huang et al. (2022), we define the variety as follows.

Firstly, we quantify cross-disciplinary impact of a publication based on citations. Cross-disciplinary impact of a publication in a discipline reflects how its knowledge is acknowledged by the publications out of the discipline, which is defined as the number of citations it receives out of the discipline the publication belongs. In MAG, a publication can be assigned to one or more fields. Using the discipline hierarchy, we recursively find the parent fields until the 0-level discipline is obtained. Then, the cross-disciplinary impact of a publication is calculated based on its 0-level discipline. If a citing paper of the publication has a different 0-level discipline, the publication will gain one cross-disciplinary citation.

To quantify the impact of a paper on a discipline, one typical method used by previous studies is to directly count the citations from the publications of the discipline to the focal paper, i.e., group the citations at the discipline level (Wang et al., 2011). Depending on whether the paper belongs to the discipline, two forms of impact can be differentiated, i.e., the intra-disciplinary influence and the crossdisciplinary influence. The intra-disciplinary influence of a paper only counts the citations from the publications of the discipline it belongs to. The cross-disciplinary influence of a paper considers the citations from the publications out of the discipline it belongs to. To reflect the degree to which a publication impacts another discipline instead of its original discipline, we quantify cross-disciplinary impact of a publication (CIP) as the ratio of its cross-disciplinary influence (CI) to its intra-disciplinary influence (II), which is formulated as: 1 IIPp=CIII \[\text{II}{{\text{P}}_{p}}=\frac{\text{CI}}{\text{II}}\]

Relying on this ratio, we rank all publications of all the disciplines in the entire dataset, and get two thresholds, i.e., the top 5% (3.92) and 95% (0.089) of the CIP. We then applied the two thresholds to categorize all publications in a discipline into three groups, including the high cross-disciplinary impact group whose CIP is greater than the top 5% threshold, the low cross-disciplinary impact group whose CIP is lesser than the 95% threshold, the normal cross-disciplinary impact group whose CIP is between the two thresholds.

Since papers are the basic component of the discipline and highly cited papers can be considered to have made core contributions to the advancement of knowledge in the discipline, we define variety as the proportion of the high cross-disciplinary impact publications over all publications in the discipline, which is formulated as: 2 variety=#ofthehighcross-disciplinaryimpactpublications #ofallthepulications \[\text{variety}=\frac{\#\text{of}\,\text{the}\,\text{high}\,\text{cross-disciplinary}\,\text{impact}\,\text{publications }}{\,\#\text{of}\,\text{all}\,\text{the}\,\text{pulications }}\]

Then, to measure the balance of cross-disciplinary impact, Shannon entropy is applied. A greater score of Shannon entropy indicates that the discipline impacts others more evenly. The indicator used in Zhang (2016) is applied to measure the disparity of cross-disciplinary impact. A greater disparity means that the discipline impacts those disciplines with more different properties. For example, a discipline that is cited by both hard and soft sciences demonstrates a greater disparity of crossdisciplinary impact than a discipline that is cited by only hard sciences. The two structural indicators are calculated as: 3 Shannon(Xi)=−∑jqjlog(qj) \[\text{Shannon}\left( {{\text{X}}_{\text{i}}} \right)=-\sum\nolimits_{\text{j}}{{{\text{q}}_{\text{j}}}\log \left( {{\text{q}}_{\text{j}}} \right)}\] 4 Disparity =1n(n−1)∑i≠j(1−Sij) \[\text{Disparity}~=\frac{1}{\text{n}(\text{n}-1)}\underset{\text{i}\ne \text{j}}{\mathop \sum }\,(1-{{\text{S}}_{\text{ij}}})\] 5 Sij=Cij+Cji(TCi+TRi)(TCj+TRj) \[{{\text{S}}_{\text{ij}}}=\frac{{{\text{C}}_{\text{ij}}}+{{\text{C}}_{\text{ji}}}}{\sqrt{\left( \text{T}{{\text{C}}_{\text{i}}}+\text{T}{{\text{R}}_{\text{i}}} \right)\left( \text{T}{{\text{C}}_{\text{j}}}+\text{T}{{\text{R}}_{\text{j}}} \right)}}\]

In the above equations, q_j refers to the ratio of the cited publications belonging to discipline j to all the cited publications. S_ij is the similarity between discipline i and j, and n is the number of discipline categories. C_ij+C_ji is the total number of crosscitations between discipline i and discipline j, TC_i denotes the total number of citations received by discipline i from the other 18 disciplines, and TR_i denotes the total number of citations given by discipline i to the other 18 disciplines.

