Exploring Scientific Collaboration Patterns from the Perspective of Disciplinary Difference: Evidence from Scientific Literature Data

Abstract

With the accelerating globalization and rapid development of science and technology, scientific collaboration has become a key driver of knowledge production, yet its patterns vary significantly across disciplines. This study aims to explore the disciplinary differences in scholars’ scientific collaboration patterns and their underlying mechanisms. Data were collected from the China National Knowledge Infrastructure (CNKI) database, covering papers from four disciplines: mathematics, mechanical engineering, philosophy, and sociology. Using social network analysis, we examined core network metrics (degree centrality, neighbor connectivity, clustering coefficient) in collaboration networks, analyzed collaboration patterns across scholars of different academic ages, and compared the academic age distribution of collaborators and network characteristics across career stages. Key findings include the following. (1) Mechanical engineering exhibits the highest and most stable clustering coefficient (mean 0.62) across all academic ages, reflecting tight team collaboration, with degree centrality increasing fastest with academic age (3.2 times higher for senior vs. beginner scholars), driven by its reliance on experimental resources and skill division. (2) Philosophy shows high degree centrality in early career stages (mean 0.38 for beginners) but a sharp decline in clustering coefficient in senior stages (from 0.42 to 0.17), indicating broad early collaboration but loose later ties due to individualized knowledge production. (3) Mathematics scholars prefer collaborating with high-centrality peers (higher neighbor connectivity, mean 0.51), while sociology shows more inclusive collaboration with dispersed partner centrality.

1. Introduction

With the rapid development of science and technology, researchers worldwide are becoming increasingly connected. The production of new knowledge is presented in more complex patterns, making it impossible for individual scientific research to adapt to the large-scale and rapid emergence of knowledge. The deepening of scientific collaboration has significantly advanced the progress of human civilization. Research shows that it can significantly promote the growth of research outcomes and drive continuous innovation [1,2,3]. Collaborative relationships in academia are crucial for scientific productivity [4,5]. The subjects of scientific research have gradually shifted from individualization to collaboration, with scientific collaboration continuously strengthening and moving towards a broader scope and more profound level. In modern society, the boundaries between different disciplines are increasingly blurred. The emergence and resolution of new problems often involve multiple disciplines, necessitating interdisciplinary collaboration. Therefore, scientific collaboration is crucial for promoting the production of new knowledge, improving research output efficiency, and meeting societal development needs [6,7,8].

In the context of the ongoing development of scientific collaboration, the analysis of it becomes particularly important. Furthermore, the patterns of scientific collaborations vary across different disciplines [9]. Different disciplines possess distinct disciplinary attributes, and the frequency of scientific collaboration as well as the demand for collaboration also vary accordingly. Thus, studying disciplinary differences in scholars’ collaboration patterns not only opens a window to understand these differences better but also offers insights for policy making [10,11]. As the most fundamental component of scientific collaboration, individuals play a pivotal role in conducting scientific collaboration, which is of significant importance for the success and outcomes of such collaborations.

This paper refines the study of scientific collaboration by focusing on individual scholars and considering disciplinary characteristics. We focus on three research questions:

• What are the differences in collaboration patterns among these four disciplines: mathematics, mechanical engineering, philosophy, and sociology?
• How do these network characteristics dynamically change with scholars’ academic ages?
• What reasons lead to the aforementioned differences?

To answer the above questions, this paper employs social network analysis methods to explore scholars’ collaboration patterns based on scientific literature data. We first categorized scholars in the four disciplines (mathematics, mechanical engineering, philosophy, and sociology) into three academic age stages: beginner, junior, and senior. Then we find that the number of scholars decreases with increasing academic age, with those at academic age 1 accounting for the highest proportion (approximately half). This is associated with the short academic careers typical of graduate students. The collaboration patterns of scholars in different disciplines are significantly different. Mechanical engineering is dominated by team collaboration, with consistent collaboration preferences across all academic age stages. The senior–beginner–senior pattern accounts for 44.4%, reflecting a “senior-led, beginner-implemented” division of labor. Beginner philosophy scholars collaborate widely, while senior philosophy scholars collaborate with beginners in approximately 50% of cases, with same-stage collaborations increasing with age. Senior mathematics scholars show a “U-shaped” collaboration pattern with high homophily among peers. Sociologists across all stages tend to collaborate with beginner scholars, while junior sociologists collaborate most frequently with their peers.

Furthermore, there also exist differences in the degree centrality (DC), clustering coefficient (CC), neighbor connectivity (NC) of collaboration networks across four disciplines. DC increases with academic age across all disciplines. Philosophy shows the highest DC in the beginner and junior stages; mechanical engineering exhibits the fastest DC growth after academic age 15; mathematics and sociology follow similar trends, with mathematics growing more rapidly in later stages. Mechanical engineering maintains high and stable CC across all stages (reflecting tight team collaboration). Philosophy experiences a sharp decline in CC in the senior stage, indicating loose collaboration due to diverging research directions. Mathematics and sociology show smaller fluctuations in CC. Mathematics scholars tend to collaborate with high-centrality peers, while sociology shows a more dispersed distribution of partner centrality. Philosophy exhibits periodic declines in NC.

