Mathematical researchers, especially those in early-career positions, face critical decisions about topic specialization with limited information about the collaborative environments of different research areas. The aim of this paper is to study how the popularity of a research topic is associated with the structure of that topic’s collaboration network, as observed by a suite of measures capturing organizational structure at several scales. We apply these measures to 1,938 algorithmically discovered topics across 121,391 papers sourced from arXiv metadata during the period 2020–2025. Our analysis, which controls for the confounding effects of network size, reveals a structural dichotomy–we find that popular topics organize into modular “schools of thought,” while niche topics maintain hierarchical core-periphery structures centered around established experts. This divide is not an artifact of scale, but represents a size-independent structural pattern correlated with popularity. We also document a “constraint reversal”: after controlling for size, researchers in popular fields face greater structural constraints on collaboration opportunities, contrary to conventional expectations. Our findings suggest that topic selection is an implicit choice between two fundamentally different collaborative environments, each with distinct implications for a researcher’s career. To make these structural patterns transparent to the research community, we developed the Math Research Compass (https://mathresearchcompass.com), an interactive platform providing data on topic popularity and collaboration patterns across mathematical topics.

We analyzed the Cornell ArXiv dataset ([1]), which contains metadata for approximately 2.7 million scientific papers. Our analysis focused on mathematics papers published between 2020 and 2025, yielding 121,391 papers across 31 mathematical subfields as classified by the ArXiv Mathematics Subject Classification system.

Research topics were identified using BERTopic [21], a state-of-the-art topic modeling approach. The BERTopic pipeline consists of three main stages. First, it generates semantic vector embeddings for each document using a Bidirectional Encoder Representations from Transformers (BERT) model [22]. Next, it applies Uniform Manifold Approximation and Projection (UMAP) to these embeddings to reduce their dimensionality while preserving local and global structure [23]. Finally, it uses Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) on the dimension-reduced embeddings to identify dense clusters of documents, which constitute the research topics [24]. We concatenated paper titles and abstracts as input text, applied preprocessing to remove common stopwords and mathematical notation artifacts, and generated semantic embeddings using the all-MiniLM-L6-v2 sentence transformer model. UMAP reduced embeddings to 5 dimensions using 15 neighbors and cosine distance, followed by HDBSCAN clustering with a minimum cluster size of 15 papers. This process identified 1,938 distinct research topics, with each paper assigned to its most coherent topic based on the highest probability score. Each topic was assigned a primary mathematical category based on the most frequent ArXiv category among its constituent papers.

A prerequisite for constructing accurate collaboration networks is robust author name disambiguation (AND). Given that raw author strings from bibliographic sources often contain variations, initials, and ambiguities that can lead to either incorrectly splitting single authors into multiple nodes or incorrectly merging distinct authors into one, we implemented a dedicated multi-stage AND pipeline AuthorDisambiguator; see Appendix B for full algorithm details.

Our multi-stage process aligns with current best practices in the field, which have converged on sequential pipelines that combine graph-based similarity with network-aware clustering to achieve high precision on large-scale datasets ([2]; [3]). We adopted a particularly conservative approach, prioritizing high precision to minimize the introduction of false connections that could artificially distort network topology ([4]; [5]).

The pipeline began by parsing and normalizing all unique author name strings (\( N \approx 120,000\) ) from the 121,391 papers. Normalization involved lowercasing, diacritic removal, and standardization of particles and punctuation. Subsequently, a graph-based approach was employed:

