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Graph rewiring and preprocessing for graph neural networks based on effective resistance

Graph neural networks (GNNs) are powerful models for processing graph data and have demonstrated state-of-the-art performance on many downstream tasks. However, existing GNNs can generally suffer from two limitations: over-smoothing and over-squashing, which can significantly undermine their learning ability for large graphs. To overcome these issues simultaneously, by utilizing the concept of effective resistances, we focus on minimizing total constrained resistance while identifying problematic edges using topological redundancy and bottleneck sparsity coefficients. We introduce a novel graph rewiring and preprocessing method guided by effective resistance (GPER), capable of edge addition or removal. Theoretical analysis validates our method's efficacy in mitigating over-smoothing and over-squashing. {In the experiments, we conduct node and graph classifications on the benchmark datasets and can achieve an average improvement of $7.8\%$ and $2.0\%$, respectively. We also conduct scalability analysis on large graphs with GCN and demonstrate that the proposed preprocess approach can reduce graph size by over $50\%$ while improve the performance.

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