Graph Neural Networks are highly effective at learning from relational data, leveraging node and edge features while maintaining the symmetries inherent to graph structures. However, many real-world systems, such as social or biological networks, exhibit complex interactions that are more naturally represented by higher-order topological domains. The emerging field of Geometric and Topological Deep Learning addresses this challenge by introducing methods that utilize and benefit from higher-order structures. Central to TDL is the concept of lifting, which transforms data representations from basic graph forms to more expressive topologies before the application of GNN models for learning. In this work, we propose a structural lifting strategy using Forman-Ricci curvature, which defines an edge-based network characteristic based on Riemannian geometry. Curvature reveals local and global properties of a graph, such as a network's backbones, i.e. coarse, structure-preserving graph geometries that form connections between major communities - most suitably represented as hyperedges to model information flows between clusters across large distances in the network. To this end, our approach provides a remedy to the problem of information distortion in message passing across long distances and graph bottlenecks - a phenomenon known in graph learning as over-squashing.

Graph neural networks (GNNs) have exhibited state-of-the-art performance across wide-range of domains such as recommender systems, material design, and drug repurposing. Yet message-passing GNNs suffer from over-squashing -- exponential compression of long-range information from distant nodes -- which limits expressivity. Rewiring techniques can ease this bottleneck; but their practical impacts are unclear due to the lack of a direct empirical over-squashing metric. We propose a rigorous, topology-focused method for assessing over-squashing between node pairs using the decay rate of their mutual sensitivity. We then extend these pairwise assessments to four graph-level statistics (prevalence, intensity, variability, extremity). Coupling these metrics with a within-graph causal design, we quantify how rewiring strategies affect over-squashing on diverse graph- and node-classification benchmarks. Our extensive empirical analyses show that most graph classification datasets suffer from over-squashing (but to various extents), and rewiring effectively mitigates it -- though the degree of mitigation, and its translation into performance gains, varies by dataset and method. We also found that over-squashing is less notable in node classification datasets, where rewiring often increases over-squashing, and performance variations are uncorrelated with over-squashing changes. These findings suggest that rewiring is most beneficial when over-squashing is both substantial and corrected with restraint -- while overly aggressive rewiring, or rewiring applied to minimally over-squashed graphs, is unlikely to help and may even harm performance. Our plug-and-play diagnostic tool lets practitioners decide -- before any training -- whether rewiring is likely to pay off.

Graph Neural Networks (GNNs) are models that leverage the graph structure to transmit information between nodes, typically through the message-passing operation. While widely successful, this approach is well known to suffer from the over-smoothing and over-squashing phenomena, which result in representational collapse as the number of layers increases and insensitivity to the information contained at distant and poorly connected nodes, respectively. In this paper, we present a unified view of these problems through the lens of vanishing gradients, using ideas from linear control theory for our analysis. We propose an interpretation of GNNs as recurrent models and empirically demonstrate that a simple state-space formulation of a GNN effectively alleviates over-smoothing and over-squashing at no extra trainable parameter cost. Further, we show theoretically and empirically that (i) GNNs are by design prone to extreme gradient vanishing even after a few layers; (ii) Over-smoothing is directly related to the mechanism causing vanishing gradients; (iii) Over-squashing is most easily alleviated by a combination of graph rewiring and vanishing gradient mitigation. We believe our work will help bridge the gap between the recurrent and graph neural network literature and will unlock the design of new deep and performant GNNs.

Graph Neural Networks (GNNs) revolutionize machine learning for graph-structured data, effectively capturing complex relationships. They disseminate information through interconnected nodes, but long-range interactions face challenges known as "over-squashing". This survey delves into the challenge of over-squashing in Graph Neural Networks (GNNs), where long-range information dissemination is hindered, impacting tasks reliant on intricate long-distance interactions. It comprehensively explores the causes, consequences, and mitigation strategies for over-squashing. Various methodologies are reviewed, including graph rewiring, novel normalization, spectral analysis, and curvature-based strategies, with a focus on their trade-offs and effectiveness. The survey also discusses the interplay between over-squashing and other GNN limitations, such as over-smoothing, and provides a taxonomy of models designed to address these issues in node and graph-level tasks. Benchmark datasets for performance evaluation are also detailed, making this survey a valuable resource for researchers and practitioners in the GNN field.

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.

Recent works have investigated the role of graph bottlenecks in preventing long-range information propagation in message-passing graph neural networks, causing the so-called `over-squashing' phenomenon. As a remedy, graph rewiring mechanisms have been proposed as preprocessing steps. Graph Echo State Networks (GESNs) are a reservoir computing model for graphs, where node embeddings are recursively computed by an untrained message-passing function. In this paper, we show that GESNs can achieve a significantly better accuracy on six heterophilic node classification tasks without altering the graph connectivity, thus suggesting a different route for addressing the over-squashing problem.