Multi-scale Hochschild spectral analysis on graph data

Topological data analysis (TDA) has experienced significant advancements with the integration of various advanced mathematical tools. While traditional TDA has primarily focused on point cloud data, there is a growing emphasis on the analysis of graph data. In this work, we proposed a spectral analysis method for digraph data, grounded in the theory of Hochschild cohomology. To enable efficient computation and practical application of Hochschild spectral analysis, we introduced the concept of truncated path algebras, along with key mathematical results that support the computation of the Hochschild Laplacian. Our study established key mathematical results, including a relationship between Hochschild Betti numbers and the Euler characteristic of digraphs, as well as efficient representations of Hochschild Laplacian matrices. These innovations enabled us to extract multiscale topological and geometric features from graph data. We demonstrated the effectiveness of our method by analyzing the molecular structures of common drugs, such as ibuprofen and aspirin, producing visualized Hochschild feature curves that capture intricate topological properties. This work provides a novel perspective on digraph analysis and offers practical tools for topological data analysis in molecular and broader scientific applications.