1. Initial Similarity Graph Construction: An author similarity graph was constructed where nodes represent unique normalized names. Edges were added between names sharing the same last name if they met strict string similarity criteria (e.g., initial expansions like ‘J. Doe’ to ‘John Doe’ or high Levenshtein distance > 0.95).
2. Cluster Merging: Connected components in this graph (clusters of potentially identical authors) were analyzed. A cluster was merged into a single canonical author profile if all full first names within it were determined to be compatible variants (e.g., ‘Robert’, ‘Rob’). Specific heuristics were applied to handle potentially ambiguous patterns, such as those common in Romanized East Asian names, by employing stricter similarity thresholds for first name compatibility. Clusters exceeding a size threshold (50 names) or containing multiple clearly distinct first names (e.g., ‘Pengcheng Xie’ and ‘Pengxu Xie’ within the same last-name block) were handled conservatively, with the algorithm only attempting to identify and merge highly confident sub-groups.
3. Network-Based Co-occurrence Merging: Remaining canonical profiles were further compared based on co-authorship patterns and shared publications. Pairs of authors sharing the same last name and first initial were evaluated using a weighted Jaccard similarity of their co-author sets and paper sets. Pairs exceeding a similarity threshold (0.5) and minimum paper count (2) were merged.

This iterative process reduced the initial  120,767 unique raw author strings to 117,883 canonical author profiles, representing a 2.39 

For each of the 1,938 research topics, we constructed an undirected co-authorship network using the disambiguated canonical author profiles generated by our AND pipeline. In these networks, nodes represent unique authors active in the topic, and edges represent a co-authorship relationship on at least one paper within that topic.

The construction process followed established protocols for creating collaboration networks from bibliographic data ([6]). For each paper with multiple authors, a complete graph (or clique) was formed among all co-authors. This means an edge was created between every pair of authors on a given paper.

• Edge Weights: The networks are weighted. If a pair of authors co-authored multiple papers within the same topic, the weight of the edge between them was incremented for each joint publication. These weights represent the strength of the collaborative tie and were used in weighted network metric calculations, such as modularity.
• Node Attributes: Each node (author) was attributed with the total count of papers they published within that specific topic.
• Self-Loops: Self-loops were prevented by ensuring the author list for each paper contained unique individuals before edge creation.

Only papers with multiple authors contributed to the formation of network edges. However, papers with a single author were retained in our dataset for the calculation of topic-level metrics like the overall collaboration rate.

To characterize the structural properties of mathematical collaboration networks, we employ a carefully selected set of 10 network metrics organized into four domains. This captures distinct yet complementary aspects of network structure, providing a comprehensive structural signature of each mathematical subfield.

2.4.1 Collaboration Dynamics (Measuring Tie Strength and Persistence)

2.4.2 Global Network Topology (Characterizing Overall Network Shape and Resilience)

2.4.3 Mesoscopic Organizational Structure (Identifying Community and Core-Periphery Patterns)

2.4.4 Nodal Positioning and Brokerage (Quantifying Individual-level Opportunity)

2.4.5 Integrated Framework Justification

Topics were classified by research popularity based on total paper count. We rank-ordered all 1,938 topics by the number of papers and designated the top 20 The 20 

Our analysis proceeded in two stages to disentangle popularity effects from network size effects:

Stage 1: Baseline Comparisons. We first assessed distributional assumptions using Shapiro-Wilk tests for normality. Since all metrics violated normality assumptions (all \( p < 0.001\) ), we employed non-parametric statistical tests throughout. This non-normality is expected in network data due to bounded metrics (e.g., collaboration rates constrained to 0-1), power-law degree distributions common in academic networks, and heterogeneous collaboration structures across scientific disciplines ([7], [8], [9]).

Group comparisons used Mann-Whitney U tests to assess statistical significance. This non-parametric test operates by ranking all observations from both groups and calculating the U statistic, which is the number of times a value from one group is larger than a value from the other. This statistic is then used to compute a p-value against the null hypothesis that the two distributions are identical. This approach is robust to non-normal distributions and outliers common in network data.

To quantify the magnitudes of these differences, we calculated the non-parametric effect size Cliff’s delta (\( \delta\) ) ([10]), which estimates the degree of separation between two distributions as the difference in dominance probabilities (i.e., \( \delta = P(X > Y) - P(Y > X)\) ). For interpretation, we use standard thresholds of 0.147 (small), 0.33 (medium), and 0.474 (large), which represent the non-parameteric equivalents of Cohen’s original effect size benchmarks ([11]).

