The study of complex systems requires models that capture a hierarchy of higher-order interactions that move beyond the pairwise representation that simple networks provide. While mathematical frameworks exist for such higher-order systems, robust geometric tools to characterize their structure and organization remain underdeveloped. Here we show that the introduction of geometric measures for these structures is achieved by leveraging the non-commutative algebra of their matrix representations through the application of Connes’ spectral triplet formalism. Within this framework, we extend the spectral distance, a metric adapted from Connes’ formalism and previously applied to graphs, to higher-order networks, and additionally propose a definition of discrete curvature which explicitly depends on the spectral dimension. These serve as characterizing features of higher-order networks and complement known topological metrics. The formalism is demonstrated on a dataset of musical compositions, revealing their latent geometric structures. Simplicial complexes provide a mathematical structure for understanding higher-order interactions in complex systems. Based on frameworks from non-commutative geometry, particularly the topological Dirac operator and Connes’ spectral triplet, the authors introduce measures for analyzing simplicial complexes, including a definition of discrete curvature, and apply these tools to a dataset of musical compositions such as J. S. Bach’s sonatas and partitas for solo violin.
Najem et al. (Wed,) studied this question.