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The Abstract Laplacian Tensor of a Hypergraph with Applications in Clustering

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Abstract

The normalized abstract Laplacian tensor of a weighted hypergraph is investigated. The connectivity of the hypergraph is associated with the geometric multiplicity of the smallest eigenvalue of the abstract Laplacian tensor. There is an inequality between the normalized cut of the hypergraph and the second smallest eigenvalue of the abstract Laplacian tensor. An optimization method of the hypergraph clustering is established and analyzed. Numerical examples illustrate that our method is effective.

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Data Availability

All datasets are publicly available.

Notes

  1. http://archive.ics.uci.edu/ml/datasets/Wine

  2. https://sci2s.ugr.es/keel/dataset.php?cod=60.

  3. https://sci2s.ugr.es/keel/dataset.php?cod=18.

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Acknowledgements

The authors would like to thank the handling editor and three reviewers for their detailed comments on our first version.

Funding

Tianhang Liu is supported by the National Natural Science Foundation of China under grant 11771099. Yimin Wei is supported by the National Natural Science Foundation of China under grant 11771099, Innovation Program of Shanghai Municipal Education Commission and Shanghai Municipal Science and Technology Commission under grant 22WZ2501900.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Tianhang Liu

  2. School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

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  1. Tianhang Liu
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Correspondence to Yimin Wei.

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Liu, T., Wei, Y. The Abstract Laplacian Tensor of a Hypergraph with Applications in Clustering. J Sci Comput 93, 7 (2022). https://doi.org/10.1007/s10915-022-01973-x

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