A Bonferroni correction was applied for the ten comparisons, setting our significance threshold at \( \alpha = 0.005\) . Confidence intervals for key estimates were computed using bootstrap resampling with 10,000 iterations.

Stage 2: Controlling for Network Size. To test whether observed differences persisted beyond size effects, we conducted multiple regression analyses. For each network metric, we fit three models:

1. Simple Model: Metric \( \sim\) Popularity (binary)
2. Size Control Model: Metric \( \sim\) Popularity + log(Network Size)
3. Interaction Model: Metric \( \sim\) Popularity \( \times\) log(Network Size)

All variables were standardized (mean=0, SD=1) before analysis to ensure comparability of coefficients across metrics. A significant popularity coefficient in the size control model indicates a robust effect that persists beyond network scale. We also examined correlations between network size and each metric to assess confounding strength.

All analyses were performed in Python 3.11 using SciPy 1.15.2 and statsmodels 0.14.0.

To ensure our findings were not artifacts of specific parameter choices, we conducted two comprehensive validation analyses for our topic modeling and classification approach.

First, to validate the BERTopic model itself, we performed a sensitivity analysis testing 72 different hyperparameter combinations on a 10,000-document subsample. This revealed that while the exact number of topics generated was moderately sensitive to the min_topic_size parameter (Coefficient of Variation = 0.44), the overall topic structure was reasonably stable (mean Adjusted Rand Index = 0.34 \( \pm\) 0.06). This analysis informed our choice of baseline parameters for the main study (min_topic_size=15, n_neighbors=15, n_components=5). Furthermore, to ensure the semantic coherence of the generated topics, we conducted a manual validation of a stratified sample of 100 topics. Appendix C (Table 9) presents representative examples from this validation, demonstrating that BERTopic successfully identified genuine mathematical research areas across all size ranges. The full validation found that 74  Second, and most crucially, we generated three additional complete topic models using alternative minimum topic size values (10, 20, and 25), resulting in four distinct topic sets ranging from 1,164 to 2,953 topics. We then re-ran our entire two-stage statistical analysis—both the baseline Mann-Whitney U tests and the full regression models—on all four topic sets (see Appendix C, Table 6). The central finding of our paper, the duality between scale-driven and popularity-driven network effects, remained consistent across all four independent analyses. The same metrics were robustly associated with popularity, and the same metrics were confounded by size in every single run (see Appendix C, Table 5 for full interaction model results across one run). This demonstrates that our conclusions reflect a fundamental phenomenon in the data, independent of the specific granularity of the topic model.

Our analysis encompassed 121,391 mathematics papers from which BERTopic identified 1,938 distinct research topics. The classification yielded 387 popular topics (mean = 314.2 \( \pm\) 345.1 papers) and 387 niche topics (mean = 13.8 \( \pm\) 1.7 papers), representing a 22.8-fold difference in mean paper counts. This extreme difference ensured distinct collaboration environments for comparison while maintaining balanced sample sizes.

Mann-Whitney U tests revealed significant differences in 9 of 10 network metrics between popular and niche topics (Table 1). All significant results survived Bonferroni correction (\( \alpha = 0.005\) ), with most metrics showing large effect sizes by conventional standards for Cliff’s delta.

Popular topics demonstrated substantially higher modularity (87.9 Niche topics exhibited contrasting organizational patterns characterized by hierarchical structures. These networks demonstrated exceptionally high coreness ratios (35.3 Only collaboration rate showed no significant difference between groups (75.5 

Network size correlations revealed substantial confounding for multiple metrics (Table 3). The strongest associations appeared for robustness ratio (\( r = -0.79\) ), coreness ratio (\( r = -0.73\) ), and small-world coefficient (\( r = 0.72\) ), confirming that a simple bivariate comparison is insufficient to isolate the effects of popularity. To visually illustrate our core finding, Figure 1 displays exemplar networks for a popular and a niche topic with identical author counts, highlighting the stark structural differences that persist even when size is held constant.

