Advancing Graph Neural Networks with HL-HGAT: A Hodge-Laplacian and Attention Mechanism Approach for Heterogeneous Graph-Structured Data

TL;DR

This paper introduces HL-HGAT, a novel graph neural network that models graphs as simplicial complexes using Hodge-Laplacian operators and attention mechanisms, enabling advanced analysis of heterogeneous, multi-dimensional graph data.

Contribution

The paper proposes HL-HGAT, a new GNN architecture that incorporates Hodge-Laplacian spectral filters, simplicial projection, and attention pooling to effectively learn from complex simplicial structures.

Findings

Demonstrates superior performance on NP-hard problems and classification tasks.

Effective in diverse domains including logistics, biology, and neuroscience.

Shows versatility and robustness across multiple graph applications.

Abstract

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.