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.

本研究提出了一个新的视角，将图形视为一个单纯复合体，其中包括节点、边、三角形和 k - 单纯形，从而能够在任意 k - 单纯形上定义图结构化数据。我们的贡献是 Hodge-Laplacian 异构图注意网络（HL-HGAT），旨在学习 k - 单纯形间的异构信号表示。HL-HGAT 包括三个关键组件：HL 卷积滤波器（HL-filters）、单纯形投影（SP）和单纯形注意池（SAP）运算符，应用于 k - 单纯形。HL-filters 通过 Hodge-Laplacian（HL）算子编码的 k - 单纯形的独特拓扑性质，在 k-th HL 算子的频谱域内进行运算。为了解决计算挑战，我们引入了 HL-filters 的多项式逼近方法，具有空间定位性质。此外，我们提出了一个池化运算符，通过 transformers 和 SP 运算符的自注意力和跨注意力机制，对 k - 单纯形进行粗化，并通过特征组合来捕获多个维度的单纯形之间的拓扑连接。HL-HGAT 在包括 NP 困难问题、图多标签和分类挑战，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中进行了全面评估。结果表明，该模型在处理各种图形场景方面具有高效性和多功能性。

用 HL-HGAT 推动图神经网络的发展：一种面向异构图结构数据的 Hodge-Laplacian 和注意机制方法

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

With the rapid advancement of AI applications, the growing needs for data
privacy and model robustness have highlighted the importance of machine
unlearning, especially in thriving graph-based scenarios. However, most
existing graph unlearning strategies primarily rely on well-designed
architectures or manual process, rendering them less user-friendly and posing
challenges in terms of deployment efficiency. Furthermore, striking a balance
between unlearning performance and framework generalization is also a pivotal
concern. To address the above issues, we propose \underline{\textbf{M}}utual
\underline{\textbf{E}}volution \underline{\textbf{G}}raph
\underline{\textbf{U}}nlearning (MEGU), a new mutual evolution paradigm that
simultaneously evolves the predictive and unlearning capacities of graph
unlearning. By incorporating aforementioned two components, MEGU ensures
complementary optimization in a unified training framework that aligns with the
prediction and unlearning requirements. Extensive experiments on 9 graph
benchmark datasets demonstrate the superior performance of MEGU in addressing
unlearning requirements at the feature, node, and edge levels. Specifically,
MEGU achieves average performance improvements of 2.7\%, 2.5\%, and 3.2\%
across these three levels of unlearning tasks when compared to state-of-the-art
baselines. Furthermore, MEGU exhibits satisfactory training efficiency,
reducing time and space overhead by an average of 159.8x and 9.6x,
respectively, in comparison to retraining GNN from scratch.

通过互相进化的图形反学习（MEGU），我们提出了一种新的互相进化模式，同时发展了图形反学习的预测和学习能力，解决了在 AI 应用中数据隐私和模型健壮性的增长需求。