Regression analysis controlling for network size revealed three distinct patterns in how popularity effects manifest (Table 4).

3.3.1 Robust Popularity Effects

3.3.2 Size-Confounded and Marginal Effects

3.3.3 Emergent Effects Through Statistical Suppression

Our analysis of 121,391 mathematical papers shows that research topics have fundamentally different collaboration environments, and these differences persist even after controlling for network size. This provides an empirical basis for understanding how topic selection shapes the context of a research career.

The distinction between the organization of popular and niche topics reflects different modes of knowledge production. Popular topics, which have significantly higher modularity (87.9 Conversely, niche topics maintain a pronounced core-periphery structure (7.0 Furthermore, the emergent property whereby popular topics show lower collaboration rates at equivalent network sizes (\( \beta = -2.33\) , \( p < 0.001\) ) adds another dimension to these considerations. This suggests that beyond topological differences, popular and niche fields may also cultivate different collaboration norms. Popular fields, with their established research programs and larger communities, might place a greater emphasis on individual contributions, whereas the tighter-knit, exploratory nature of niche fields may necessitate more interdependent teamwork.

Our analysis of structural constraint reveals a complex relationship with popularity. While a simple bivariate comparison shows researchers in popular topics face less constraint, this relationship reverses after controlling for network size; they instead face significantly higher structural constraints (\( \beta\) = 0.46, \( p < 0.004\) ). This result extends Burt’s theory of structural holes by incorporating a temporal, career-stage dimension [29].

The modular organization of popular fields creates numerous structural holes between communities, a configuration Burt identifies as advantageous for innovation. However, our data shows that the researchers occupying these brokerage positions are predominantly those with higher publication counts and seniority. This suggests a “brokerage ladder”: a career progression from a constrained position within a module toward a boundary-spanning role.

This progression can be understood as a network-based mechanism for the “Matthew Effect” [33], a dynamic formalized in network science as “preferential attachment” [13]. This principle posits that new collaborations are more likely to form with already-prominent researchers. Thus, when a valuable brokerage opportunity arises, it is the senior, high-status researchers who are most likely to attract that connection. The social capital required to span these boundaries is therefore a resource that accumulates over a career.

As [30] note, network size is an intrinsic part of the constraint measure. By statistically controlling for size, our analysis isolates the effect of structural position relative to a researcher’s peers. The high average constraint in popular fields is therefore not paradoxical; it reflects the typical early-career experience. A researcher must first embed within a specific community to build reputation, with brokerage opportunities emerging only as their career matures.

These structural differences carry practical implications for researchers at various career stages, though we emphasize that our analysis characterizes collaboration environments rather than predicting career outcomes. The Math Research Compass operationalizes these insights by providing transparent access to collaboration structures across 1,938 identified topics.

For early-career mathematicians, our findings suggest that topic selection involves an implicit choice between contrasting collaborative environments. Those drawn to popular areas should anticipate working within modular communities where establishing position within a specific research cluster becomes paramount. Success in such environments likely requires different strategies than in niche fields, where direct access to central experts provides clearer mentorship pathways but potentially fewer alternative routes should initial connections prove unproductive.

The emergent property whereby popular topics show lower collaboration rates at equivalent network sizes (\( \beta = -2.33\) , \( p < 0.001\) ) adds another dimension to these considerations. This suggests that beyond structural differences, popular and niche fields may cultivate different collaboration norms, with popular fields potentially emphasizing individual contributions within established research programs.

