Introduction

Throughout history, science and technology (S&T) have progressed through a delicate balance of competition and co-operation. This interaction, occurring at various levels—among individuals, teams, organisations, countries and economic regions—has created unique value and propelled modern civilisation forward. While S&T has brought many benefits, it has also posed various threats, risks, and disruptions. These challenges have been addressed and tackled through co-operation at different academic levels and coordinated efforts led by governments.

Not all S&T domains have followed the same developmental trajectory throughout this progression. This also holds for international research collaboration, which has undergone substantial changes across various disciplines and over time (Adams, 2013; Dong et al., 2017; Fortunato et al., 2018; Okamura, 2023;2024; Wagner et al., 2017). Many factors have influenced these changes, including the demands and needs of the era, the unique characteristics of specific fields, the research and development (R&D) capabilities of various institutions, national objectives and missions, international strategies, their connections to national security and industrial policies, as well as the availability of abundant S&T talent (Chen et al., 2019; Chinchilla-Rodríguez et al., 2018; Leydesdorff et al., 2013; Makkonen and Mitze, 2016). These factors have intertwined in complex ways, collectively shaping the current scientific and technological landscape.

Today, we are firmly entrenched in an information society where vast volumes of data flow through the internet, marking the era of data-driven and AI-empowered team science (Cooke and Hilton, 2015; The Organisation for Economic Co-operation and Development OECD, 2015). As we reflect on the multifaceted and international history that S&T and society have traversed, we unearth valuable lessons from the dynamics of competition and co-operation, especially at the national level. These lessons are significant for the future, motivating researchers to analyse international collaborative research across various disciplines using modern technologies and resources. This analysis also provides policymakers with useful insights and recommendations (Dong et al., 2017; Kwiek, 2021; National Science Board NSB and National Science Foundation NSF, 2022; The Organisation for Economic Co-operation and Development OECD, 2017; The Organisation for Economic Co-operation and Development OECD and SCImago Research Group CSIC, 2016; Whetsell, 2023). Amid the irreversible surge of Open Science (Burgelman et al., 2019; Miedema, 2022), the scope and depth of the ‘Science of Science’ (Fortunato et al., 2018; Lin et al., 2023; Liu et al., 2023; Zeng et al., 2017) are also expanding.

Efforts to analyse and visualise international co-operation in S&T have been extensive. Some studies have focused on specific countries or economic regions, while others have examined particular research disciplines (Chen et al., 2019; Chinchilla-Rodríguez et al., 2018; Gazni et al., 2012; Gök and Karaulova, 2024; Haupt and Lee, 2024; Liu et al., 2023). For instance, Chen et al. (2019) conducted a systematic review of international collaborative research, suggesting that such collaboration between 1957 and 2015 consisted of three phases: ‘emergence’, ‘fermentation’ and ‘take-off’. Moreover, bibliometric characteristics such as patterns, networks, effects, measurements, and driving factors have been identified by country and research discipline.

Previous studies have also revealed that the trend of international collaborative research has been steadily deepening globally over the past few decades, including recent years (Dong et al., 2017; Kwiek, 2021; National Science Board NSB and National Science Foundation NSF, 2022; The Organisation for Economic Co-operation and Development OECD, 2017; Okamura, 2024). This trend can be described as a ‘Shrinking World’ in the context of international collaborative research. A recent study by Okamura (2023) provided a global-scale quantitative demonstration of this concept. The study conducted hierarchical clustering based on international collaboration relationships among countries and analysed the clustering by examining the heights of branch points in the dendrogram. This analysis suggested, through a bibliometric lens, that the world has been shrinking over the past half-century across various disciplines.Footnote 1

Reflecting on the past few decades, it is particularly interesting to focus on five key players: the US, China, EU27,Footnote 2 the UK and Japan. Figure 1 illustrates and quantifies the distances between these entities over the past half-century, with the methodology detailed in the next section. In this context, EU27 and the UK are combined and treated as a single entity. The size of each sphere in the figure represents the volume of scientific output from each player, which has grown over time. Simultaneously, the lengths of the edges connecting these spheres, symbolising the distances between them, have markedly reduced. Consequently, while global scientific output has dramatically increased, the overall volume of tetrahedron has shrunk. This phenomenon effectively encapsulates the concept of the ‘Shrinking World’.

Fig. 1: Changes in the mutual distance between the four parties over time.
figure 1

The four parties included in the analysis are the US, China, EU27&UK and Japan. The changes over the period of 1970 to 2021 are displayed in five-year intervals. Each coloured sphere’s volume is proportional to the number of scientific publications produced by each party, and the sizes can be compared within and across snapshots. The distance between the centres of each sphere represents the level of closeness in international collaboration between the corresponding parties, with shorter distances indicating closer collaboration. The calculation methodology for these distances is described in the main text.

Full size image

Regarding knowledge production in S&T at the national level, the US historically held a dominant position as the sole superpower across many disciplines until roughly a decade or fifteen years ago. However, recent years have witnessed substantial transformations in the global landscape of scientific and technological competition and collaboration. The primary catalyst for this transformation has been the remarkable rise of China over the past two decades (Baker, 2023; National Science Board NSB and National Science Foundation NSF, 2022; Okamura, 2023; 2024; Woolston, 2023), as evident in Fig. 1.

