Community detection on elite mathematicians’ collaboration network

In this study, we leverage the extensive dataset from OpenAlex, a key resource for our analysis. Our study collected data on scholars who fulfilled specific requirements: they were prolific authors with more than ten publications between 1960 and 2021. We classified authors primarily active in mathematics and identified collaborative links among them. Figure 1(a) shows a significant rise in collaboration among mathematicians after 2005, reflecting the necessity to tackle complex problems requiring interdisciplinary cooperation. Additionally, we compiled information on renowned mathematicians who have received awards in their field from platforms including Wikipedia, Wikidata, and official award websites. This supplementary dataset encompasses 341 unique award categories and includes detailed information on the mathematicians’ demographics and achievements. Of these distinguished award recipients, 395 individuals were successfully matched with profiles in the OpenAlex database.

Figure 1.

Data characteristics of mathematicians.

This data collection methodology enabled us to gain insights into the collaborative networks of 79,016 mathematicians affiliated with 8,833 institutions across 172 countries. We were also able to categorize their main areas of research within mathematics, including fields like pure mathematics, mathematical analysis, combinatorics, and statistics, as shown in Figure 1(b). Following this, we constructed an undirected weighted collaborative network, designated as G, where mathematicians are represented as nodes v₁, v₂, ⋯, v_N. The edges lvivj\[{{l}_{{{v}_{i}}{{v}_{j}}}}\] and their respective weights wvivj\[{{w}_{{{v}_{i}}{{v}_{j}}}}\] represent the collaborative connections and the frequency of collaborations, respectively, over the period from 1960 to 2021 (Priem et al., 2022).

In our network G, the average degree 〈k〉 is lower than the logarithm of the number of nodes lnN, indicating that the network’s interconnections are relatively sparse. This is further supported by the network density being less than 0.002, suggesting sparser connectivity than one might expect in a denser network. This finding implies that the collaboration network among mathematicians is characterized by less dense interconnections, with fewer links connecting the nodes than in a more interconnected network. We then examined the structural attributes of network G using various analytical methods, with the results visualized in Figure 2. Initially, we applied kernel density estimation (KDE) to assess the distribution of connected component sizes within G. The visualization of this analysis distinctly illustrates a significant polarization in component sizes, indicating stark differences in the community structures present. Next, we analyzed node degree ranking to understand the relative importance or centrality of nodes in G. Figure 2(b) highlights critical nodes or ‘hubs’ in the network, suggesting the presence of highly connected individuals. Furthermore, we generated a histogram to depict the frequency distribution of node degrees in G, providing insights into the range and scarcity of specific node degrees and enhancing our understanding of the network’s connectivity patterns. These comprehensive analyses collectively shed light on the structural properties of the collaboration network G, offering valuable insights into its network dynamics.

Figure 2.

Network characteristics.

In network analysis and statistical modeling, centrality metrics are crucial for evaluating the significance or influence of nodes within a network. These metrics are instrumental in deciphering the network’s structural nuances and the dynamics of influence and information flow. This paper highlights four primary centrality measures, each offering unique insights into nodes’ roles within weighted networks: Betweenness, Closeness, Harmonic centrality, and Eigenvector centrality. For this analysis, we employed the Python interface of igraph (Csardi & Nepusz, 2006) to execute and calculate four key centrality metrics. Here, d^w (u, v) denotes the weight-adjusted shortest path between nodes u and v.

The study of elite mathematicians within academic communities is instrumental in unraveling the complexities of knowledge sharing, collaborative patterns, and the structure of scholarly networks. This exploration, particularly through the lens of network analysis, has become a vital aspect of modern research (Asif & Islam, 2016; Gaskó et al., 2016; Izquierdo et al., 2018). Our research identifies these elite mathematicians based on their significant career achievements, namely their receipt of notable awards in mathematics. We then segment collaboration networks into distinct communities using two algorithms and focus on calculating the four aforementioned centrality metrics within these communities. This method provides a deeper insight into the dynamics of centrality and the critical role these eminent figures play in the formation and evolution of academic networks.

