$p$-Laplacian Based Graph Neural Networks

• URL: http://arxiv.org/abs/2111.07337v1
• Date: Sun, 14 Nov 2021 13:16:28 GMT
• Title: $p$-Laplacian Based Graph Neural Networks
• Authors: Guoji Fu and Peilin Zhao and Yatao Bian
• Abstract summary: Graph networks (GNNs) have demonstrated superior performance for semi-supervised node classification on graphs. We propose a new $p$-Laplacian based GNN model, termed as $p$GNN, whose message passing mechanism is derived from a discrete regularization framework. We show that the new message passing mechanism works simultaneously as low-pass and high-pass filters, thus making $p$GNNs effective on both homophilic and heterophilic graphs.
• Score: 27.747195341003263
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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