As Cole (1983) argued, hierarchy is an intrinsic characteristic of scientific system, which can be observed that different disciplines have different degrees of hardness and consensus. However, scholars have different opinions and measurements about the hardness of disciplines, and proposed different versions of HOS by including varying disciplines. Most of them investigated high-level disciplines, rather than finer-grained research fields. Lodahl and Gordon (1972) invited scholars to rank the disciplines according to their development level, and the results dexterously reflected the traditional scientific hierarchy, namely physics, chemistry, biology, psychology, and sociology. Cole (1983) asserted that the disciplines from hardest to softest are physics, chemistry, biology, psychology, and sociology. Simonton (2004) applied quantitative methods to derive HOS for the first time, and obtained the hierarchical structure of physics, chemistry, biology, psychology, and sociology. Klavans and Boyack (2010) obtained a non-central consensus map: mathematics is at the top, and the clockwise order is physics, engineering, chemistry, earth science, biology, biochemistry, infectious diseases, medicine, health services, brain research, psychology, humanities, social science, and computer science. Smith et al. (2000) also verified the correctness of the hierarchical structure from the perspective of the utilization rate of graph, and obtained the order of physics, chemistry, biology, medicine, psychology, economics, and sociology. Fanelli and Scalas (2010) verified the existence of scientific hierarchy based on the frequency of positive results, and got the order of physics, biology, and social science. Further, Fanelli and Glänzel (2013) obtained the order of mathematics, physics, biology, sociology, and Humanities by measuring a set of parameters reflecting consensus. John (2020) verified the order of astronomy, physics, chemistry, biology, psychology and sociology from the perspective of vocabulary sharing. Integrating different versions of HOS, the most frequently mentioned and agreed discipline rank of HOS is physics, chemistry, biology, psychology and sociology, which is used in this study.

We hypothesize that there is either a positive or negative association between the ranking of disciplines based on their variety of cross-disciplinary impact and their positions in the HOS. To test it, we conduct a Kendall’s tau-b correlation analysis on the two ranks of disciplines. Then, we divide the 19 0-level disciplines into four groups according to their balance and disparity of cross-disciplinary impact. The distribution of the five disciplines in the four groups will be analyzed. On the basis of the cross-discipline citations, we create a map of the relationships among the five disciplines to explore the interactions between the disciplines of different HOS layers.

We aggregate a few bibliometric features of publications to the discipline level and conduct a descriptive statistical analysis of the bibliometric features of the disciplines. An ordinal logistic regression model is applied to investigate the relationship between the bibliometric features and the structure of cross-disciplinary impact of disciplines.

Based on the citation relationship between disciplines, we obtained the mutual citation heat map of the 19 0-level disciplines, as shown in Figure 4. The value of each cell is calculated as the percentage of a publication in the column citing a publication in the row, and the darker color indicates higher values. It can be seen that the color of the main diagonal cell in each row is the deepest, that is, the self-citation rate of a certain discipline is generally higher than the cross-citation rate of disciplines.

Figure 4.

After grouping the publications into high cross-disciplinary impact ones, normal ones, and low ones, we counted the proportion of three types of publications in the disciplines. The distribution of the publications in the 19 disciplines over the three types is shown in Figure 5. Overall, the low cross-disciplinary impact group of publications occupies the most, exceeding 50% in all disciplines, while the high cross-disciplinary impact group is relatively low in most of the disciplines. Specifically, sociology has the largest proportion of the high cross-disciplinary impact group of publications (33.25%), which is consistent with the findings of many studies (Wang, 2011). In addition to sociology, economics (32.7%), business (30.83%), and political science (29.57%) also have a large proportion of high crossdisciplinary impact publications. Regarding the low cross-disciplinary impact group, it can be observed that environmental sciences (61.46%), arts (59.7%), and history (59.66%) have the largest proportions.

Figure 5.

According to the proportion of high cross-disciplinary impact publications (the variety of cross-disciplinary impact), an ascending ranking of cross-disciplinary impact was obtained for 19 disciplines: chemistry, materials science, physics, medicine, biology, engineering, psychology, geology, mathematics, computer science, philosophy, art, geography, history, environmental science, political science, business, economics, and sociology. We demonstrate the disciplines by the percentage of high cross-disciplinary impact publications as shown in Figure 6. Based on the rank, it could be inferred that the cross-disciplinary impact of a discipline may be related to its inherent features.

Figure 6.