2. Literature Review

2.1. Scientific Collaboration Patterns

Collaboration is a fundamental form of human society, and scientific collaboration is a common phenomenon in academia [12,13]. In recent years, the presentation of research outcomes has shifted from individual researchers to multiple researchers, from single institution to multiple institutions, and from national to international levels [14]. Research on scientific collaboration patterns falls into two main categories; the first focuses on different subjects of collaboration, and the second analyzes disciplinary differences in scientific collaboration.

Based on the different subjects of collaboration, scientific collaboration can be divided into international collaboration, inter-institutional collaboration, and collaboration among scholars. From the perspective of international collaboration, studies have shown that smaller and less developed countries have higher collaboration rates [15]. The world has formed a “quadrilateral” network of scientific collaboration, with the United States, Western Europe, China, and Australia as the vertices. Its degree centrality has formed a “one superpower, multiple powers” pattern, with the United States as the core, and Canada, Australia, China, and Western European countries as the sub centers. Concurrently, the global scientific network is quietly reshaping the world order, with rapid growth in China, South Africa, India, and Brazil. The robust research development in the Middle East, Southeast Asia, and North Africa accelerates the multipolarity of global science and technology research centers, fostering a new global scientific innovation landscape [16]. From the perspective of cross-institutional collaboration, Liu et al. analyzed the collaboration network among Chinese universities, discovering that cross-university collaboration is continuously increasing [17]. Using social network analysis methods, Chen concluded that cross-institutional collaboration exhibits a significant central effect and geographical characteristics [18]. Marijan and Sen conducted an in-depth analysis of cross-institutional collaboration, finding severe polarization in research dominance [19]. They also found that some crucial factors, including cognitive background, institutional environment and cultural environment, significantly impact scientific collaboration. Additionally, there is a correlation between academic influence and cross-institutional collaboration.

At the micro level, scholars are the primary subjects of scientific collaboration and the smallest units of such collaboration. From the perspective of ages, in co-authored papers, the average age of the first authors in fundamental disciplines, such as mathematics and physics, is between 40 and 50 years old, while in management disciplines, the peak age range is 22–45 years, indicating that young and middle-aged scholars are the main force in scientific collaboration. In the field of computer science, junior scholars in the early stages of their careers are more willing to collaborate with senior scholars [20]. When classified by gender, there are differences in how male and female scholars select their collaborators and ways of collaboration [21]. Female scholars tend to engage more in intra-institutional collaborations, whereas male scholars frequently engage in international collaborations [22,23,24]. In the field of economics, female scholars tend to collaborate primarily in the first year of their career, whereas male colleagues focus more on building a solid reputation through independent publications to facilitate future collaborations [25,26]. In addition, female academic scientists do not face obstacles in collaboration, whether in the size of their collaboration network or in crossing specified boundaries [27]. Meanwhile, different disciplines exhibit different collaboration patterns. Over the past decade, the field of physics has maintained over 60% of collaborations, while collaboration in other disciplines continues to increase. Research indicated that the proportion of international collaboration is higher in fundamental research fields compared to non-fundamental disciplines [28]. Overall, there is an increasing trend in co-authorship of scientific papers, but there are significant differences in collaboration patterns among different disciplines. The co-authorship rate of humanities and social sciences is generally lower than that of natural sciences.

2.2. Influencing Factors of Scientific Collaboration

Scientific collaboration is a complex behavior. Generally, when the interests of scholars align or each scholar’s needs are met, the scientific collaboration occurs. The factors influencing collaboration are numerous, and discussions about these factors can mainly be divided into three aspects: economic factors, knowledge production, and social reputation:

• Economic factors: Economic factors are the foundation for conducting academic research and are also a significant consideration for researchers when engaging in collaboration. During the process of academic research, the costs, such as the purchase of large experimental equipment, must be considered [29]. Research shows that scientific collaboration among scholars can effectively reduce research costs, and sharing experimental equipment is an important driving force in the collaboration [30]. Birnholtz found that scientific collaborations, such as establishing joint laboratories and setting up joint projects, can provide collaborators with facilities and platform support [31]. Lee conducted a survey of members of the National Academy of Sciences and Engineering in the United States and found that the economic benefits and other related resources obtained are important factors in promoting the scientific collaboration [32].
• Knowledge production: An important reason for engaging in scientific collaboration is to promote knowledge production. In recent decades, knowledge has become increasingly complex, necessitating collaboration among scholars from multiple fields to produce new knowledge [33]. Authors publishing in high-quality journals tend to have stronger common research interests with their co-authors. Core authors are experts in their fields, and they prefer to collaborate with experts who have similar research interests to delve into specialized research topics rather than broadly studying multiple fields superficially [34]. Additionally, with the refinement of knowledge and disciplinary classification, interdisciplinary integration has become a new research trend. Wu et al. believe that to address the needs of interdisciplinary research, collaboration among scholars from different disciplines can help them inspire each other, leading to the generation of new ideas [35]. A theory of collaborative knowledge production has been proposed, indicating that the knowledge production efficiency is highest when collaborators’ shared and differing knowledge is at a balanced level [36]. Research also indicates that the achievements of scientific research remain a determining factor in scientific collaboration [37].
• Social reputation: Scientific collaboration is an important means of maintaining social relationships and is also a necessary way to enhance social reputation. Scholars can maintain existing social relationships and create new ones through scientific collaboration, thereby accumulating human capital [38]. Research suggests that scholars tend to collaborate with peers from external organizations, which can publish academic articles in high-quality journals, and gain a good academic reputation [39]. Brands et al. found that scientists’ engagement in scientific collaborations is driven by utilitarian motivation, aimed at enhancing their reputation or gaining professional advantages through collaboration [40]. When researchers have achieved a certain social status, their motivation to publish papers increasingly stems from the pursuit of social reputation [10].

