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 1,938 algorithmically discovered topics spanning 121,391 papers 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 (87.9%) and low centralization (6.2%), while niche topics maintain hierarchical core-periphery structures with high coreness ratios (35.3%) and centralization (14.2%). 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.

To establish causality, we present follow-up work leveraging ChatGPT's November 2022 release as a natural experiment. Using difference-in-differences analysis on computer science topics, we find that the sudden popularity shock causally increased network modularity by 23.0% and decreased hierarchical structure by 41.5%, providing strong causal validation of our cross-sectional findings.

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. Our findings suggest that topic selection represents an implicit choice between fundamentally different collaborative environments, each with distinct implications for career development and research strategy.