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.

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