When considering the future of S&T and the prospects for international collaboration, the current trend will continue in the foreseeable future. In this scenario, a critical key factor that will influence S&T policy decisions, much like the conditions shaping the global economy, will be the relationship between the US and China (Cao et al., 2020; Haupt and Lee, 2024; Schmid and Edenfield, 2023; Toney and Flagg, 2021; Zhao, 2019). It is widely acknowledged that both countries have rapidly increased their international collaborative research across various disciplines over the past few decades. For instance, Okamura (2023) visualises how the distance between the US and China has been decreasing since 2000 in all 15 fields of interest (Fig. 3, Okamura 2023).Footnote 3

However, several studies have indicated a possible decline in collaboration between these two major powers or a deterioration in their cooperative relationship, particularly since around 2019 (Aghion et al., 2023; Jia et al., 2022; Okamura, 2023; Tang, 2024; Van Noorden, 2022; Wagner and Cai, 2022; Xie et al., 2023). The ‘China Initiative’, initiated by the US Department of Justice in November 2018 to address the national security threat posed by China and concluded in February 2022, is often cited as a significant factor underlying these developments (Gilbert, 2023; Mervis, 2023; Wagner and Cai, 2022).

If the production of human knowledge and intelligence were confined to specific clusters of countries with less effective cross-border collaboration than possible, it could heighten risks for the global economy and civilised society. In such scenarios, addressing global challenges like preventing pandemics, enhancing resilience against natural disasters, establishing a decarbonised society, and fostering a world that embraces diverse perspectives would be less efficient. Signs of decoupling between the US and China present an opportunity for science diplomacy (Ambrozaite et al., 2024; Gluckman et al., 2017; Flink, 2021; Ruffini, 2020), which must be grounded in modern understanding and realism. Promoting international collaboration based on research integrity and employing diplomatic efforts when necessary is a significant responsibility for all current S&T leaders.

This paper aims to provide valuable evidence for policy deliberation to S&T policymakers, diplomats, researchers, and practitioners by addressing the following research questions: How has the distance between the US and China evolved over the past few decades? How does this compare to the behaviour of other country pairs, and has it exhibited any distinctive patterns? If so, what might be the underlying causes of these phenomena? To explore these questions, the paper analyses and visualises changes in the relationships between major scientific powerhouses—specifically the US, China, the EU27, the UK, and Japan—over the past decades across various research disciplines. By examining these trends, we highlight how the relationship between the US and China has exhibited a notably dynamic pattern of convergence and divergence, setting it apart among these leading scientific powerhouses.

Methods

This section describes the data used in this study. All data were analysed and visualised with Python software (version 3.6), STATA/IC software (version 13; StataCorp LP, TX, USA) and Mathematica (version 13.2; Wolfram Research, Champaign, IL, USA).

Data preparation and implementation

Data source

The data used in this study was obtained through the OpenAlex API (Priem et al., 2022), an Open Bibliometrics platform launched in 2022 as a replacement for Microsoft Academic Graph (MAG) (Sinha et al., 2015). The data was retrieved on the 19th of May, 2023. OpenAlex collects information on various types of scientific publications, including journal articles, non-journal articles, reviews, letters, preprints, conference papers, books, theses and datasets. These publications are collectively referred to as ‘works’, and OpenAlex indexes over 240 million of them.

To analyse changes in international research collaboration over time, we focus on works that include information on the year of publication among the relevant data. The publication year typically lags behind the actual time the research was conducted. When aggregated, this lag can be several months to years on average, depending on the R&D discipline (Björk and Solomon, 2013; Okamura, 2022). The number of works included in the OpenAlex data used in this study by country is the US (18,927,016), China (9,422,895), Germany (17,116,787), the UK (4,957,347) and Japan (4,165,540), for a total of approximately 55 million works, including crossovers between countries (Supplementary Table S1). Here, the figures for each country are counts of the number of papers whose authors are affiliated with institutions located in that country.

The advantages and potential drawbacks of using OpenAlex data for bibliometric studies are extensively discussed in Okamura (2023), which is relevant to the present study. A key advantage is the timely and efficient access to a substantial volume of freely available data. This accessibility facilitates the identification of broad bibliometric trends, making it particularly valuable for policy-related objectives. Policymakers often require up-to-date data that can be quickly accessed, analysed, and visualised as needed. However, a potential drawback is that the quality of such data may not be comparable to that of commercial databases. For the purposes of the current study, as will be demonstrated later, this limitation does not pose an issue, and the advantages outweigh it.

R&D disciplines

An appropriate classification scheme for various R&D disciplines is essential to ensure meaningful policy implications for international research collaboration. OpenAlex has devised a classification scheme called the ‘concepts’, which encompasses field categories at multiple levels of granularity, spanning from level 0 to level 6.Footnote 4 Higher levels within this scheme denote more specific concepts, while lower levels encompass broader ones. These concepts serve as a framework for categorising research areas.

