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

Permutation Equivariant Graph Framelets for Heterophilous Graph Learning