A key methodological contribution of this work is our two-stage analytical approach, which rigorously separates universal scaling effects from genuine popularity-driven organizational patterns. By first comparing groups and then using regression to control for network size, we demonstrate how univariate analyses can be misleading. The dramatic attenuation or reversal of several effects after size control (Table 4) underscores the necessity of this method for future studies of scientific collaboration.

While our findings are robust within this framework, several limitations define the scope of our conclusions. Our reliance on arXiv data, while comprehensive for many theoretical fields, may incompletely represent mathematical collaboration in applied areas where conference proceedings or industry partnerships are more common. Furthermore, our cross-sectional design establishes strong associations but precludes direct causal inference; future longitudinal work is needed to track how network structures evolve as topics gain or lose popularity. Finally, this study characterizes collaborative environments but does not link them to career outcomes like academic placement or grant success, which remains a valuable avenue for subsequent research.

We proactively addressed two other potential methodological concerns. First, while our author name disambiguation (AND) pipeline achieves a high validated precision of 86.0 Second, to ensure our findings were not an artifact of the COVID-19 pandemic, we performed a full temporal sensitivity analysis, splitting our dataset into “peak pandemic” (2020–2021) and “post-peak” (2022–2025) periods. As detailed in Appendix C (Table 8), the modular-hierarchical duality remained stable and highly significant across both eras. This provides strong evidence that the observed structures are fundamental features of mathematical collaboration, not transient phenomena. Interestingly, the only metrics that showed instability (degree centralization and average constraint) were not significant during the pandemic, suggesting that some hierarchical features may have been temporarily flattened by the global shift to remote work.

Future work should build on this foundation by applying our validated pipeline to longitudinal data and linking structural patterns to concrete career outcomes. Examining how researchers successfully transition between communities within modular fields could also provide actionable insights for those seeking to expand their collaborative reach.

Our comprehensive analysis reveals that mathematical collaboration networks exhibit predictable structural variations based on topic popularity, with these differences persisting after controlling for network size. The duality between universal scaling laws and field-specific organizational patterns suggests that while some network properties emerge mechanically with growth, others reflect genuine adaptations to different research contexts.

By documenting these patterns systematically and making them accessible through the Math Research Compass, we aim to democratize knowledge about collaboration structures that previously remained tacit within mathematical communities. While we stop short of prescriptive career advice, we believe that transparency about these structural realities enables more informed decision-making as researchers navigate the mathematical landscape. Understanding whether one is entering a modular, school-based field or a hierarchical, expert-centered domain represents valuable context for calibrating expectations and developing appropriate collaborative strategies.

This metric was calculated as the ratio of unique author pairs with more than one joint publication to the total number of unique collaborating pairs:

To ensure a robust calculation for potentially disconnected networks, our custom implementation first identified the LCC of each topic network. The actual clustering coefficient \( (C)\) and average shortest path length (\( L\) ) were computed on this LCC. The expected values for an equivalent Erdős-Rényi random graph (\( C_{\text{rand}}\) and \( L_{\text{rand}}\) ) were then calculated based on the number of nodes and edges of the LCC, providing a consistent null model. For networks with over 200 nodes, path lengths were estimated using random sampling to ensure computational feasibility.

This metric was calculated by simulating and comparing two node removal scenarios. We measured the size of the LCC after removing the top 10 

Networks with fewer than 3 nodes typically returned default values (e.g., 0 or NaN depending on the metric) or were handled to ensure computational stability; such cases were minimal.

For each of the 121,391 papers, author lists were parsed from the raw ArXiv metadata format. All unique author strings were then subjected to a rigorous normalization procedure to create a consistent representation for comparison:

1. Lowercase Conversion: All strings were converted to lowercase.
2. Diacritic Removal: Diacritics were removed by normalizing to the NFD Unicode form and encoding to ASCII (e.g., “François” becomes “francois”).
3. Particle Standardization: Common name particles (e.g., ’van’, ’der’, ’de’, ’von’) were standardized to prevent them from being treated as separate name components during splitting.
4. Punctuation and Character Removal: All punctuation (except for internal hyphens and apostrophes, which are handled by later tokenization) and non-alphanumeric characters were removed.