The outcomes, outlined in Table 2 and depicted in Figure 3, reveal a distinct trend when comparing award-winning mathematicians with their peers. Significant differences in Betweenness, Closeness, and Harmonic Centrality between these two groups were identified, showing statistically meaningful variations (p < 0.05) through a t-test when computing centralities focused on the sub-network formed by the community division via the community detection algorithm. The heightened Betweenness Centrality among award recipients suggests their pivotal role as vital links or intermediaries among diverse nodes within the communities. Their elevated status implies a substantial influence over the shortest paths connecting other nodes. Conversely, the lower Betweenness Centrality among other mathematicians indicates a potential lack of intermediary roles, signaling a different function or position within the network’s structure. Moreover, awardees exhibit superior Closeness and Harmonic Centrality metrics. The increased Closeness implies that awardees are more swiftly connected to other nodes, maintaining shorter average distances compared to non-recipients. This suggests enhanced and more direct communication channels within the group of awardees. Additionally, the heightened Harmonic Centrality among awardees highlights not only their proximity to others but also shorter harmonic mean distances, indicating more effective communication routes and a stronger sense of interconnectedness within their subgroup.

Figure 3.

Four centrality metrics were computed within communities identified by both GMM and Infomap, and subsequently compared between awardees and other mathematicians. The metrics assessed were: (a), (e) Betweenness; (b), (f) Closeness; (c), (g) Harmonic Centrality; and (d), (h) Eigenvector Centrality.

The analysis of eigenvector centrality for awardees shows contrasting results between communities identified by GMM (negative) and Infomap (positive). As previously discussed, although both algorithms uncover community structures, they operate on different principles and exhibit distinct strengths. GMM is straightforward but may encounter resolution limits, hindering its ability to detect very small or loosely connected communities. In contrast, Infomap is adept at identifying community structures in large, modular networks, typically resulting in more balanced community sizes. This is the consequence that in larger networks, eigenvector centrality tends to exhibit greater consistency. The complexity and sheer number of nodes in these extensive systems reduce the influence of minor connectivity changes on individual nodes’ eigenvector centrality, thereby enhancing their stability. Conversely, in smaller networks, certain nodes might display heightened eigenvector centrality due to simpler connection patterns, potentially leading to concentrated control of information flow among a few nodes. An examination of eigenvector centrality across the entire collaboration network reveals no substantial difference between awardees and their peers (-0.00127 with p-value of 0.4771). This implies that both award-winning and non-award-winning mathematicians exert a similar level of influence throughout the network, pointing to a comparable effect on the network’s structural dynamics and their capacity to direct the flow of information.

Researchers frequently collaborate with familiar individuals, such as mentors, past research colleagues, or established experts in their academic domain (Singh et al., 2021; Yu et al., 2021). These collaborations form complex networks characterized by well-established connections, notably evident in scholarly co-authorship. Understanding these connections is crucial in exploring scientists’ intricate collaboration patterns and identifying cohesive clusters or communities within academic networks.

In our study of mathematicians’ collaboration network, we employed two community detection algorithms, namely GMM and Infomap. The first one identified 2,629 communities within network G, while the last one delineated 8,962 communities. Among these structures, the positioning of 395 award-winning mathematicians did not display an even distribution. As depicted in Figure 4(a) where each circle represents a community, and the radius is positively proportional to the number of awardees in this community. The community sizes identified by GMM showed a partially positive correlation with the presence of awardees. Notably, the community hosting the most awardees did not align with its overall size. Conversely, Infomap exhibited a modest positive link between community size and the number of awardees, notably observed in the largest community harboring the most awardees. This finding prompts consideration of an “elite mathematician” accumulation effect, suggesting that awardees tend to collaborate within their elite circle or with potential awardees in the field of mathematics. This encourages a further exploration of how award recipients are distributed across various communities by involving the statistical measure of skewness (Groeneveld & Meeden, 1984), formally defined by the following equation: 5 η=1n∑i=1n(xi−x¯)3(1n∑i=1n(xi−x¯)2)3/2, $$\eta = {1 \over n}{{\sum\nolimits_{i = 1}^n {{{\left( {{x_i} - \overline x } \right)}^3}} } \over {{{\left( {{1 \over n}\sum\nolimits_{i = 1}^n {{{\left( {{x_i} - \overline x } \right)}^2}} } \right)}^{3/2}}}},$$ where we denote the sizes of detected communities as c₁, c₂, ⋯, c_n, and the tally of awardees within these communities as a₁, a₂, ⋯, a_n. Here, xi=aici−∑i=1nai∑i=1nci\[{{x}_{i}}=\frac{{{a}_{i}}}{{{c}_{i}}}-\frac{\sum\nolimits_{i=1}^{n}{{{a}_{i}}}}{\sum\nolimits_{i=1}^{n}{{{c}_{i}}}}\] signifies the deviation between the actual proportion of awardees in the i-th community and the average awardee ratio across the entire network of collaborations. The skewness η quantifies the asymmetry of the data distribution around its mean. In essence, for the array of discrepancies x₁, x₂, ⋯, x_n reflecting the variation between the observed and expected ratios of awardees in each community skewness measures how asymmetric these deviations are from their mean.