3. Data and Research Methodology

3.1. Dataset

This study investigates the differences in scientific collaboration patterns among scholars across various disciplines. We select mathematics, mechanical engineering, philosophy, and sociology as representative disciplines, and analyze these differences. The data for this study is acquired from the China National Knowledge Infrastructure (CNKI) database, which is the largest continuously updated journal database in China. CNKI includes a wide range of Chinese academic resources, such as domestic journals, dissertations, newspapers, and yearbooks, covering multiple disciplines, including basic sciences, engineering technology, agricultural science, medical and health science, philosophy and humanities, social sciences, information technology, and economics and management science. The article data used in this study is derived from the CNKI database spanning from 2000 to 2020, including information on authors, publication dates, and journals’ information, which meets the needs of this study. The dataset includes 156,164 papers in mathematics, 142,010 papers in mechanical engineering, 67,264 papers in philosophy, and 122,945 papers in sociology. Based on these literature raw data, we construct four collaboration networks. The specific information for each network is as follows. The mechanical engineering collaboration network contains 77,407 nodes and 207,412 edges; the sociology collaboration network contains 50,383 nodes and 32,148 edges; the philosophy collaboration network contains 12,357 nodes and 8916 edges; and the mathematics collaboration network contains 91,057 nodes and 69,753 edges.

Our dataset, derived from the China National Knowledge Infrastructure (CNKI) database covering the period 2000–2020, differs from many existing collaboration network datasets in three key aspects:

Focus on Chinese Academic Literature: While numerous studies have explored collaboration networks using databases like Web of Science or Scopus, our dataset centers on Chinese-language scholarly publications across four disciplines (mathematics, mechanical engineering, philosophy, and sociology). This focus allows us to capture collaboration patterns within a distinct academic ecosystem that reflects China’s research landscape, which has been underrepresented in international studies.

Cross-Disciplinary Scope with Contrasting Knowledge Attributes: Our dataset intentionally includes disciplines spanning natural sciences (mathematics, mechanical engineering) and humanities/social sciences (philosophy, sociology). This design enables direct comparisons between fields with differing knowledge production modes—such as the experimental, team-based nature of mechanical engineering versus the more individualistic, theoretical focus of philosophy—revealing how disciplinary attributes shape collaboration networks.

Integration of Academic Age: Unlike many existing datasets that focus solely on static network structures, our data links collaboration ties to scholars’ academic age (defined as the span from their first to last publication). This allows us to analyze how collaboration patterns evolve across career stages (beginner, junior, and senior scholars), a dimension rarely emphasized in previous network studies.

3.2. Research Methodology

This study first defines the academic age of scholars and categorizes them into three phases based on their academic productivity: beginner scholars, junior scholars, and senior scholars. We investigate the scientific collaboration patterns and collaboration intensity, and compare the collaboration patterns over different periods among scholars from various disciplines. Additionally, we examine the characteristics of scholars’ collaboration networks, including degree centrality, neighbor connectivity, and clustering coefficient. These characteristics comprehensively describe the network properties of scholars’ ego networks, facilitating the study of scholars’ collaboration patterns. Regarding the use of VOSviewer in many current studies, our initial analytical focus was on exploring and verifying basic collaboration patterns. Therefore, in terms of tool selection, we used Python 3.13 to process the data and failed to introduce VOSviewer in a timely manner for term clustering and map analysis.

In this study, we first define the age of papers. For a given scholar, if someone has p papers numbered 1, 2, 3, … p, then we can calculate the age of the $j^{th}$ paper as follows:

$${\mathrm{AP}}_j=t_j-t_1$$

(1)

where $t_j$ is the publication year of the $j^{th}$ paper, and $t_1$ is the publication year of the first paper.

Then academic age of scholar is defined as:

$$\mathrm{AA}=t_p-t_1$$

(2)

Based on this definition, we categorize scholars into three career stages for our analysis:

• Beginner scholars: Academic Age ≤ 5 years.
• Junior scholars: 5 < Academic Age ≤ 15 years.
• Senior scholars: Academic Age > 15 years.

This approach allows us to dynamically track the evolution of collaboration patterns throughout a scholar’s career.

Generally, as academic age increases, scholars accumulate more knowledge, expand their collaboration networks, diversify their choice of collaborators, and exhibit more complex collaboration patterns, leading to increased academic productivity [41]. However, scholars’ academic productivity varies at different stages of their careers, and these stages impact their scientific collaboration patterns differently. Categorizing academic careers based on scholars’ academic productivity provides a new perspective for directly reflecting their scientific collaboration patterns.

