The nature of heterophilous graphs is significantly different with that of homophilous graphs, which causes difficulties in early graph neural network models and suggests aggregations beyond 1-hop neighborhood. In this paper, we develop a new way to implement multi-scale extraction via constructing Haar-type graph framelets with desired properties of permutation equivariance, efficiency, and sparsity, for deep learning tasks on graphs. We further design a graph framelet neural network model PEGFAN (Permutation Equivariant Graph Framelet Augmented Network) based on our constructed graph framelets. The experiments are conducted on a synthetic dataset and 9 benchmark datasets to compare performance with other state-of-the-art models. The result shows that our model can achieve best performance on certain datasets of heterophilous graphs (including the majority of heterophilous datasets with relatively larger sizes and denser connections) and competitive performance on the remaining.

翻译：异配图的本质与同配图存在显著差异，这导致早期图神经网络模型难以处理此类数据，并提示需要超越1跳邻域的聚合操作。本文通过构造具有置换等变性、高效性和稀疏性等理想性质的Haar型图框架let，提出了一种实现多尺度特征提取的新方法，用于图上的深度学习任务。我们进一步基于所构造的图框架let设计了图框架let神经网络模型PEGFAN（置换等变图框架let增强网络）。在合成数据集和9个基准数据集上进行了实验，以与其他最先进模型进行性能比较。结果表明，我们的模型在特定异配图数据集（包括大多数规模较大且连接更密集的异配数据集）上取得了最佳性能，并在其余数据集上展现了具有竞争力的表现。