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.
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.
Subgraph sampling reduces the training cost of large-scale graph neural networks, but sampling criteria may overlook the geometric roles of edges. We propose a resistance-curvature-guided sampling framework built on ERC-LG, a curvature approximation method for large-scale graphs. ERC-LG combines Johnson-Lindenstrauss projections with regularized multi-GPU batched conjugate gradient solvers, avoiding explicit Laplacian pseudoinverse computation and full embedding storage. The resulting curvature informs node- and edge-sampling probabilities for constructing GNN training subgraphs. Experiments show numerical agreement with pseudoinverse-based curvature and reduced runtime compared with CG-only computation. ERC-LG-based sampling variants achieve the highest mean accuracy on six of seven real-world datasets in downstream node classification.
Oversmoothing is a fundamental limitation of deep graph neural networks (GNNs), where repeated message passing causes node representations to become increasingly similar, eventually collapsing toward a low-dimensional subspace. This phenomenon limits the effective depth of message-passing architectures and motivates the search for mechanisms that preserve representation diversity. In this paper, we study a recurrent graph neural network in which independent Gaussian noise is injected after every propagation step and analyze the resulting architecture as a stochastic dynamical system. Under a standard global contraction assumption on the deterministic update, we prove that the hidden representations form a geometrically ergodic Markov chain admitting a unique invariant probability measure. Our main theoretical result establishes an explicit positive lower bound on the expected stationary Dirichlet energy, proportional to both the noise variance and the spectral gap of the underlying graph. Consequently, the stationary representations cannot collapse onto the constant manifold, providing a rigorous guarantee that asymptotic oversmoothing is prevented in the sense of non-vanishing Dirichlet energy. Our analysis reveals persistent stochastic perturbations as a fundamentally different mechanism for combating oversmoothing, complementing existing deterministic approaches based on residual connections, normalization, and graph rewiring. Finally, numerical experiments on both linear and nonlinear recurrent graph neural networks closely match the theoretical predictions, illustrating the emergence of a stationary distribution and the predicted dependence of the limiting Dirichlet energy on the noise intensity.