This study specifically focuses on the 19 level-0 concepts from the OpenAlex classification. These concepts cover various disciplines, including humanities and social sciences (HSS) and natural sciences. The level-0 concepts we consider are Mathematics, Computer science, Geography, Biology, Physics, Chemistry, Engineering, Materials science, Environmental science, Medicine, Psychology, Business, Economics, Sociology, Political science, Geology, History, Philosophy and Art. By including these diverse disciplines of study, we aim to capture a comprehensive view of international research collaboration across different domains.

‘Nationality’ of works and the counting method

To streamline the counting process, we use the term ‘work of nationality X’ to refer to the work produced by contributors from institutions in country X. In cases where contributors from both country X and country Y collaborate on a work, the work is considered to have dual nationality, thus being counted as both a work of nationality X and a work of nationality Y. While we use the terms ‘country’ or ‘nationality’ in this context, it is essential to note that the discussion in this paper is equally applicable to situations where it encompasses a group of countries, such as EU27.

This counting method allows works to have multiple nationalities in our analysis. To count the number of works for each country, we employ a binary counting method based on nationality (Okamura, 2023). In this method, if a work has dual nationality (X and Y), it is counted as one work output for each country. Even if there are multiple contributors from country X, the work is only counted as one in the production volume for country X. If the country information is unknown for all contributors of a particular work, it is categorised as ‘unknown’ and excluded from the analysis conducted in this study.

Countries

This paper focuses on five key parties: the US, China, EU27, the UK and Japan. These countries were selected based on their significant contributions to work production across all science disciplines from 1971 to 2020. When excluding works with unknown nationality, the top five countries in terms of work production are as follows: the US (24,425,579), China (12,106,744), the UK (6,403,998), Germany (5,288,254) and Japan (4,768,966). Following closely are France (4,318,698), Canada (3,154,311), India (3,001,958) and Italy (2,766,564) (as of 12th August 2022). These selections align with the countries recognised as the ‘Big 5 science nations’ (the US, China, Germany, the UK and Japan) in the Nature Index (Crew, 2022). While other countries following those listed above also provide interesting insights, limiting the analysis to these five parties provides clarity in the results. It is worth noting that the methodology described in this paper can be applied to include additional parties or focus on different countries, maintaining its applicability and relevance.

The ‘Collaboration Distance’ approach

There can be various approaches to quantifying bilateral distance based on coauthorship relationships between two countries. A naïve approach may consider that the greater the number of coauthored papers, the closer the distance between the countries. However, adopting such a simplistic approach fails to satisfy a crucial mathematical property of distance: triangle inequality. This limitation can be a drawback for specific formal analyses, such as clustering analysis. Therefore, a certain level of refinement is necessary in this context. Any modifications made during this refinement process must not be overly complex or costly. Considering the significant interest of S&T policymakers in the cooperative and distance relationships within academia between countries, it is necessary to adopt a practical and easily interpretable method that suits their needs. Still, the definition of distance should ideally possess the property of enabling cluster analysis at the level of individual countries or research institutions. This is particularly significant because policymakers are always highly concerned about the existence of collaborative relationship clusters at the international level and the positioning of their own country within those clusters. To meet all these requirements, this paper adopts the same simple set-theoretic distance method developed and employed in the prior studies (Okamura, 2023; 2024).

Let \({S}_{\text{X}|\alpha ,{\tau }}\) denote the set of works of nationality X published in discipline α during a given period τ. Then, the distance \({D}_{\alpha ,{\tau }}\left(\text{X},\text{Y}\right)\equiv {D}_{\text{XY}}\) between countries X and Y in the given discipline and period is defined as:

$${D}_{\alpha ,\tau }({\rm{X}},{\rm{Y}})=1-\frac{|{S}_{{\rm{X}}|\alpha ,\tau }\cap {S}_{{\rm{Y}}|\alpha ,\tau }|}{|{S}_{{\rm{X}}|\alpha ,\tau }\cup {S}_{{\rm{Y}}|\alpha ,\tau }|}.$$
(1)

This distance metric is known as the Jaccard distance and ranges from 0 to 1. A value of 0 indicates that X and Y are identical, while 1 indicates that they are entirely distinct. The Jaccard distance satisfies the mathematical definition of distance in the set-theoretic sense, including the triangle inequality.

Using the distance measure, Eq. (1), which adheres to the axioms of distance functions, we calculate the distances between countries, facilitating comparisons between the distances of different country pairs. In this paper, we define ‘convergence’ as a tendency for the distance between two specific countries to decrease over time, and ‘divergence’ as a tendency for this distance to increase. Figure 2 provides schematic diagrams to illustrate this concept. First, compare diagrams (a) and (b). Given \(\left|{S}_{\text{X}}\right|\) and \(\left|{S}_{\text{Y}}\right|\), representing the number of works produced by countries X and Y, respectively, if \(\left|{S}_{\text{X}}\cap {S}_{\text{Y}}\right|\) is large, then \({D}_{{\rm{XY}}}\) is small (close to 0). Conversely, if \(\left|{S}_{\text{X}}\cap {S}_{\text{Y}}\right|\) is small, then \({D}_{{\rm{XY}}}\) is large (close to 1). Next, compare diagrams (b) and (c). Considering another country X’, if \(\left|{S}_{\text{X}}\right| > \left|{S}_{{\text{X'}}}\right|\) and \(\left|{S}_{\text{X}}\cap {S}_{\text{Y}}\right|=\left|{S}_{{{\text{X'}}}}\cap {S}_{\text{Y}}\right|\), it follows that \({D}_{{\rm{XY}}} > {D}_{{{\text{X'}}}{\rm{Y}}}\). In the limiting case where the number of works produced by one country becomes negligible compared to the other, the distance between the two countries approaches one (the maximum distance), indicating no mutual affinity. Based on this formulation, diagram (d) conceptualises that countries tend to converge when their distance decreases (‘Shrinking World’), and diverge when their distance increases.

