Advancing Graph Neural Networks with HL-HGAT: A Hodge-Laplacian and
  Attention Mechanism Approach for Heterogeneous Graph-Structured Data
Jinghan Huang Qiufeng Chen Yijun Bian Pengli Zhu Nanguang Chen Moo K. Chung Anqi Qiu
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.