A scholar’s ego network represents the connections between a central scholar and other scholars in the academic network, clearly indicating the relationships and collaboration intensity among scholars. A scholar’s ego network, consisting of the scholar (ego), their direct collaborators (alters), and the ties among these collaborators, is a fundamental tool for this analysis. By examining properties of these ego networks, such as the ego’s degree and the local clustering coefficient, we can uncover personal collaboration patterns [42,43]. We utilize ego networks to study changes in collaboration relationships among scholars of different academic ages. In the constructed collaboration network, node $N_i$ represents scholar i, and edge $E_{ij}$ indicates the cooperation relationship between scholar i and scholar j. In social network theory, degree centrality, neighbor connectivity, and clustering coefficient are frequently used metrics for studying collaboration relationships. These metrics are introduced as follows:

• Degree Centrality (DC). This metric quantifies the number of direct collaborators a scholar has. We represent the collaboration network as a graph $G=(V,E)$, where V is the set of scholars (nodes) and E is the set of collaboration ties (edges). The degree centrality of a scholar i, denoted as ${\mathrm{DC}}_i$, is simply its degree, $k_i$.

  $${\mathrm{DC}}_i=k_i$$

  (3)

  where $k_i$ counts the number of scholars with whom scholar i has directly collaborated.
• Neighbor Connectivity (NC). This metric measures the average centrality (degree) of a scholar’s collaborators, reflecting their tendency to collaborate with high- or low-degree peers. The neighbor connectivity for scholar i, ${\mathrm{NC}}_i$, is defined as the average degree of the nodes in its neighborhood.

  $${\mathrm{NC}}_i={\displaystyle \frac{1}{k_i}}\sum _{j\in \mathcal{N}\left(i\right)}{k}_j$$

  (4)

  where $\mathcal{N}\left(i\right)$ is the set of neighbors of node i, $k_i$ is the degree of node i and $k_j$ is the degree of a neighboring node j.
• Clustering Coefficient (CC). We use the local clustering coefficient to measure how tightly clustered a scholar’s collaborators are (i.e., the probability that any two of their collaborators also collaborate with each other). For a scholar i with degree $k_i\ge 2$, its local clustering coefficient, ${\mathrm{CC}}_i$, is the ratio of the number of actual edges between its neighbors ($e_i$) to the maximum possible number of edges between them.

  $${\mathrm{CC}}_i={\displaystyle \frac{2e_i}{k_i(k_i-1)}}$$

  (5)

  where $e_i$ is the number of edges connecting the $k_i$ neighbors of scholar i to each other. A higher ${\mathrm{CC}}_i$ indicates a more cohesive or tightly knit collaboration circle around scholar i.

4. Results

4.1. Distribution of Scholars’ Academic Ages

To categorize scholars into different academic age groups, we use their academic productivity within each age segment as a criterion. The metric measured in this study is the average annual publication count, i.e., the number of papers published by the scholar per year. As shown in Figure 1a, the average annual publication trends for scholars in the four disciplines—mathematics, mechanical engineering, philosophy, and sociology—are similar. Initially, or scholars with $AcademicAge<5$, the productivity sharply increases at academic age 2, then shows a downward trend. During the 5–15 academic age phase, the productivity of scholars in all disciplines remains stable. When $15\le AcademicAge$, the productivity shows a sharp increase, reaching the highest value during the entire academic career phase. This indicates that academic productivity varies across different academic age stages. To better explore the scientific collaboration patterns from the perspective of academic age, the academic career can be divided into three stages: early stage, mid-career stage, and senior stage. Based on these stages, scholars can be grouped as follows: beginner scholars ($AcademicAge<5$), junior scholars ($5\le AcademicAge<15$), and senior scholars ($15\le AcademicAge$). By analyzing the characteristics and collaboration patterns of scholars during their academic careers and conducting comparative analyses across disciplines, we can identify differences and investigate the underlying reasons.

We then calculate the academic ages of authors in these four disciplines. As shown in Figure 1b, the number of scholars decreases with the increasing academic ages. In Chinese academic papers, scholars with an academic age of 1 comprise the majority. During the 2–12 academic age phase, the number of scholars in each discipline shows a gradual decline. In the 12–20 academic age phase, the number of mathematics scholars decreases rapidly, while the numbers in mechanical engineering and philosophy stabilize. The number of sociology scholars is the lowest at the academic age of 18. It can be seen that scholars with an academic age of 1 account for about half of the total number of scholars, suggesting that most scholars have a short academic career. It is well known that graduate students constitute a significant group in academic research. Many of them publish academic papers during their graduate education and submit their dissertations upon graduation. However, many students no longer engage in academic research after completing their graduate degrees, resulting in a relatively short academic career. This also explains why the number of scholars peaks when their academic age is 1, then is followed by a rapid decline.

4.2. Scholars’ Network Characteristics

Network characteristics can describe scholars’ attributes within the academic network, illustrating their positions and relationships. As academic age changes, scholars’ network characteristics also shift, and these shifts vary across disciplines. Figure 2 demonstrates the trends in DC, NC, and CC as academic age changes.

DC, representing a scholar’s degree, directly measures their number of unique collaborators. As shown in Figure 2, DC increases significantly with academic age across all four disciplines. Generally, the increase is most rapid during the first two years of a scholar’s career (from Academic Age 1 to 2), and followed by a steady rise. Philosophy scholars consistently exhibit higher DC across all stages. Mechanical engineering scholars have slightly lower DC than philosophy scholars before academic age 15, after which it fluctuates but generally increases. The trends for sociology and mathematics scholars are quite similar, with sociology scholars showing higher DC. However, the gap narrows with the increase in academic age, indicating a faster increase in DC for mathematics scholars.

