Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering

• URL: http://arxiv.org/abs/2208.07457v1
• Date: Mon, 15 Aug 2022 22:29:00 GMT
• Title: Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering
• Authors: Yu Zhu and Santiago Segarra
• Abstract summary: We study p-Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent weights (EDVWs) We convert hypergraphs with EDVWs into submodular hypergraphs for which the spectral theory is better developed. Existing concepts and theorems such as p-Laplacians and Cheeger inequalities proposed under the submodular hypergraph setting can be directly extended to hypergraphs with EDVWs.
• Score: 29.400924845055865
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study p-Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent vertex weights (EDVWs). These weights can reflect different importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity and flexibility. By constructing submodular EDVWs-based splitting functions, we convert hypergraphs with EDVWs into submodular hypergraphs for which the spectral theory is better developed. In this way, existing concepts and theorems such as p-Laplacians and Cheeger inequalities proposed under the submodular hypergraph setting can be directly extended to hypergraphs with EDVWs. For submodular hypergraphs with EDVWs-based splitting functions, we propose an efficient algorithm to compute the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the vertices, achieving higher clustering accuracy than traditional spectral clustering based on the 2-Laplacian. More broadly, the proposed algorithm works for all submodular hypergraphs that are graph reducible. Numerical experiments using real-world data demonstrate the effectiveness of combining spectral clustering based on the 1-Laplacian and EDVWs.

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