Fig. 2: Conceptualisation and quantification of bilateral distances using the Jaccard distance.
figure 2

Focusing on a specific country pair, X and Y, the Collaboration Distance (\({D}_{{\rm{XY}}}\)) decreases as the number of coauthored works produced by researchers affiliated with institutions in these countries increases, and increases as the number of such collaborations decreases (see diagrams (a) vs. (b)). Considering another country, X’, if the number of coauthored works between X and Y is the same as that between X’ and Y, but the number of works produced by X is greater than that of X’, then \({D}_{{\rm{XY}}}\) will be larger than \({D}_{{{\text{X'}}}{\rm{Y}}}\). (see diagrams (b) vs. (c)). Based on this formulation, it is conceptualised that countries tend to converge when their distance decreases, and diverge when their distance increases (diagram (d)).

Full size image

Based on the resulting n × n distance matrix \({{\boldsymbol{D}}}_{\alpha ,{\tau }}\) for each discipline and period, where n represents the number of countries (parties), it is possible to embed the position coordinates \({{\boldsymbol{x}}}_{{i|}\alpha ,{\tau }}\) of each country Xi (i = 1, …, n) in an (n−1)-dimensional Euclidean space.Footnote 5 Hereafter, the distance measure (1) is transformed through rescaling: \(D\,\mapsto\, {D}_{\text{resc}}=-\ln D\), aiming to improve visual effectiveness.

The ‘Knowledge Flow’ approach

Coauthorship relationships in works discussed above can be viewed as an ‘undirected-graph’ approach, where the symmetrical relationship between two countries (parties), X and Y, is expressed through a distance function. However, it is equally valuable and highly relevant for S&T policymakers, diplomats and practitioners to capture unilateral movements of researchers from one country to another, such as inflows or outflows based on affiliations. To address this point, we also take a ‘directed-graph’ or ‘asymmetric’ approach, focusing on author (researcher) identifiers instead of work (paper) identifiers. This direction leads us to introduce the concept of ‘knowledge flow’, which conceptually extends the conventional idea of researcher mobility (Chinchilla-Rodríguez et al., 2018; Guthrie et al., 2017; Robinson-García et al., 2019); Schmid and Edenfield, 2023; Sugimoto et al., 2017; Vaccario et al., 2021). The term ‘knowledge flow’ used here does not necessarily ensure that specific knowledge will propagate. Rather, it represents the existence of a potential channel for knowledge to travel between affiliated organisations due to an author’s multiple affiliations. This is a useful indicator for understanding international collaborative relationships when those organisations span different countries. Below, for convenience, we consider the temporal flow on a yearly basis. Still, it is essential to emphasise that the concept of knowledge flow is applicable to any appropriate time unit.

Let us consider a researcher, P, affiliated with universities in both country X and country Y in a certain year τ. This fact can be inferred from P’s produced work’s ‘affiliation’ field, where universities from countries X and Y are listed. In the subsequent year (τ + 1), let us assume that researcher P remains affiliated with universities in both countries X and Y. Consequently, we interpret that academic knowledge flows between the two countries through the mediator, researcher P, across consecutive years (τ → τ + 1), implying potential knowledge exchange from X to Y and vice versa. This new concept of knowledge flow introduced in the current study contrasts with what is called researcher mobility, which often assumes physical relocation (Chinchilla-Rodríguez et al., 2018; Guthrie et al., 2017; Robinson-García et al., 2019; Schmid and Edenfield, 2023; Sugimoto et al., 2017; Vaccario et al., 2021). Instead, we focus on how researchers’ affiliations change within what we term the affiliation space without necessarily involving the physical migration of researchers.

This leads to our ‘directed-graph’ or ‘asymmetric’ approach, considering researcher identifiers and diverging from the ‘undirected-graph’ or ‘symmetric’ approach, which centres on coauthorship relationships based on work identifiers. Now, let us consider the following year (τ + 2), where researcher P is exclusively affiliated with a university in country X. We interpret that academic knowledge flows from country Y to country X through researcher P across years (τ + 1 → τ + 2), but not in the opposite direction. The rationale behind this interpretation is rooted in the fact that researcher P is no longer affiliated with country Y, resulting in the discontinuation of any academic knowledge previously held by P while affiliated with country X from flowing to country Y through P. Likewise, in the subsequent year (τ + 3), assuming that researcher P is exclusively affiliated with a university in country Z, we can infer the knowledge flow from country X to country Z through researcher P over the years (τ + 2 → τ + 3), but not in the direction from Z to X or Y.