NC represents the average degree of a scholar’s neighbors, indicating whether scholars tend to connect with others who are themselves highly or sparsely connected. Between academic ages 1 and 2, it rises sharply. Thereafter, trends for mechanical engineering, mathematics, and philosophy are similar, showing a steady increase, suggesting that scholars prefer collaborating with those having higher DC. Philosophy scholars show notable drops at academic ages 3, 8, and 14. Sociology scholars’ NC steadily increases, indicating a consistent preference for collaborating with high-DC scholars.

CC, the local clustering coefficient, reflects the cohesiveness of a scholar’s immediate collaboration circle. A high CC means a scholar’s partners are also likely to collaborate with each other. Mechanical engineering scholars maintain a relatively high and stable CC across all ages, with a slight drop at academic ages 19–20. In the first two years of academic career, the CC for the other three disciplines rises rapidly. Between academic ages 2 and 20, mathematics scholars have a stable CC, while sociology and philosophy scholars exhibit more fluctuations and a declining trend. Sociology scholars show smaller fluctuations, whereas philosophy scholars’ CC drops sharply at academic age 17, reaching the lowest point at academic age 18 before a slight rise.

These network characteristics reveal that as academic age increases, scholars’ collaboration scope expands, and the number of collaborators increases significantly. Scholars increasingly tend to collaborate with central scholars in the network, particularly in mechanical engineering. Among the four studied disciplines, mechanical engineering scholars exhibit the closest collaboration relationship, frequently co-authoring papers with two or more collaborators. Mathematics and sociology scholars show similar trends in CC, indicating comparable intensity in scientific collaboration. Philosophy scholars’ CC is similar to mathematics and sociology in the early and mid-career stages but differs significantly in the senior stage, where it is the lowest, suggesting less collaboration among senior philosophy scholars.

4.3. Scientific Collaboration Patterns

Figure 3 reveals the significant differences in scientific collaboration patterns among scholars in different disciplines. In the field of mathematics, beginning scholars have a low proportion of collaboration with other beginner scholars. Their collaborations with junior scholars increase as the academic age of the collaborators rises, peaking in collaborations with partners who have an academic age of 11. The proportion of collaboration between beginner scholars and senior scholars rises sharply during the 15–20 academic age range of the collaborators. Senior scholars show a “U-shaped” pattern of collaboration, with the highest proportions at academic ages 1 and 20. However, junior scholars collaborate more frequently with other junior scholars than senior scholars do, and junior scholars have a lower proportion of collaboration with senior scholars compared to senior–senior scholar collaboration. This indicates a higher homophily among young and senior scholars in mathematics, where those with high DC prefer collaborating with similar scholars.

In mechanical engineering, the collaboration patterns differ significantly from those in other disciplines. Beginner, young, and senior scholars exhibit almost identical preferences when choosing scientific collaborators. This phenomenon is likely due to the high CC value in this field, where research often involves teamwork. The overall trend shows a rapid initial decline in collaboration, followed by a stable period and a subsequent increase. Scholars in all three career stages show a higher proportion of collaboration with beginner scholars compared to other disciplines, indicating a stronger willingness to collaborate with beginner scholars. The increase in collaboration with senior scholars also suggests a preference for choosing experienced collaborators.

In philosophy, the proportion of collaborations between beginner scholars and scholars of various academic ages fluctuates. Clearly, beginner scholars collaborate most frequently with scholars who have an academic age between 18 and 20 years. Junior scholars show high collaboration with those at academic ages 2 and 20. This pattern indicates close collaboration between junior scholars and both beginner and senior scholars. Collaboration with beginner scholars is most common among senior scholars, making up about half of all collaborations. The collaboration frequency with junior scholars decreases with academic age but peaks with those at academic age 10, while collaboration among senior scholars increases steadily, indicating a growing preference for collaborating with peers.

In sociology, scholars in all three career stages exhibit similar preferences for choosing collaborators. Beginner scholars collaborate the most with those at academic age 1. Junior scholars show fluctuating collaboration frequencies with the increase in academic age, reaching the lowest point at age 18 before peaking again at age 20. Senior scholars exhibit similar collaboration patterns to beginner and junior scholars, with the highest collaboration proportion with beginner scholars. Junior scholars primarily collaborate with those academics aged 5–10, highlighting their importance and appeal within the sociology discipline.

The above analysis reveals several key differences:

• Beginner scholars in mathematics and philosophy collaborate less with peers, while those in mechanical engineering collaborate more. This indicates the varying preferences for choosing collaboration partners across disciplines.
• Junior scholars’ collaboration preferences differ by discipline. In mathematics, sociology, and mechanical engineering, junior scholars collaborate more with peers, while philosophy scholars prefer collaborating with beginner and senior scholars.
• Generally, senior scholars collaborate the most with peers. However, in philosophy, senior–senior collaborations are relatively low, potentially due to different research interests and directions among senior scholars.

To further explore differences in collaboration patterns, we examine the tripartite relationships in scientific collaborations. Due to the variety of collaboration patterns, this study focuses on the most common relationships. Figure 4 shows significant differences in collaboration patterns across disciplines. In mathematics and mechanical engineering, beginner scholars are almost involved in all collaborative relationships. In mathematics, the most common tripartite relationship is SBB, reflecting the typical advisor–student collaboration model. In mechanical engineering, beginner scholars are very active, with the most common pattern being BSB. Here, senior scholars take central roles, with beginner scholars handling task division and implementation. In philosophy, the SBB pattern dominates at 39.4%, consistent with the advisor–student model. In sociology, the highest proportion is BBB, reflecting collaboration among beginner scholars who split tasks to achieve research goals.

