Graph convolutions have been a pivotal element in learning graph representations. However, recursively aggregating neighboring information with graph convolutions leads to indistinguishable node features in deep layers, which is known as the over-smoothing issue. The performance of graph neural networks decays fast as the number of stacked layers increases, and the Dirichlet energy associated with the graph decreases to zero as well. In this work, we introduce a framelet system into the analysis of Dirichlet energy and take a multi-scale perspective to leverage the Dirichlet energy and alleviate the over-smoothing issue. Specifically, we develop a Framelet Augmentation strategy by adjusting the update rules with positive and negative increments for low-pass and high-passes respectively. Based on that, we design the Energy Enhanced Convolution (EEConv), which is an effective and practical operation that is proved to strictly enhance Dirichlet energy. From a message-passing perspective, EEConv inherits multi-hop aggregation property from the framelet transform and takes into account all hops in the multi-scale representation, which benefits the node classification tasks over heterophilous graphs. Experiments show that deep GNNs with EEConv achieve state-of-the-art performance over various node classification datasets, especially for heterophilous graphs, while also lifting the Dirichlet energy as the network goes deeper.

通过引入framelet系统到Dirichlet能量分析并采用多尺度视角来缓解过度平滑问题，我们设计了Energy Enhanced Convolution（EEConv），它是一种有效且实用的操作，被证明严格增强Dirichlet能量，并通过对异种图中的节点分类任务受益于多跳聚合特性。实验结果表明，具有EEConv的深度GNN在各种节点分类数据集上实现了最先进的性能，特别是对于异种图，同时增加了网络的Dirichlet能量。

通过小波框架增强图神经网络的迪利克莱能量