An introspection of graph structure learning: A graph skeleton extraction via minimum dominating set

Graph structure learning (GSL) is a data-driven learning approach that has garnered widespread attention in recent years. Nevertheless, the insufficient understanding of latent graph properties poses various challenges for effective graph modeling. This raises the following question: What type of graph skeleton can preserve the most crucial latent properties that significantly impact the performance of graph neural networks (GNNs) in downstream tasks? To this end, we have conducted a comprehensive study on three key graph properties: homophily, degree distribution, and connected components, and determined how these factors influence semi-supervised node classification tasks. Specifically, the influence of homophily on GNN performance is rigorously assessed. Motivated by the analysis, a dual-sparsity graph extraction method, based on the minimum dominating set (MDS), is proposed, to intelligently select informative edges under a given edge sampling ratio. This method effectively captures the scale-free characteristics of the degree distribution, and prioritizes the preservation of node connectivity. Experimental results show that homophily is a key factor in achieving high GNN accuracy. Additionally, the degree distribution and connected components describe the connectivity patterns of the graph from both local and global topological perspectives, which are highly correlated with node classification performance under the GNN message-passing mechanism. This work reveals the necessity of considering the graph skeleton and provides a stepping stone for facilitating GSL using these latent graph properties.