Recently, researchers have proposed a graph pooling operation, akin to the pooling process in conventional convolutional neural networks (CNN), aimed at reducing the computation cost of Graph convolutional neural networks (GCNNs). While most GCNN-based methods treat graph pooling as a node clustering problem and propose learning a cluster assignment matrix, existing clustering-based pooling methods tend to focus solely on the rough topology information of graphs, neglecting the exploitation of higher-order mutual connections among neighbors. In terms of message passing on graph, the ease of information passing on edges reflects the closeness between neighboring nodes, which significantly relies on the interconnectivity among neighbors. In this study, we address this gap by considering such local connection information and introducing a novel graph pooling method named RicciPool. We introduce discrete graph curvature, particularly Ollivier-Ricci curvature, as a measure of higher-order connectivity around an edge. Subsequently, we construct an Ollivier-Ricci flow formula to reweigh edge weights, leveraging the crucial information provided by Ricci curvature, particularly vital for extracting clusters in graphs. Building upon this foundation, we utilize the spectral clustering technique to learn a new cluster assignment matrix. Experimental results on multiple bioinformatics protein datasets and social networks underscore the effectiveness of our proposed method.
Graph Neural Networks (GNNs) have emerged as a powerful paradigm for learning on graph-structured data by iteratively propagating and aggregating information across edges. However, conventional message passing schemes often suffer from over-squashing, whereby exponentially large neighborhoods are compressed into fixed-dimensional embeddings, impeding effective long-range dependency learning. In this work, we introduce Ramanujan Propagation, a graph rewiring strategy that leverages Ramanujan graphs to alleviate topological bottlenecks in GNNs. We first establish that suitably chosen Ramanujan graphs guarantee non-negative resistance curvature, which mitigates over-squashing and facilitates efficient information flow. We then propose an algorithmic framework to construct a Ramanujan rewired graph that preserves the local connectivity of the original graph. Our experiments demonstrate that our method outperforms nine state-of-the-art rewiring techniques. These results establish Ramanujan graphs as a rigorous structural prior for scalable, topology-aware message passing in GNNs.
Training Graph Neural Networks on large graphs is challenged by the memory cost of storing all node representations across layers. We show that several existing scalable approaches can be written as structured modifications of the GNN propagation matrix, providing a unified perspective that exposes their respective limitations. In particular, graph coarsening replaces it by a low-rank approximation that enables spectral guarantees but assigns uniform representations to clustered nodes, while Cluster-GCN restricts the propagation matrix to intra-cluster connections that allow efficient batching but sever long-range information. These are complementary failures of the \emph{same} decomposition of the graph into groups of nodes. To obtain the best of both worlds, we propose \textbf{CoRe-GNN}, which performs both propagations in parallel at each layer: a coarsened inter-cluster term capturing long-range structure, and a local intra-cluster term preserving per-node discriminability. We prove that CoRe-GNN inherits analogous approximation guarantees to those of graph coarsening, and introduce a natural cluster-based \emph{batching scheme} that scales to graphs with millions of nodes. On node classification benchmarks spanning homophilic, heterophilic, large-scale, and long-range graphs, CoRe-GNN outperforms both graph coarsening and Cluster-GCN baselines. Notably, CoRe-GNN reaches competitive accuracy on \emph{long-range} tasks, while remaining memory-efficient through batching.
Curvature notions on graphs, particularly Ollivier-Ricci and Forman, have emerged as powerful tools for addressing fundamental issues in Graph Neural Networks (GNNs) such as oversmoothing and oversquashing, but rely almost exclusively on local edge-level comparisons and therefore fail to certify how information actually propagates over long distances. We introduce Entropic Curvature, a global, transport-based curvature obtained by extending the Lott-Sturm-Villani framework to graphs through the displacement convexity of entropy along Wasserstein geodesics. We define a tractable Weak Entropic Curvature proxy that lower-bounds the global entropic curvature, and from it derive (i) a Poincare-type inequality controlling oversmoothing, (ii) a transport-entropy generalization bound, and (iii) an expansion paradox proving that sparsity, strong spectral expansion, and positive entropic curvature cannot coexist in large graphs, unifying oversmoothing and oversquashing as opposite ends of a single curvature spectrum. We translate the theory into three practical mechanisms, the E-Gate aggregator, the ENT structural encoding, and Midpoint-Completion Rewiring (MCR), and benchmark them against SDRF, FoSR, BORF, LCP, and Graph Ricci Flow on six node-classification benchmarks, and graph-classification.