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（GNN）已被证明能有效捕捉图中节点间的关系。本研究引入一种新视角，将图视为包含节点、边、三角形及$k$-单纯形的单纯复形，从而可在任意$k$-单纯形上定义图结构数据。我们的贡献在于提出霍奇-拉普拉斯异构图注意力网络（HL-HGAT），旨在学习跨$k$-单纯形的异质信号表示。HL-HGAT包含三个关键组件：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，均应用于$k$-单纯形。HL-滤波器利用由霍奇-拉普拉斯（HL）算子编码的$k$-单纯形独特拓扑结构，在第$k$个HL算子的谱域内运行。为应对计算挑战，我们引入HL-滤波器的多项式近似，具有空间局部化特性。此外，我们提出一种池化算子，通过基于Transformer的自注意力和交叉注意力中的单纯形注意力机制，结合SP算子对$k$-单纯形进行粗化，从而捕捉跨多个维度单纯形的拓扑互连。HL-HGAT在多种图应用场景中得到全面评估，包括NP困难问题、图多标签与分类挑战，以及物流、计算机视觉、生物学、化学和神经科学中的图回归任务。结果表明，该模型在处理广泛的基于图的问题时具有高效性和通用性。