Modular Versus Hierarchical: A Structural Signature of Topic Popularity in Mathematical Research

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Summary: 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. This study analyzes how research topic popularity relates to collaboration network structure across algorithmically discovered topics from arXiv mathematics metadata (2020–2025). Our analysis, which controls for confounding effects of network size, reveals a structural dichotomy: popular topics organize into modular "schools of thought" with high modularity and low centralization, while niche topics maintain hierarchical core-periphery structures with high coreness ratios and centralization. This divide represents a size-independent structural pattern, not an artifact of scale. We also document a "constraint reversal"—researchers in popular fields face greater structural constraints on collaboration opportunities after controlling for size, contrary to conventional expectations. Time permitting, we will discuss replications of these findings on the computer science and quantum physics subject arXiv datasets, and present follow-up work which aims to establish causality by leveraging ChatGPT's November 2022 release as a natural experiment. To make these structural patterns transparent to the research community, we developed the Math Research Compass (mathresearchcompass.com), an interactive platform providing data on topic popularity and collaboration patterns across mathematical research areas.