Connected Theorems: A Graph-Based Approach to Evaluating Mathematical Results

Abstract

The evaluation of mathematical results plays a central role in assessing researchers' contributions and shaping the direction of the field. Currently, such evaluations rely primarily on human judgment, whether through journal peer review or committees at research institutions. To complement these traditional processes, we propose a data-driven approach. We construct a hierarchical graph linking conjectures, theorems, papers, authors and fields to capture their citation relationships. We then introduce a PageRank-style algorithm to compute influence scores for these entities. Using these scores, we analyze the evolution of field rankings over time and quantify the impact between fields. We hope this framework can contribute to the development of more advanced, quantitative methods for evaluating mathematical research and serve as a complement to expert assessment.

Figures (6)

• Figure 1: The five-level graph.
• Figure 2: Journal rankings for the years 1897, 1950, 1970, 1974, 1994, and 2022. In 1897, only two journals appear in the ranking; this does not imply that only two journals existed at that time. Rather, we include only journals that received more than 50 ICM citations and published more than 100 papers in the 15 years preceding each respective year.
• Figure 3: Influence scores of fields over time. $(\alpha_T, \beta_T, \gamma_T, \alpha_P, \beta_P, \gamma_P, \delta_P, \alpha_A, \beta_A, \gamma_A, \alpha_F) = (0.8, 0.1, 0.05, 0.6, 0.2, 0.1, 0.05, 0.6, 0.2, 0.1, 0.85)$.
• Figure 4: Evolution of paper category ratios over time. For each year, the ratio for each category is calculated by dividing the cumulative number of papers in that category from 1991 up to the end of that year by the total number of papers across all categories during the same period.
• Figure 5: Field-to-field impact.