p-Laplacian Based Graph Neural Networks

Graph neural networks (GNNs) have demonstrated superior performance for semi-supervised node classification on graphs, as a result of their ability to exploit node features and topological information. However, most GNNs implicitly assume that the labels of nodes and their neighbors in a graph are the same or consistent, which does not hold in heterophilic graphs, where the labels of linked nodes are likely to differ. Moreover, when the topology is noninformative for label prediction, ordinary GNNs may work significantly worse than simply applying multi-layer perceptrons (MLPs) on each node. To tackle the above problem, we propose a new p-Laplacian based GNN model, termed as p GNN, whose message passing mechanism is derived from a discrete regularization framework and can be theoretically explained as an approximation of a polynomial graph filter defined on the spectral domain of p-Laplacians. The spectral analysis shows that the new message passing mechanism works as low-high-pass filters, thus rendering p GNNs effective on both homophilic and heterophilic graphs. Empirical studies on real-world and synthetic datasets validate our findings and demonstrate that p GNNs significantly outperform several state-of-the-art GNN architectures on heterophilic benchmarks while achieving competitive performance on homophilic benchmarks. Moreover, p GNNs can adaptively learn aggregation weights and are robust to noisy edges.