Self-supervised learning has shown its promising capability in graph representation learning in recent work. Most existing pre-training strategies usually choose the popular Graph neural networks (GNNs), which can be seen as a special form of low-pass filter, fail to effectively capture heterophily. In this paper, we first present an experimental investigation exploring the performance of low-pass and high-pass filters in heterophily graph classification, where the results clearly show that high-frequency signal is important for learning heterophily graph representation. On the other hand, it is still unclear how to effectively capture the structural pattern of graphs and how to measure the capability of the self-supervised pre-training strategy in capturing graph structure. To address the problem, we first design a quantitative metric to Measure Graph Structure (MGS), which analyzes correlation between structural similarity and embedding similarity of graph pairs. Then, to enhance the graph structural information captured by self-supervised learning, we propose a novel self-supervised strategy for Pre-training GNNs based on the Metric (PGM). Extensive experiments validate our pre-training strategy achieves state-of-the-art performance for molecular property prediction and protein function prediction. In addition, we find choosing the suitable filter sometimes may be better than designing good pre-training strategies for heterophily graph classification.
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