We present a new effective and scalable framework for training GNNs in node classification tasks, based on the effective resistance, a powerful tool solidly rooted in graph theory. Our approach progressively refines the GNN weights on an extensive sequence of random spanning trees, suitably transformed into path graphs that retain essential topological and node information of the original graph. The sparse nature of these path graphs substantially lightens the computational burden of GNN training. This not only enhances scalability but also effectively addresses common issues like over-squashing, over-smoothing, and performance deterioration caused by overfitting in small training set regimes. We carry out an extensive experimental investigation on a number of real-world graph benchmarks, where we apply our framework to graph convolutional networks, showing simultaneous improvement of both training speed and test accuracy over a wide pool of representative baselines.

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