To facilitate a quantitative approach, let us introduce some notations. For each year (τ), we quantify the number of knowledge flows from country \({{\rm{X}}}_{i}\) to country \({{\rm{X}}}_{j}\) in the subsequent year (τ + 1). We denote this count as the (i, j)-th element of the K-matrix, referred to as \({K}_{\mathrm{X}_{i}\to \mathrm{X}_{j}}({\tau})\), or more concisely as \({K}_{ij}({\tau })\), defined for year τ. Here, the labels i, j and so on represent distinct countries (parties). Since each element of the calculated K-matrix is influenced by researchers’ overall mobility trend and the total number of works they produce, it is crucial to scale each element appropriately relative to such absolute quantities. Our specific interest lies in the knowledge flows between countries, which correspond to the off-diagonal elements of the K-matrix. Therefore, for each year τ, we calculate the sum of the off-diagonal elements of K, denoted as \({\Sigma }_{\text{off}}\left(K\right)={\sum }_{i\ne j}{K}_{ij}\), and utilise this off-diagonal sum to derive the off-diagonal elements of the scaled K-matrix, \({{\mathcal{K}}}_{ij}={K}_{ij}/{\Sigma }_{\text{off}}\left(K\right)\) (i ≠ j). This quantity represents our focal point, the Knowledge Flow Rate (KFR).Footnote 6 Note that this quantity is normalised over all combinations of distinct pairs of i and j, i.e. \({\sum }_{i\ne j}{{\mathcal{K}}}_{ij}=1\). In other words, each \({{\mathcal{K}}}_{ij}\) quantitatively represents the proportion of knowledge flow from Xi to Xj among all unique pairs of countries within the total knowledge flow occurring during a particular time transition.

By observing the temporal changes in \({{\mathcal{K}}}_{ij}({\tau })\), we can gauge the relative intimacy of countries \({{\rm{X}}}_{i}\) and \({{\rm{X}}}_{j}\). It is crucial to note that the K-matrix in this context is an asymmetric matrix, i.e., \({{\mathcal{K}}}_{ij}\,\ne\, {{\mathcal{K}}}_{ji}\) in general. To demonstrate the implications of this characteristic, let us consider a scenario during a particular time transition from τ → τ + 1, where \({{\mathcal{K}}}_{ij} \,>\, {{\mathcal{K}}}_{ji}\) holds for a given i and all j for simplicity. In this case, country \({{\rm{X}}}_{i}\) is suggested to experience more knowledge flowing out to other countries than from other countries during this period. Furthermore, when focusing on a specific pair of i and j, a situation where the value of \({{\mathcal{K}}}_{ij}\) increases (or decreases) over the years indicates a relative upward (or downward) trend in the proportion of knowledge flowing from country \({{\rm{X}}}_{i}\) to country \({{\rm{X}}}_{j}\). Here, it should be noted that \({{\mathcal{K}}}_{ji}\) may not necessarily follow the same trend of temporal changes. In cases where there is a significant asymmetric knowledge absorption potential between countries \({{\rm{X}}}_{i}\) and \({{\rm{X}}}_{j}\), it is possible that \({{\mathcal{K}}}_{ji}\) shows a decreasing (increasing) trend while \({{\mathcal{K}}}_{ij}\) simultaneously exhibits an increasing (decreasing) trend.

The approach above might appear somewhat heuristic, but as we will see, it remains effective in quantifying and analysing the flow of knowledge over time. It proves to be a valuable tool for understanding the circulation of knowledge among researchers, as will be demonstrated subsequently.

Results

This section provides a detailed analysis of the two approaches introduced in the previous section: the ‘Collaboration Distance’ approach and the ‘Knowledge Flow Rate (KFR)’ approach. These methods centre around distinct identifiers, specifically work (paper) identifiers and author (researcher) identifiers. Interestingly, both approaches yield consistent quantitative evidence and insights from different perspectives, particularly concerning the relationship between the US and China.

Change in the ‘Collaboration Distance’

Evolution of collaboration distance by field

Based on the methodology described earlier, we computed and visually represented the distances (D) between pairs of the five parties: the US, China, EU27, the UK and Japan, across different periods and the 19 focused disciplines. See Supplementary Fig. S1, where the graph’s vertical axis represents the bilateral distance. Among the three parties—the US, EU27 and the UK—there have been generally similar trends in their distances across various disciplines, with some variations in absolute levels. The distances between these parties have consistently decreased nearly linearly, and they have maintained the shortest distances throughout the analysed periods.

On the contrary, the distances between China and any other parties have typically been more significant than those between Western countries within each period. Nonetheless, there has been a similar trend of distance reduction over time. Of particular interest is the rapid decline in distance between the US and China since the beginning of the 21st century, with a steep decline in the early 2010s. This phenomenon is noteworthy not only in natural science disciplines but also in HSS disciplines, such as History, Philosophy and Art.

