Higher-order graph neural networks (HOGNNs) are an important class of GNN models that harness polyadic relations between vertices beyond plain edges. They have been used to eliminate issues such as over-smoothing or over-squashing, to significantly enhance the accuracy of GNN predictions, to improve the expressiveness of GNN architectures, and for numerous other goals. A plethora of HOGNN models have been introduced, and they come with diverse neural architectures, and even with different notions of what the "higher-order" means. This richness makes it very challenging to appropriately analyze and compare HOGNN models, and to decide in what scenario to use specific ones. To alleviate this, we first design an in-depth taxonomy and a blueprint for HOGNNs. This facilitates designing models that maximize performance. Then, we use our taxonomy to analyze and compare the available HOGNN models. The outcomes of our analysis are synthesized in a set of insights that help to select the most beneficial GNN model in a given scenario, and a comprehensive list of challenges and opportunities for further research into more powerful HOGNNs.

翻译：高阶图神经网络（HOGNNs）是一类重要的图神经网络模型，其利用超越普通边的顶点间多元关系。这类模型已被用于消除过度平滑或过度挤压等问题，显著提升图神经网络预测的准确性，增强图神经网络架构的表达能力，并实现众多其他目标。目前已有大量高阶图神经网络模型被提出，它们具有多样化的神经架构，甚至对“高阶”的定义也存在差异。这种丰富性使得对高阶图神经网络模型进行恰当分析和比较、并确定在何种场景下使用特定模型变得极具挑战性。为缓解这一问题，我们首先设计了一套深入的高阶图神经网络分类体系与蓝图框架，这有助于设计出性能最大化的模型。随后，我们运用该分类体系对现有高阶图神经网络模型进行分析与比较。分析成果凝练为一组指导性见解，可帮助在给定场景中选择最有效的图神经网络模型，并形成一份全面的挑战与机遇清单，为开发更强大的高阶图神经网络指明未来研究方向。