$p$-Laplacian Based Graph Neural Networks
This addresses a key limitation in GNNs for node classification on heterophilic graphs, which is important for applications like social networks or recommendation systems where node connections may not imply similarity.
The paper tackles the problem of graph neural networks (GNNs) performing poorly on heterophilic graphs where linked nodes have different labels, by proposing a $p$-Laplacian based GNN model ($^p$GNN) that works as low-pass and high-pass filters. The result shows that $^p$GNN significantly outperforms state-of-the-art GNNs on heterophilic benchmarks and achieves competitive performance on homophilic benchmarks.
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 simultaneously. 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. Hence, when the topology is non-informative 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 could 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 simultaneously as low-pass and high-pass filters, thus making $^p$GNNs are 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.