Over-Squashing in Riemannian Graph Neural Networks

Most graph neural networks (GNNs) are prone to the phenomenon of over-squashing in which node features become insensitive to information from distant nodes in the graph. Recent works have shown that the topology of the graph has the greatest impact on over-squashing, suggesting graph rewiring approaches as a suitable solution. In this work, we explore whether over-squashing can be mitigated through the embedding space of the GNN. In particular, we consider the generalization of Hyperbolic GNNs (HGNNs) to Riemannian manifolds of variable curvature in which the geometry of the embedding space is faithful to the graph's topology. We derive bounds on the sensitivity of the node features in these Riemannian GNNs as the number of layers increases, which yield promising theoretical and empirical results for alleviating over-squashing in graphs with negative curvature

FMGNN: Fused Manifold Graph Neural Network

Graph representation learning has been widely studied and demonstrated effectiveness in various graph tasks. Most existing works embed graph data in the Euclidean space, while recent works extend the embedding models to hyperbolic or spherical spaces to achieve better performance on graphs with complex structures, such as hierarchical or ring structures. Fusing the embedding from different manifolds can further take advantage of the embedding capabilities over different graph structures. However, existing embedding fusion methods mostly focus on concatenating or summing up the output embeddings, without considering interacting and aligning the embeddings of the same vertices on different manifolds, which can lead to distortion and impression in the final fusion results. Besides, it is also challenging to fuse the embeddings of the same vertices from different coordinate systems. In face of these challenges, we propose the Fused Manifold Graph Neural Network (FMGNN), a novel GNN architecture that embeds graphs into different Riemannian manifolds with interaction and alignment among these manifolds during training and fuses the vertex embeddings through the distances on different manifolds between vertices and selected landmarks, geometric coresets. Our experiments demonstrate that FMGNN yields superior performance over strong baselines on the benchmarks of node classification and link prediction tasks