4.4. Academic Age Distribution of Collaborator Pairs

The number of scholars at different academic ages varies across disciplines, and the network characteristics of scholars at different career stages also differ. When engaging in research activities and collaborations, scholars of different academic ages tend to choose collaborators based on their characteristics. Analyzing the academic age distribution of collaborator pairs can provide a clear insight into the differences in how scholars at various stages select their research partners. This deepens the understanding of scientific collaboration patterns. Due to significant variations in the number of collaborative papers, the data was normalized, and the final results are shown in Figure 5. In the figure, the spectrum from yellow to blue represents the distribution of collaborators’ academic ages, with deeper colors indicating more frequent collaboration and a higher number of collaborative activities at that academic age.

In the field of mathematics, when the academic age is 1, scholars collaborate frequently with those of all academic ages except for ages 1 and 18. As the academic age increases, collaboration with scholars at the academic age of 1 also increases. Additionally, junior scholars have significant collaboration across various academic ages. This suggests that junior scholars actively seek collaborators and establish partnerships during their research activities. In mechanical engineering, a prominent blue spectrum appears at academic age 1, indicating that scholars at this stage are highly active in scientific collaboration. The figure also reveals a noticeable bright diagonal line with darker colors at both ends and lighter colors in the middle. This suggests that there is a tendency for scholars to collaborate with others of similar academic ages. This corresponds to the actual scientific research mode in mechanical engineering, in which scientific research activities are mostly carried out by teamwork. Within the research team, senior scholars often act as the “leader” of the team, determining the research direction, guiding team members to carry out scientific research, and beginners and junior scholars are mainly responsible for carrying out specific research tasks. In philosophy, the green spectrum is more prominent in the academic age range of 15–20, with collaborators’ academic ages mostly concentrated in the 3–10 range. This indicates that as academic age increases, collaboration with scholars at academic age 1 also increases. In sociology, a common feature with the other three disciplines is that scholars at academic age 1 collaborate with scholars of all other academic ages. In academia, scholars with an academic age of 1 are mostly fresh graduate students who have just begun their academic research. Collaboration with junior scholars and senior scholars continues to increase, and this phenomenon is similar to the collaboration model between advisors and students. However, the overall light color in the figure indicates fewer collaborative activities in sociology, suggesting weaker scientific collaboration relationships among scholars.

4.5. Comparison of Network Characteristics Across Different Time Periods

Through the above analysis of scholars’ scientific collaboration behaviors, we can see that significant differences exist between disciplines. As scientific collaboration has accelerated in recent years, it is essential to examine whether the network characteristics of collaboration behaviors differ across periods. To this end, we compare the network characteristics of scholars’ scientific collaborations across two time periods: 2000–2002 and 2018–2020.

In the field of mathematics (see Figure 6), the degree centrality, neighbor connectivity, and clustering coefficient of scholars in 2000–2002 are significantly lower than those in 2018–2020. This indicates that modern scholars are more inclined towards collaboration, with a higher degree of centrality suggesting a greater number of collaborators. The differences in neighbor connectivity and clustering coefficient are even more pronounced. In 2000–2002, the neighbor connectivity was 0 at an academic age of 3, while in 2018–2020, it showed a significant increase. Similarly, the clustering coefficient was 0 in 2000–2002, indicating minimal scientific collaboration, whereas it significantly increased in 2018–2020. This demonstrates a marked increase in collaborative behavior among scholars over time. Additionally, as academic age increases, the differences in network characteristics between the two periods diminish, suggesting that for senior scholars, temporal differences in network characteristics are less significant.

In mechanical engineering (see Figure 7), the overall changes in scholars’ network characteristics are not obvious. However, there is a noticeable change among beginner scholars. The degree centrality of new-generation beginner scholars has increased significantly, indicating a higher level of engagement in collaboration. The neighbor connectivity for beginner scholars in 2018–2020 is lower than that in 2000–2002, reflecting a more diversified pattern of collaboration. In 2000–2002, beginner scholars had missing degree centrality and clustering coefficients, indicating minimal collaboration and lack of clustering. By 2018–2020, the clustering coefficient remained stable across academic ages, indicating a well-established and stable team collaboration model in mechanical engineering.

Philosophy scholars (see Figure 8) show significant changes in network characteristics across the two periods. Degree centrality was significantly lower in 2000–2002 compared to 2018–2020, reflecting increased collaborative activities in recent years. The neighbor connectivity for beginner and junior scholars in their initial stages also shows a significant change, with near-zero values in 2000–2002, and rising considerably in 2018–2020. This suggests that beginner and junior scholars are more inclined to collaborate with highly central scholars. Senior scholars show stable neighbor connectivity across both periods, indicating steady collaboration patterns. The most notable change is in the clustering coefficient; in 2000–2002 senior scholars chose to collaborate only during academic maturity time, with beginner and senior scholars primarily working independently. By 2018–2020, the clustering coefficient increased across all academic ages, indicating a rise in collaborative behavior and closer relationships among the newer generations of scholars.