According to the Hierarchy of Science (Cole,1983), the disciplines from hardness to softness include physics, chemistry, biology, psychology, and sociology. Intuitively, the order of disciplines by the variety of cross-disciplinary impact is almost consistent with that by HOS, except for physics and chemistry (Figure 6). Kendall’s tau-b correlation coefficient between the two ranks of disciplines (-0.900, p<0.05) further verifies the strong and significant negative correlation between the ranks by the variety of cross-disciplinary impact and HOS, which can be preliminarily considered that the less hard the discipline, the higher the cross-disciplinary impact.

We divide the 19 0-level disciplines into four quadrants according to the mean of the balance (1.47) and disparity (0.89) of cross-disciplinary impact, namely “high balance-high disparity”, “high balance-low disparity”, “low balance-high disparity”, and “low balance-low disparity”, as shown in Figure 7. The distribution of the five disciplines corresponds to the HOS structure. Physics and chemistry in the top layer, the two hardest disciplines, are assigned to the “low balance-low disparity” category. It means that the publications in the two disciplines are mainly cited by a few similar disciplines, rather than by all the disciplines equally. For example, according to Figure 4, the majority of cross-disciplinary citations to chemistry are from materials science and biology. Biology and psychology in the middle layer, the less hardness, are assigned to the “low balance-high disparity” category and “high balance-low disparity” category respectively. The low balance of biology indicates that its publications are mainly cited by a few disciplines, reflecting its specialized knowledge often influences related disciplines. While the high balance of psychology indicates that it is more evenly cited by other disciplines, reflecting that its knowledge can be widely borrowed by other disciplines. Sociology in the bottom layer that is the softest discipline is assigned to the “high balance-high disparity” category. It means that its publications are often generally cited by all the other disciplines, which can also be inferred from Figure 4 that the citations to sociology are more evenly distributed over all the other disciplines. It can also be concluded that the less hard the discipline, the higher balance and disparity of cross-disciplinary impact.

Figure 7.

To further explore the interactions between the disciplines of different HOS layers, we plot the relationship map among the five disciplines, as shown in Figure 8. Taking physics as an example, it can be seen that physics has the greatest cross-disciplinary impact on chemistry, followed by biology, psychology and sociology. This order is consistent with the HOS structure. It shows that the cross-disciplinary impact diminishes as the distance of hierarchy between disciplines increases. The same conclusion can be drawn for the other four disciplines, namely that the proximity of disciplines in HOS will determine the degree to which they impact each other. Therefore, each discipline exerts the greatest cross-disciplinary impact on the disciplines adjacent to it in the hierarchy of science, and the cross-disciplinary impact decreases as the hierarchical distance increases. The result is consistent with the findings of some previous studies, e.g., some scholars regard that from the perspectives of co-citation, one discipline tends to influence the disciplines closer to it more easily and faster (Thed & Robert, 2000).

Figure 8.

Table 2 shows the descriptive statistics of the nine bibliometric features of the 19 0-level disciplines. It can be seen that the average number of authors exceeds six, which verifies the current phenomenon of increasing collaboration among scholars (Chen et al., 2021). The number of references is generally high, but the standard deviation is also relatively large. Most of the publications follow a typical citation pattern, i.e., they are rarely cited in the first year, and reach a citation peak a few years later, then decline slowly. Figure 9 shows the distribution of bibliometric indicators for the four groups of disciplines.

Figure 9.

As shown in Figure 9, the bibliometric features have different distributions over the 19 disciplines, and the scores of the disciplines in “high balance-high disparity” group are either the highest or the lowest among the four groups. Except for the “low balance-low disparity” group, the number of authors shows a decreasing trend followed by an increasing trend. For the number of references and Price index, the variation in the “high balance-high disparity” group is small, while the variation in the other three groups is large. For the number of cited disciplines and citation distribution, the differences within the “high balance-high disparity” group are negligible, while the other three groups show a clear trend of decreasing crossdisciplinary impact. For the year of the first citation and the first citation count, the trends of the four groups are similar. For the year of the first peak citation and the first peak citation count, the trend within the “high balance-high disparity” group is generally decreasing as the decrease of cross-disciplinary impact. It can be seen that the “high balance-high difference” group shows distinct characteristics from other groups of disciplines.

