Graph neural networks (GNNs) have proven effective in capturing relationships among nodes in a graph. This study introduces a novel perspective by considering a graph as a simplicial complex, encompassing nodes, edges, triangles, and $k$-simplices, enabling the definition of graph-structured data on any $k$-simplices. Our contribution is the Hodge-Laplacian heterogeneous graph attention network (HL-HGAT), designed to learn heterogeneous signal representations across $k$-simplices. The HL-HGAT incorporates three key components: HL convolutional filters (HL-filters), simplicial projection (SP), and simplicial attention pooling (SAP) operators, applied to $k$-simplices. HL-filters leverage the unique topology of $k$-simplices encoded by the Hodge-Laplacian (HL) operator, operating within the spectral domain of the $k$-th HL operator. To address computation challenges, we introduce a polynomial approximation for HL-filters, exhibiting spatial localization properties. Additionally, we propose a pooling operator to coarsen $k$-simplices, combining features through simplicial attention mechanisms of self-attention and cross-attention via transformers and SP operators, capturing topological interconnections across multiple dimensions of simplices. The HL-HGAT is comprehensively evaluated across diverse graph applications, including NP-hard problems, graph multi-label and classification challenges, and graph regression tasks in logistics, computer vision, biology, chemistry, and neuroscience. The results demonstrate the model's efficacy and versatility in handling a wide range of graph-based scenarios.

翻译：图神经网络（GNNs）已被证明能有效捕捉图中节点间的关联关系。本研究引入全新视角，将图视为由节点、边、三角形及k-单纯形构成的单纯复形，从而可在任意k-单纯形上定义图结构化数据。我们提出的主要贡献为Hodge-Laplacian异构图注意力网络（HL-HGAT），该网络专门设计用于学习跨k-单纯形的异构信号表示。HL-HGAT包含三个核心组件：HL卷积滤波器（HL滤波器）、单纯形投影（SP）和单纯形注意力池化（SAP）算子，三者均应用于k-单纯形。HL滤波器利用Hodge-Laplacian（HL）算子编码的k-单纯形独特拓扑结构，在第k阶HL算子的谱域内运行。为应对计算挑战，我们引入具有空间局部化特性的HL滤波器多项式近似方法。此外，我们提出池化算子以粗化k-单纯形，通过基于Transformer的自注意力与交叉注意力机制结合SP算子构建的单纯形注意力机制融合特征，从而捕捉多维度单纯形间的拓扑互连性。HL-HGAT在多种图应用场景中进行了全面评估，涵盖NP-hard问题、图多标签分类挑战，以及物流、计算机视觉、生物学、化学和神经科学领域的图回归任务。实验结果证明了该模型在处理各类图场景时的有效性和泛化能力。