In sociology (see Figure 9), all three network characteristics have changed. From 2018–2020, the degree centrality of scholars increased across all academic ages, though the gap between the two periods narrows with increasing academic age. The trend in neighbor connectivity is similar, with scholars preferring choosing collaborators of similar centrality. The clustering coefficient shows significant differences between the two periods. In 2000–2002, beginner and junior scholars had near-zero clustering coefficients, indicating low collaboration willingness and minimal collaborative behavior. In 2018–2020, clustering coefficients were higher, with beginner and junior scholars having higher coefficients than senior scholars, indicating significant changes in collaborative behaviors within sociology across the two periods.

5. Discussion

Through the above analysis, it is evident that network characteristics and scientific collaboration patterns differ among scholars from different disciplines. These differences are significant for disciplinary development and knowledge production, prompting further exploration into the reasons behind these inter-disciplinary variations. We summarize the reasons for the differences from two perspectives: (1) attribute differences across disciplines; (2) development differences of disciplines.

5.1. Attribute Differences Across Disciplines

5.1.1. Different Knowledge Attributes

According to “the representation of reality” [44], which refers to the differences between disciplinary research characteristics and the real world, scientific knowledge can be classified into pure hard sciences, pure soft sciences, applied hard sciences, and applied soft sciences. This classification highlights the distinctive features of various disciplines based on their knowledge attributes, which in turn affect scholars’ scientific collaboration. For pure hard sciences like mathematics, the high number of senior scholars indicates that research in mathematics requires sufficient knowledge accumulation. In collaboration relationships, the pattern often involves senior scholars leading beginner scholars, reflecting the rigorous logical framework and clear knowledge boundaries inherent in mathematics. Knowledge production in mathematics is systematic and holistic, making the decomposition of the research process challenging, resulting in relatively less collaboration and more homogeneous collaboration patterns.

In applied hard sciences like mechanical engineering, the frequency of collaboration is higher, with a high clustering coefficient. Research is typically conducted in teams, with a collaboration pattern involving senior, young, and beginner scholars. Mechanical engineering knowledge must adhere to both practical and deductive logic, balancing application with internal consistency. Knowledge production is driven by social needs, utilizing mathematics, physics, and mechanics as foundational disciplines, with interdisciplinary collaboration promoting continuous knowledge advancement. In pure soft sciences like philosophy, scientific collaboration is less frequent, often involving small, low-frequency collaborative groups. Philosophy, as a discipline that guides human existence and development, relies heavily on personal experiences and reflections, leading to a lower intensity of collaboration. Applied soft sciences like sociology, which studies social behavior and human groups, have a broad research scope that includes both micro-level interpersonal interactions and macro-level social systems. The dynamic and complex nature of social issues necessitates the integration of knowledge from psychology, history, political science, and economics, continuously evolving to address societal needs.

5.1.2. Different Organizational Attributes

Disciplines have two connotations: the knowledge aspect and the organizational aspect. The organization of disciplines is the main form of university disciplines. Universities organize multiple units around disciplines, with these units serving as functional entities that fulfill the university’s mission through academic activities. As disciplines emerge and develop, so do their organizational structures. University discipline organizations have their unique life cycles, characterized by three main elements: scholars, academic information, and academic resources.

The organizational attributes of disciplines significantly influence scientific collaboration behaviors. Factors, such as the basic elements of discipline organization, material conditions, academic culture, degree conferral, knowledge development, and social needs, impact the lifecycle of discipline organizations, resulting in different collaboration behaviors among scholars at different stages. In mathematics, while the essential elements are present resources, such as manpower, material, and financial support, are relatively weak, placing the discipline in a formative stage where senior scholars play a crucial role in collaborations. Mechanical engineering requires research platforms and academic resources to meet societal needs, showing a team-based collaboration model involving senior, young, and beginner scholars. Philosophy, an ancient discipline with a unique academic culture but limited resources, exhibits collaboration patterns primarily between senior and beginner or junior scholars. In sociology, junior scholars are the main drivers of collaborations, supported increasingly by resources and driven by the need to address current social issues.

5.2. Development Differences of Disciplines

5.2.1. Different Social Development Needs

Disciplines are responsible for producing new knowledge, and aligning with societal development and needs. Disciplines actively adapt to societal trends while maintaining their internal knowledge logic, solving societal problems proactively. The mutual influence of social progress and disciplinary development fosters a symbiotic relationship where social advancements push disciplinary growth, which in turn promotes societal progress.

The relationship between disciplinary development and social needs significantly impacts scholars’ scientific collaboration patterns. The selected disciplines of mathematics, mechanical engineering, philosophy, and sociology vary in their alignment with social development needs, resulting in distinct collaboration behaviors. Mathematics, with clear boundaries and evaluation criteria, provides foundational knowledge for other disciplines, rarely engaging directly with societal demands. Consequently, scholars in mathematics often work independently, leading to lower collaboration frequencies. Mechanical engineering, a typical applied hard science, must rapidly respond to societal needs, accelerating knowledge transformation into tangible products. This necessity drives frequent, large-scale collaborations led by senior scholars, involving young and beginner scholars. Philosophy’s development is driven by the intrinsic logic of knowledge, focusing on the value of knowledge and its influence on human values. Therefore, collaborations in philosophy are typically smaller. Sociology, addressing complex social issues, requires frequent collaborations among scholars to provide insights and solutions, resulting in diverse collaboration patterns.