We further examine how the hardness of each discipline evolved over time. Figure 10 shows the evolution of the five bibliometric features of the hardness for the five disciplines from 1970 to 2020. For the number of authors, it shows an upward trend as a whole, but physics shows great fluctuation. For the number of references, the four disciplines show slight changes for a period of time except sociology, while sociology has had a slight downward trend in recent years. For the number of cited disciplines, biology shows a small change. sociology and psychology fluctuate greatly but the overall trend is still increasing. For the citation distribution, the whole shows an early rise but decline in recent years. For price index, sociology and psychology fluctuate greatly but show an upward trend. While biology declines slightly. The results demonstrate that the properties of disciplines are not static, but keep changing continuously over time. Therefore, the hardness/softness of the disciplines could be understood from a dynamic perspective.

Figure 10.

The main finding is that the less hard the discipline, the higher the crossdisciplinary impact, the higher the balance and the lower the disparity. Some studies have argued that hard and soft sciences are different social systems with different patterns of citation, communication, and publication (Shapin, 2022). It is generally accepted that the hard science has a higher consensus and a more uniform paradigm within the discipline than the soft science (Fanelli & Glänzel, 2013). Therefore, their scientific findings have more impact on their original disciplines, rather than other disciplines, that is, they generate less cross-disciplinary impact. It has also been noted that the hard sciences will show stronger group bias, that is, it is more inclined to refer to other hard disciplines than soft disciplines. On the contrary, soft sciences may have some shared theories, research objectives, methodologies, etc., which leads to more communications among them (Wu et al., 2022).

First, the conclusion is evidenced by the statistically negative correlation between the ranking of disciplines by the variety of cross-disciplinary impact and the ranking at the HOS structure, e.g., from hardness to softness. Additionally, it can also be concluded that a discipline will have a greater cross-disciplinary impact on a discipline with which it shares a lesser degree of proximity. Previous studies have found that knowledge of one discipline tends to influence its closer disciplines more easily and faster, from the standpoints of co-citation, text semantic analysis, and network metrics (Thed & Robert, 2000). From the perspective of scientific hierarchy, we redefine the distance and adjacent relationship between disciplines, and our findings are congruent with these researches. The distribution of disciplines over the four categories based on the balance and disparity of cross-disciplinary impact also coincides with the HOS.

Secondly, the finding is also supported by the relationships between bibliometric features of disciplines and their cross-disciplinary impact, revealed by the regression models. The bibliometric features we evaluated, including the number of authors, the number of references, the number of cited disciplines, citation distribution, and the Price index, can quantify the hardness of disciplines. Among these features, the number of references, the number of cited disciplines, the citation distribution, and the Price index, have a significant positive effect on the variety of cross-disciplinary impact. As discussed in Section 3.4, the value of these features may be greater in the soft sciences. Therefore, the positive correlation indicates that the less hard the discipline, the higher the cross-disciplinary impact. We also demonstrate that the Price index and the citation distribution are the most important features. What’s more, the number of references, the number of cited disciplines, and the citation distribution have significant positive and negative effects on the balance and disparity of cross-disciplinary impact, respectively.

In the empirical analysis of HOS, we only included five broad disciplines, which are concluded from previous studies. In theory, the hierarchy could be enhanced by adding more broad disciplines or finer-grained disciplines such as the Level-1 disciplines in MAG. Whether the findings in this study can be generalized to an enhanced hierarchy of science needs further exploration.

This study also has some biases caused by the data source. First, we only include the journal papers in this study, which may suffer from the issue of low coverage for some disciplines, such as engineering disciplines and computer science. Second, although MAG has already collected a quite large number of social science articles, many social science records are still absent. The reasons may lie in: a) Many social science research articles are not written in English and published in a variety of countries, and b) MAG tended to collect more nature science records. Therefore, it should be noted that the results reported in this study are solely based on the collection of MAG, which might have some biases to social science. Second, we deleted the data with missing references in the data cleaning stage, however, there are incomplete reference data of articles in MAG. For this problem, we simply considered that the distribution of articles with incomplete references is uniform over all disciplines, and we adopted the indicator of average citation to undergrade the influence of missing references as much as possible. In the future, as the initiative of open citation, we could use the databases like COLI to enhance the citation data of MAG (Liang et al., 2021).

About the bibliographic features, we only use five indicators to reflect the hardness of disciplines. It needs further examinations to explore a more comprehensive set of indicators that could quantify the harness of disciplines. About the research method, we only use regression models to analyze the influence relationship, and other methods like machine learning can be used to further explore the causal relationship. In the future, we attempt to further investigate the cross-disciplinary impact of patents. And, we will also combine citation network analysis and text mining methods to design new indicators, with the purpose of unearthing how crossdisciplinary impact forms and evolves.