What is it about?
We developed a new method for analyzing directed graphs (digraphs) using a mathematical framework called Hochschild cohomology. To make the theory computationally feasible, we introduced the concept of truncated path algebras, which enables efficient construction of a Hochschild spectral. This allows us to extract rich topological and geometric features from graph-structured data.
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Why is it important?
Many real-world systems such as molecules, neural pathways, and social networks are best described as directed graphs rather than point clouds. Our method helps extract rich, multiscale topological and geometric features from graph-structured data. For example, we applied our method to analyze molecular structures of common drugs like ibuprofen and aspirin, generating curves that capture subtle topological traits.
Perspectives
This work creates an exciting bridge between pure mathematics and applied data science. It offers promising new avenues for analyzing structured data in chemistry, biology, and beyond, providing fresh insights into the underlying “shape” of data across diverse fields.
Jian Liu
Chongqing University of Technology
Read the Original
This page is a summary of: Multi-scale Hochschild spectral analysis on graph data, AIMS Mathematics, January 2025, Tsinghua University Press,
DOI: 10.3934/math.2025064.
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