To efficiently identify potential merge candidates, we first constructed an undirected similarity graph where nodes represent unique normalized author strings.

• Blocking: To ensure computational feasibility across \( \approx\) 120,000 unique names, we employed a blocking strategy. Name pairs were only considered for similarity comparison if they shared the same last name.
• Edge Creation: An edge was added between two nodes (authors) in the graph if they satisfied strict string similarity criteria, defined as either:
1. Initial Expansion: One name is a clear initialism of the other (e.g., “J. Doe” and “John Doe”).
2. High String Similarity: The names have a Levenshtein distance ratio greater than 0.95, as calculated by Python’s difflib.SequenceMatcher.

The connected components of the similarity graph, representing clusters of potentially identical authors, were then analyzed using a set of conservative, rule-based heuristics to decide whether a merge was safe.

• Conservative Handling of Large Clusters: Clusters containing more than 50 name variants were flagged for cautious treatment. The algorithm did not attempt to merge the entire cluster, but instead sought to identify and merge only highly confident subgroups within it, leaving ambiguous cases unresolved. This prevents cascading errors in clusters corresponding to common last names.
• First Name Compatibility Logic: The core of the merging process relied on assessing the compatibility of full first names within a cluster. A cluster was approved for merging only if all extracted first names were determined to be plausible variants of each other. This was assessed using the following rules:
1. Abbreviation/Diminutive Check: One name is a standard abbreviation or diminutive of another. This includes initialism matching (e.g., ‘Rob’ and ‘Robert’) and a pre-defined dictionary of common Western variants (e.g., ‘William’/‘Bill’,‘Christopher’/‘Chris’).
2. Context-Sensitive String Similarity: For names not covered by the above, a high string similarity was required. Crucially, to handle the ambiguity common in Romanized East Asian names, we applied a stricter similarity threshold. Based on a list of common pinyin syllables (e.g., ‘wei’, ‘ming’, ‘xiao’, ‘zhang’), if a name pair was identified as potentially Asian, a similarity threshold of 0.92 was required; otherwise, a threshold of 0.87 was used.

A cluster was only merged into a single canonical profile if all its first-name variants were compatible under this logic. For ambiguous clusters containing multiple distinct and incompatible first names, the algorithm attempted to partition the cluster into smaller, internally consistent subgroups for merging.

The final stage used co-authorship and publication patterns to resolve remaining ambiguities among canonical profiles that were not merged by string similarity alone.

• Blocking: Candidate pairs for this stage were limited to authors who shared the same last name and the same first initial.
• Weighted Jaccard Similarity: Each candidate pair was evaluated using a weighted Jaccard similarity score based on their sets of co-authors and their sets of publications:
• Let \( C_1, C_2\) be the sets of co-authors for authors 1 and 2.
• Let \( P_1, P_2\) be the sets of papers for authors 1 and 2.

• The combined similarity was calculated as:

  \[ \begin{equation} S = w \cdot \frac{|C_1 \cap C_2|}{|C_1 \cup C_2|} + (1-w) \cdot \frac{|P_1 \cap P_2|}{|P_1 \cup P_2|} \end{equation} \]

  (1)

• Thresholding: A pair was merged if \( S > 0.5\) and each author had a minimum of 2 papers in the dataset. The co-author weight was set to \( w = 0.6\) to prioritize shared collaborative circles, a strong signal of identity.

After all merging stages, the resulting author-to-canonical-name mapping was resolved transitively to ensure that all names in a merge chain pointed to a single, final canonical profile. The representative name for each merged cluster was chosen as the most complete name variant available, prioritizing names with more components and fewer initials. This process reduced the 120,767 unique raw author strings to 117,883 canonical author profiles, with a manually validated precision of 86.0 

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