However, since 2019, there has been a reversal in this trend, with distances tending to widen or remain relatively stable. This trend is particularly pronounced in the natural science disciplines, exhibiting an observable ‘time-reversed J-curve’ pattern. This observation aligns with a phenomenon reported in a prior study (Okamura, 2023) using a different set of field categories (specifically, some level-1-related natural science concepts in OpenAlex). Although the curvature of distance convergence is less prominent in the HSS disciplines compared to the natural science disciplines, there is still a tendency for deceleration. In contrast to the natural sciences, particular disciplines within humanity often require more time for international collaborative research to reach maturity (Kwiek, 2021). Therefore, there is a possibility that, with some time delay, we may also witness an increasing divergence between the US and China in these disciplines. Although there are other intriguing implications in the relationships between various pairs of parties, this paper concentrates explicitly on the US–China relationship and underscores its relative distinctions compared to other pairs.

‘Shrinking (and-possibly-Polarising) World’

Figure 3 presents the temporal evolution of pairwise distances in the natural science disciplines for the same five parties mentioned earlier (the US, China, EU27, the UK and Japan). The natural science disciplines encompass all the level-0 categories of natural science concepts in OpenAlex, including Mathematics, Computer science, Geography, Biology, Physics, Chemistry, Engineering, Materials science, Environmental science and Medicine. The figure provides an analysis and visualisation of the changing distances between each pair of parties from 1970 onwards. The overall trend observed in the figure represents an average depiction of the trends observed in each field, including the ‘time-reversed J-curve’ characteristic observed in the US–China relationship.

Fig. 3: Change in the Collaboration Distance among the scientific powerhouses in natural sciences over time.
figure 3

In addition to the observed values for 1990–2021, simulation results until 2030 based on specific scenarios (A–C) are shown for the US–China pair.

Full size image

Regarding the distances between the US and China, the data from 1970 to 2021 is based on actual observed values, while the graph from 2022 to 2030 is a simulation result based on certain assumptions. Three scenarios—A, B and C—are presented. Scenario A depicts a situation where the distance between the two countries diverges at the same rate as it had been rapidly converging, almost like rewinding a clock. In other words, it represents a scenario where the distance between the two countries widens at the same pace as it had been narrowing. By contrast, Scenario B shows a moderate progression where the distance increases but at a slightly slower growth rate. Lastly, Scenario C suggests that although the distance between the US and China will continue to increase for the next few years, it will soon transition back to a decreasing trend.

Figure 4 illustrates the ‘Shrinking World’ of science collaboration in the natural science disciplines over the past 50 years, focusing specifically on the US and China and simplifying the world into three parties: the US, China and the Rest of the World (RoW). The RoW represents the collective scientific output of the top 50 countries, following the US and China regarding work production.Footnote 7 The visual representation depicts the world as a triangle, with snapshots provided at five-year intervals. The triangle’s base is formed by two points representing the US and China, and the length of the line segment connecting these two points represents the distance between the US and China (\({D}_{\text{US}-\text{CN}}\)). The apex of the triangle represents the RoW, and the lengths of the line segments connecting the apex to the endpoints of the base (\({D}_{\text{US}-\text{RoW}}\), \({D}_{\text{CN}-\text{RoW}}\)) represent the distances between the RoW and the US and China, respectively. The height (H) of the triangle is also plotted along the vertical axis.

Fig. 4: ‘Shrinking (and-possibly-Polarising) World’.
figure 4

The data prior to 2021 represents actual values obtained from observational data, while the data from 2022 onwards illustrates results derived from simulations based on Scenario B. The abbreviation ‘RoW’ stands for ‘Rest of the World’s top 50’, comprising the top 50 countries, following the US and China in terms of work production. The decreasing trend in the area of the triangle formed by the US, China and RoW signifies the visualisation of the ‘Shrinking World’.

Full size image

From the figure, it is evident that over the past half-century, the area of the ‘US–China–RoW Triangle’ has rapidly decreased, with some fluctuations, as indicated by the decreasing lengths of all three sides (as shown in Fig. 3). The widening distance between the US and China from 2019 to 2022 is reflected in the expansion of the base (\({D}_{\text{US}-\text{CN}}\)) of the triangle. The plot from 2022 to 2030 in Fig. 4 is a simulation based on Scenario B from Fig. 3, provided for reference. While \({D}_{\text{US}-\text{RoW}}\), \({D}_{\text{CN}-\text{RoW}}\), and the height H decrease almost linearly, \({D}_{\text{US}-\text{CN}}\) shows an increasing trend, resulting in the flattening and stretching of the triangle. If Scenario A were adopted, the triangle would become even more elongated. Conversely, if Scenario C were adopted, all three sides would become shorter, resulting in a smaller triangle with a smaller perimeter and area. The realisation of Scenarios A to C in the future US–China relationship remains uncertain. Nevertheless, this triangular diagram remains an intriguing focal point, potentially offering a unique bibliometric lens to observe the US–China relationship. The discussions in this section will benefit future deliberations on international collaboration in the policy arena.

Change in the ‘Knowledge Flow’ trend

We also analysed the Knowledge Flow Rate (KFR) for each discipline at level 0 in OpenAlex. Our analysis focused on a group of the top 199 researchers in work production each year.Footnote 8 In contrast to the broader sample used in the Collaboration Distance analysis, our study on the KFR necessitated a narrower focus on the period between 2000 and 2021. This particular time frame was chosen due to the stabilisation of the sum of off-diagonal elements (\({\Sigma }_{\text{off}}\)), ensuring the reliability of our analysis. Cases where \({\Sigma }_{\text{off}}\) was less than 10 for a given year were excluded from the calculation of the KFR index. Similar to the Collaboration Distance analysis detailed in the preceding section, our investigation also centred on the identical set of five parties: the US, China, EU27, the UK and Japan.