Figure 4.

Distributed characteristics of mathematicians and mathematical awardees within communities.

In examining whether awardees are evenly distributed across communities, we postulate a straightforward, linear correlation between the number of awardees and the size of each community. To test this idea, we implemented a statistical simulation for randomly allocating awards among different communities. This simulation allocated awardee numbers across communities using a multinomial distribution F~Multinomial(n_awardee, p₁, p₂, …, p_n), where n_awardee = 395 represents the total number of awardees. The probabilities p1=c1N,p2=c2N,⋯,pn=cnN\[{{p}_{1}}=\frac{{{c}_{1}}}{N},{{p}_{2}}=\frac{{{c}_{2}}}{N},\cdots ,{{p}_{n}}=\frac{{{c}_{n}}}{N}\], correspond to each community’s size as a fraction of the total size of the collaboration network G. This method helps us understand the potential random distribution of awardees across the network’s communities.

Next, we ran 1,000 random simulations using the described multinomial distribution F to evaluate the connection between community size and awardee distribution. These simulations allowed us to compute skewness values for the synthetically distributed awardees according to the community structures detected by GMM and Infomap, notated as η_j¹ and η_j² for each j of the 1,000 simulations, respectively. For comparative purposes, we determined the actual skewness values, η¹ and η², drawn from the real network’s community structures as segmented by GMM and Infomap. Our goal in comparing the simulated and real data distributions was to assess the homogeneity between what is expected and what is observed. As shown in Figure 4 (c) and (d), both η¹ and η² values notably surpass the range of random η distributions. The skewness of awardees distribution in real network communities is more pronounced than in the randomized models. This indicates a tendency for award-winning mathematicians in actual networks to cluster more densely in certain communities rather than being evenly spread across the network, resonating with the principles of club theory in economics (Potts et al., 2017).

Furthermore, we explore the complex interplay between the research orientations of mathematicians and the communities identified within their collaboration network, categorizing them into 16 distinct mathematics sub-fields (see Figure 1(b)). To evaluate the congruence between these sub-fields and the communities delineated by two different algorithms, we apply the NMI (Details mentioned in the Appendix). For the purpose of robustness verification, this section designs a random experiment where discipline labels are randomly assigned to all nodes in the network, and the NMI values between community labels and random discipline labels are calculated for different algorithms. Through 1,000 repeated experiments, the study obtains the range of NMI values between community labels and discipline labels under random experiments. Comparing the communities identified by both algorithms against a random reassignment of sub-field labels for each mathematician, Table 3 reveals a significant positive correlation between the communities and the sub-fields. This implies that the identified communities inherently reflect aspects of these sub-fields, suggesting a natural tendency for mathematicians working in similar areas to collaborate more frequently.

These insights lead us to question whether the community information, encapsulating aspects of mathematical sub-fields, could be effectively represented using different methodologies. In particular, we explore the relationship between the number of award recipients in a community, the community’s disciplinary makeup, and its size. To analyze the structured nature of field labels within these communities, we introduce the Simpson index (Simpson, 1949; Somerfield et al., 2008). Our linear regression analysis, examining the relationship between the number of awardees, community sizes, and the Simpson index (indicative of disciplinary composition), yields compelling findings. The results displayed in Table 4 indicate that larger communities tend to have more award recipients, consistent with earlier observations in this study. Conversely, there is an inverse correlation between the Simpson index and the number of awardees. This suggests that communities with higher diversity in their field labels or greater disorderliness are less likely to include award recipients. This finding implies that mathematicians who specialize in specific subdomains and collaborate intensively within those domains are more likely to achieve notable success in their field.