5.2.2. Different Material Conditions

The dependency of knowledge production activities on material resources significantly influences scholars’ scientific collaboration behaviors. Scientific research requires basic conditions, including tangible resources and financial support, which vary across disciplines. Function-dependent disciplines demand high levels of material resources, such as large experimental equipment and precise instruments, necessitating collaborations to share resources and reduce costs. These collaborations often lead to interdisciplinary projects and teams, enhancing research output and securing further funding. Conversely, function-independent disciplines require less material support, leading to fewer collaborations.

In mathematics, the low dependency on material resources emphasizes scholars’ intellectual resources and research enthusiasm, resulting in stable and relatively infrequent collaborations with low intensity. In contrast, mechanical engineering relies heavily on material resources, necessitating frequent collaborations to meet societal needs and support new technologies and inventions. Research in mechanical engineering requires large-scale equipment, promoting frequent, and high-intensity collaborations to optimize resources. Philosophy, with minimal material dependencies, involves spontaneous research activities requiring few resources, leading to low collaboration frequency and intensity. Sociology, while relying mainly on intellectual resources, involves extensive fieldwork and large-scale surveys that necessitate collaborative efforts, resulting in a higher frequency of collaborations compared to philosophy but less than mechanical engineering.

5.2.3. Different Individual Characteristics of Scholars

As the main body in research activities, scholars’ individual characteristics influence their research behaviors. These characteristics include academic traits (e.g., academic level and interests) and social traits (e.g., collaboration willingness, cultural background, and social skills). Academic level, determined by the quantity and quality of scholarly output, generally increases with academic age. Shared research interests reduce communication costs and improve collaboration efficiency. Social traits include scholars’ willingness to collaborate, cultural background, and social skills. Some scholars prefer solitary research, presenting their findings as solo work despite engaging in academic exchanges. Others prefer collaborative research, producing joint academic outputs. Cultural background subtly shapes academic behaviors, while social skills facilitate integration into research groups and strengthen collaboration willingness.

In mathematics, the number of co-authored papers remains stable, indicating consistent collaboration trends influenced by scholars’ collaboration willingness. Junior scholars often collaborate with peers, while senior scholars frequently collaborate with each other. This suggests that personal collaboration preferences impact collaborator selection. In mechanical engineering, the strong inclination towards collaboration results in frequent team-based research. Scholars with high degree centrality engage in more frequent collaborations. In philosophy, similar to mathematics, the independent nature of philosophical inquiry, diverse theoretical frameworks, and varied evaluation standards result in low collaboration willingness. In sociology, research interests and academic levels significantly impact collaboration behaviors, with collaborations often involving senior scholars, enhancing the depth of analysis and quality of research outputs.

6. Conclusions

Scientific collaboration exists across all disciplines and has significantly contributed to the production of knowledge and the development of various fields. However, the patterns of scientific collaboration among university scholars vary across disciplines, primarily influenced by the nature of the discipline and its stage of development. In mathematics, during the early stage, collaboration predominantly occurs between senior scholars. As the discipline matures, collaboration patterns often involve senior scholars collaborating with beginner scholars, and also include collaborations among senior scholars, junior scholars, and beginner scholars. In mechanical engineering, during the initial establishment of the discipline, the typical mode of scientific collaboration is between beginner scholars. As the discipline continues to develop, the most common collaboration pattern becomes that of senior scholars, junior scholars, and beginner scholars working together. In philosophy, the frequency of scientific collaboration is lower compared to other disciplines. However, as the discipline continues to develop and collaboration increasingly becomes important, the collaboration behavior among scholars in philosophy is also gradually increasing and diversifying. In sociology, during the early stage, the primary mode of scientific collaboration is between beginner scholars. As the discipline matures, the collaboration patterns primarily involve beginner scholars collaborating with junior scholars.

The study still has some shortcomings that need to be addressed in future research. Firstly, regarding data acquisition, this study only selected four primary disciplines within the categories of natural sciences, engineering, humanities, and social sciences, spanning 20 years. This time frame is relatively short in the context of the entire history of these disciplines. Ideally, a broader range of disciplines and a longer period should be considered to provide a more comprehensive analysis of scientific collaboration patterns among scholars across various disciplines and over time. Furthermore, we failed to distinguish more clearly between internal mechanisms of collaboration (e.g., supervision, homophily, epistemic compatibility) and external forces such as competition for funding or institutional prestige. In our subsequent work, we will construct a dual-perspective analytical framework of “internal mechanisms-external factors”. On one hand, we will delve into the micro-dynamics of collaboration within disciplines, such as the collaborative bonds formed among scholars due to the alignment of research paradigms (epistemic compatibility) or similarities in academic backgrounds (homophily). On the other hand, we will systematically examine the impact of the external environment, such as how funding allocation mechanisms incentivize or constrain interdisciplinary collaboration, and how competition for institutional prestige shapes interaction patterns between disciplines. Through this distinction, we will further theorize how these two types of factors jointly influence the formation and evolution of collaborative networks, with particular focus on the reproduction logic of disciplinary norms amid the interweaving of internal and external factors, as well as the differences in behavioral strategies among various fields in competitive environments.