Multi-scale Hochschild spectral analysis on graph data

Yunan He; Jian Liu; Yunan He; Jian Liu

doi:10.3934/math.2025064

AIMS Mathematics

2025, Volume 10, Issue 1: 1384-1406. doi: 10.3934/math.2025064

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Multi-scale Hochschild spectral analysis on graph data

Yunan He ¹,
Jian Liu ^{1,2
,
,}

1.
Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, China
2.
Department of Mathematics, Michigan State University, MI 48824, USA

Received: 25 October 2024 Revised: 15 January 2025 Accepted: 16 January 2025 Published: 21 January 2025
MSC : 05C20, 55N31, 62R40

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.
- topological data analysis,
- Hochschild cohomology,
- Hochschild Laplacian,
- multi-scale,
- spectral analysis
Citation: Yunan He, Jian Liu. Multi-scale Hochschild spectral analysis on graph data[J]. AIMS Mathematics, 2025, 10(1): 1384-1406. doi: 10.3934/math.2025064

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Abstract

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.

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