The analysis results for each discipline are summarised in Supplementary Fig. S2. A key observation emerges: over the past two decades, two KFR levels between the US and China (i.e. \({{\mathcal{K}}}_{\text{US}\to \text{CN}}\) and \({{\mathcal{K}}}_{\text{CN}\to \text{US}}\)) consistently outperformed other pairs, a trend evident across all disciplines. Additionally, an intriguing trend becomes apparent across various natural science disciplines. Since the transition from 2018 to 2019, there has been a notable decrease in the KFR from China to the US. This shift implies that China is redirecting the outflow of knowledge away from the US towards other entities. Indeed, Supplementary Fig. S3 shows that the KFR from China to the EU27 has been increasing in recent years. This observation aligns effectively with the results obtained from the Collaboration Distance analysis (Supplementary Fig. S1). Furthermore, the relatively subdued trend observed in the HSS disciplines, as opposed to the natural sciences, in terms of the decreasing trend post-2018, also aligns with the findings from our Collaboration Distance analysis.

Given the consistent trends across various natural science disciplines, it is valuable to combine these disciplines and derive an average representation of knowledge flow within the domain of natural sciences. Such an analysis can provide insights into overall trends across all level-0 natural science disciplines. To avoid overwhelming data, Fig. 5 focuses on the average KFR between the US and China, while Supplementary Fig. S3 presents the remaining results. In Fig. 5, the KFRs within natural science disciplines between the US and China have shown a marked upward trend since 2000. Specifically, both \({{\mathcal{K}}}_{\text{US}\to \text{CN}}\) and indicators, which were below 20% of total knowledge flow among the five parties in 2001, and exceeding 30% by 2018. This highlights a steady growth in knowledge exchange between the US and China facilitated by productive researchers, surpassing exchanges between other pairs of countries. This trend reaffirms the deepening connection over the past two decades. However, as mentioned earlier, this trend ceased to persist after the transition from 2018 to 2019.

Fig. 5: Change in the Knowledge Flow Rate (KFR) between the US and China in natural sciences over time.
figure 5

The KFR index is multiplied by 100 to express it as a percentage.

Full size image

Noteworthily, the asymmetrical nature of the KFR matrix, where \({{\mathcal{K}}}_{ij}\ne {{\mathcal{K}}}_{ji}\), offers the potential for more detailed insights compared to Collaboration Distance analysis. As an example, during the period from 2018 to 2021, many natural science disciplines demonstrated a decline in \({{\mathcal{K}}}_{\text{CN}\to \text{US}}\), whereas \({{\mathcal{K}}}_{\text{US}\to \text{CN}}\) displayed a distinct zigzag pattern: it increased from 2018 to 2019, decreased from 2019 to 2020, and then increased again from 2020 to 2021. This phenomenon is consistently observed across various natural science disciplines (Supplementary Fig. S2), especially in Mathematics, Computer science, Biology, Physics, Chemistry, Environmental science and Medicine. With its significance on a global scale, the asymmetrical trend of knowledge flow between the US and China also holds intriguing implications across various domains.

Summary, conclusions and discussions

Summary and conclusions

This paper aimed to provide S&T policymakers, diplomats, relevant researchers and practitioners with substantial quantitative evidence on the evolution and current state of international research collaboration. This examination focused on the US, China, EU27, the UK and Japan, encompassing various research domains. The following quantitative evidence was presented:

  1. 1.

    The world has consistently experienced a shrinking trend attributed to the deepening of international collaborative research—the ‘Shrinking World’.

  2. 2.

    Over the past few decades, there has been a rapid decrease in the distance between the US and China compared to other pairs of countries.

  3. 3.

    However, starting in 2019, signs of divergence have emerged between the US and China.

  4. 4.

    This divergence trend is not limited to any specific field but spans a wide range of disciplines. It is particularly prominent in the natural sciences and relatively less pronounced in the Humanities and Social Sciences (HSS).

  5. 5.

    The observed divergence between the US and China is unique to their relationship and is not kept in other pairs of parties.

Considering the fourth and fifth characteristics outlined above, it becomes evident that the observed divergence between the US and China cannot be solely attributed to universal factors like the impact of COVID-19, which affected all countries. Had these factors been the sole drivers, we would anticipate similar effects in other pairs of countries. Furthermore, this divergence does not stem solely from the characteristics of specific research disciplines but rather represents unique dynamics between the US and China. Hence, it is reasonable to associate this divergence with underlying geopolitical factors that significantly influence the dynamics of international research collaboration. These findings are consistent with findings in prior studies (Aghion et al., 2023; Jia et al., 2022; Tang, 2024; Van Noorden, 2022; Wagner and Cai, 2022; Xie et al., 2023) and complement the preliminary discussion presented in Okamura (2023), which suggested the concept of a ‘Polarising World’ of science collaboration from a bibliometric perspective.

