CauchyGCN: Preserving Local Smoothness in Graph Convolutional Networks via a Cauchy-Based Message-Passing Scheme and Clustering Analysis

In Graph Convolutional Networks (GCNs), a message-passing scheme explicitly learns and reasons node representations via aggregation and propagation of neighboring information over the graph topology. Most existing message-passing schemes are grounded in Laplacian smoothing, which seeks to maintain the similarity of node representations in the hidden feature space (local smoothness) among neighboring nodes, ensuring their labeling consistency (global smoothness). This often leads to Laplacian smoothing imposing strict penalization on distant neighbors. Because some distant neighbors are inter-class or represent some necessary intra-class patterns, strict penalization of distant neighbors can fail to preserve local smoothness effectively as expected thus introducing noise, mixing representations, and failing to capture valuable hidden patterns. Although recent research has introduced various strategies, including graph filters, k-hop jumps, and bounded penalties to tackle this issue, these methods often fall short of explicitly capturing and preserving the local smoothness over the original topology. In this paper, we present CauchyGCN, which enhances preserving local smoothness in a more interpretable approach. CauchyGCN comprises two key components: 1) a Cauchy smoothing message-passing scheme that explains and preserves local smoothness in each hidden layer, and 2) an unsupervised clustering analysis that simultaneously improves the classifier’s capacity to learn both local and global smoothness. We conduct comprehensive experiments using five benchmark datasets to assess the performance of CauchyGCN in semi-supervised node classification tasks compared to state-of-the-art GCNs.