We propose a flexible framework for defining the 1-Laplacian of a hypergraph
that incorporates edge-dependent vertex weights. These weights are able to
reflect varying importance of vertices within a hyperedge, thus conferring the
hypergraph model higher expressivity than homogeneous hypergraphs. We then
utilize the eigenvector associated with the second smallest eigenvalue of the
hypergraph 1-Laplacian to cluster the vertices. From a theoretical standpoint
based on an adequately defined normalized Cheeger cut, this procedure is
expected to achieve higher clustering accuracy than that based on the
traditional Laplacian. Indeed, we confirm that this is the case using
real-world datasets to demonstrate the effectiveness of the proposed spectral
clustering approach. Moreover, we show that for a special case within our
framework, the corresponding hypergraph 1-Laplacian is equivalent to the
1-Laplacian of a related graph, whose eigenvectors can be computed more
efficiently, facilitating the adoption on larger datasets.

提出了一种灵活的超图 1-Laplacian 定义框架，包括依赖于边的顶点权重，以反映超边内不同顶点的重要性，增强了超图模型的表现能力。利用超图 1-Laplacian 的第二个最小特征向量进行聚类，可以实现比传统 Laplacian 更高的聚类精度，而且该方法可以在现实数据集上得到验证。在特定情况下，超图 1-Laplacian 等效于相关图的 1-Laplacian，可以更有效地计算特征向量，方便应用于更大的数据集。

基于 1-Laplacian 的边相关节点权重的超图谱聚类

Hypergraphs with Edge-Dependent Vertex Weights: Spectral Clustering  based on the 1-Laplacian

We develop a nonlinear spectral graph theory, in which the Laplace operator
is replaced by the 1-Laplacian ?$\Delta_1$. The eigenvalue problem is to solve
a nonlinear system involving a set valued function. In the study, we
investigate the structure of the solutions, the minimax characterization of
eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the
eigenvectors are computed for several elementary graphs. The graphic feature of
eigenvalues are also studied. In particular, Cheeger's constant, which has only
some upper and lower bounds in linear spectral theory, equals to the first
non-zero ?$1\Delta_$ eigenvalue for connected graphs.

基于 1-Laplacian 的非线性谱图理论中，我们研究了特征值的解决方案结构、特征值的极小极大特征和重复定理等多个方面，并计算了几个基本图的特征值及特征向量，同时研究了特征值的图形特征。特别地，对于连通图，Cheeger 恒量等于第一个非零 1-Laplacian 特征值。