The above findings were derived from an analysis of Collaboration Distance, computed based on coauthorship relationships between each pair of countries (parties). This paper also centred on a group of highly productive authors to offer supplementary validation from a different perspective. The investigation analysed the Knowledge Flow Rate (KFR) established among the five parties, examining how authors’ affiliations changed over time. The observed outcomes aligned with the findings from the Collaboration Distance approach, thus reinforcing their reliability.

Implications

While an increasing body of literature addresses the recent divergence between the US and China and the decline in research collaboration, the number of quantitative studies illustrating these trends remains limited. One notable study by Van Noorden (2022), which utilised data from the Scopus database, showed a substantial decrease of over 20% in the number of scholars affiliating with both China and the US in their research papers after 2019. The study also highlighted a reduction of research papers resulting from collaboration between the US and Chinese authors. Another study by Xie et al. (2023), employing data from the MAG database, observed that Chinese scientists based in the US have been increasingly inclined to leave the country, with a steady rise in return migration. Additionally, these researchers were less motivated to apply for US federal government grants. Furthermore, a study by Jia et al. (2022) focused on the life sciences field, utilising data from PubMed and Dimensions, and underscored the adverse effects on US-based scientists collaborating with Chinese coauthors. This impact was particularly evident in diminished perceived influence, measured by the number of citations their published papers received. Complementing these findings, an additional study by Aghion et al. (2023), based on the Scopus database, demonstrated that the China Initiative had a noteworthy negative impact on the average quality of publications (measured by citation counts) and on the coauthors of Chinese researchers with prior collaborations in the US (measured by the average H-index).

The current paper utilised data from OpenAlex, an Open Bibliometrics platform, to offer consistent insights that align with the studies above highlighting the declining trend in the US–China relationship in recent years. Our analysis is significant in capturing the convergence–divergence dynamics across a diverse range of research disciplines. We provided quantitative evidence using two distinct methodologies: The Collaboration Distance approach based on work (paper) identifiers and the KFR approach based on author (researcher) identifiers.

Limitations and future perspectives

Finally, we address the limitations of this study and propose potential avenues for future research. The use of the OpenAlex dataset, while advantageous, also presents certain limitations and considerations regarding the validity of the conclusions drawn, as noted by Okamura (2023; 2024). OpenAlex provides extensive data in a user-friendly format and continues to enhance its service quality. However, commercial databases may excel OpenAlex in specific aspects of data quality. Recent studies (Khanna et al., 2022; Liu et al., 2023; Nguyen et al., 2022) have assessed the data quality and utility of OpenAlex for bibliometric analysis in comparison with other major bibliometric databases, such as MAG, Web of Science (WoS), Dimensions and Google Scholar. These studies highlight the relative strengths of OpenAlex, depending on the context and perspective. Accordingly, comparing the findings of this study with those derived from other databases is a valuable exercise. To validate the reliability of the OpenAlex dataset, we performed the same analyses described in this study using data obtained from WoS and Scopus. The results demonstrated that the findings based on OpenAlex data were consistent and closely aligned with those derived from WoS and Scopus (see Supplementary Fig. S4). This validation confirms that the Open Bibliometrics approach employed in this study, utilising OpenAlex data, is both reliable and credible.

In addition, the validity of the Collaboration Distance and the KFR index employed in this paper as representations of bilateral relations and temporal shifts within academia raises an important question. The concept of a ‘close relationship’ among academic entities can be interpreted in various ways, suggesting the need for additional methods to examine these connections. Besides the undirected-graph approach of coauthorship analysis, there is interest in using a directed-graph technique that focuses on researcher-level analysis, as preliminarily demonstrated in this study. Enhancing the accuracy of the original data source is crucial to refine the precision of Knowledge Flow analysis. Moreover, broadening the scope to encompass more researchers and countries (parties), thereby increasing the sample size, would contribute to more robust outcomes. This study used the OpenAlex API, which imposed a maximum cap of 199 authors for simultaneous analysis. Nevertheless, future efforts should apply a similar methodology to a more extensive and comprehensive database to validate the findings established in this paper.

Furthermore, the division of roles within author teams in international coauthored papers, particularly the leadership role of the corresponding author, is highly significant. This division is considered a factor influencing citation counts and productivity, and it is also linked to issues of hierarchy in international collaboration for scientific knowledge production. Several previous studies have addressed and discussed this topic (de Moya-Anegon et al., 2018; Miao et al., 2024). In our current paper, however, our primary focus is on the distance between countries, defined by Eq. (1), and the specific allocation of roles within author teams was not included as a variable in our quantitative analysis. Indeed, the actual distribution of roles and the nature of leadership by corresponding authors can vary greatly depending on the research field. Nonetheless, future work could incorporate such factors, examining the roles within author teams at a finer level of detail. Such an approach could offer additional insights into the conclusions of this study from another perspective.

Finally, the approach presented in this paper offers a way to uncover the factors underlying the observed results, particularly the phenomenon of the ‘time-reversed J-curve’ between the US and China. It is crucial for future international research collaboration on a global scale to examine which of the three scenarios (A–C) outlined in this paper aligns most closely with the observed trend and to explore the reasons behind this. Such detailed analysis would provide valuable insights for both policymakers and academics.