Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge

TL;DR

This paper introduces a novel simplicial complex approach to analyze word co-occurrences in mathematical research, revealing that topological holes in the data can indicate gaps in knowledge and potential areas for future breakthroughs.

Contribution

The paper pioneers the use of simplicial complexes and topological methods to study higher-order relationships in scientific knowledge, specifically in mathematics.

Findings

Homological holes are common in mathematical research data.

Holes tend to close when related concepts co-occur in articles.

Larger holes may indicate potential for significant scientific advances.

Abstract

In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. Holes die when a subset of their concepts appear in the same article, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the dimension of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We also show that authors' conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.