curvature-gated sheaf diffusion

This work addresses the limitations of traditional unsupervised community detection methods on heterophilic graphs, where connected nodes often belong to different classes, and existing approaches either ignore node features or rely on opaque mechanisms. To overcome these challenges, the authors propose Curvature-Guided Stratified Diffusion (CGSD), a novel framework that uniquely leverages discrete Forman–Ricci curvature as the sole topological signal throughout an end-to-end unsupervised pipeline. Key innovations include a curvature-gated stratified diffusion encoder, a curvature-aware spectral clustering module (CSpec), and the first unified benchmark tailored for fully unsupervised methods. Evaluated on five heterophilic graph benchmarks, CGSD substantially outperforms nine baselines, achieving particularly strong results on Wisconsin and Chameleon datasets. CSpec yields a 15% average improvement in Normalized Mutual Information (NMI, p=0.008), and the curvature distribution demonstrates high interpretability.

Existing neural layer diffusion models rely on SVD-based normalization and dense edge-restriction mappings, leading to high computational overhead, gradient instability, and performance degradation as stalk dimension increases. To address these issues, we propose Polynomial Neural Stratified Diffusion (PolyNSD): a spectral diffusion framework leveraging K-th order orthogonal polynomials of the normalized layer Laplacian to enable explicit K-hop propagation over cell complexes. PolyNSD introduces trainable convex-combination spectral responses, spectral rescaling, and residual/gated pathways, while employing only diagonal restriction mappings—thereby decoupling model performance from stalk dimension. This design effectively mitigates heterophily and oversmoothing, enhancing training stability and inference efficiency. Evaluated on both homophilic and heterophilic graph benchmarks, PolyNSD achieves state-of-the-art accuracy with significantly reduced memory consumption and runtime, eliminating reliance on SVD decomposition and dense parameterization.

This work addresses the limitations of deep graph neural networks (GNNs), which often suffer from representational collapse and signal attenuation due to repeated neighborhood aggregation, thereby failing to leverage deeper architectures effectively. To overcome these issues, the authors propose a neural diffusion mechanism based on layer-wise adjacency operators, incorporating matrix-valued edge functions in place of scalar attention weights, node representation normalization, odd nonlinear activations, and gating strategies. This design preserves information fidelity during deep propagation and successfully alleviates the depth bottleneck inherent in conventional GNNs. Empirical evaluations demonstrate that the proposed method achieves up to a 30-percentage-point improvement in accuracy on synthetic long-range dependency benchmarks and consistently outperforms standard GNNs and non-local spectral diffusion (NSD) baselines across multiple real-world graph datasets.

This work addresses the over-smoothing problem in neural layer diffusion—where representation degradation leads to loss of discriminative information—by introducing a novel perspective grounded in quiver representation theory. It models cellular layer structures as representations of associated incidence quivers and uncovers the algebraic structure of the harmonic space in the diffusion limit. Over-smoothing is formally characterized for the first time as a degeneration phenomenon in representation geometry. To promote balanced geometric configurations, a moment map regularizer from geometric invariant theory is incorporated. The study further identifies a structural obstruction inherent in equidimensional stalk architectures and demonstrates that breaking symmetry via non-uniform stalk dimensions significantly reduces variance or improves validation performance on heterophilic benchmarks, while also achieving superior adaptive stability under rectangular settings.

Standard message-passing paradigms in Graph Neural Networks (GNNs) suffer from inherent limitations including over-smoothing, over-compression, and restricted node-level expressivity. To address these issues, we propose Bundle Neural Networks (BuNN), the first GNN framework that incorporates flat vector bundles—concepts from differential geometry—into graph representation learning. BuNN establishes a vector-bundle-based message diffusion mechanism, enabling continuous-dynamic modeling via diffusion-type partial differential equations (PDEs). We theoretically prove that BuNN achieves universal node-level expressivity and alleviates over-compression from a geometric perspective. Methodologically, BuNN integrates orthogonal mapping-driven graph augmentation, PDE-inspired feature evolution, a sheaf-compatible discretization scheme, and injective positional encoding. Under both transductive and inductive settings, BuNN attains state-of-the-art performance across multiple standard benchmarks. Synthetic experiments further demonstrate its strong robustness against over-smoothing and over-compression.

This work presents the first systematic evaluation of Sheaf Neural Networks (SNNs) in inductive learning, establishing a comprehensive benchmark across a design space that includes diffusion mechanisms, restriction map parameterizations, stalk dimensions, and underlying GNN architectures. To enable scalable batch training, the authors introduce a message-passing implementation that avoids explicit construction of the sheaf Laplacian. Based on 1,890 experiments across 14 datasets, the study reveals that restriction maps are pivotal—generic learnable mappings outperform specialized structures—and that the overall GNN architecture exerts a stronger influence on performance than sheaf-specific components. While SNNs demonstrate effective transfer to inductive tasks and cross-dataset generalization, they do not surpass the strongest baselines; moreover, optimizing peripheral architectural choices proves more beneficial than fine-tuning the sheaf operator itself.

Existing sheaf neural networks lack an effective hierarchical pooling mechanism for multiscale graph modeling. This work proposes HiSP, a framework that introduces, for the first time, a learnable sheaf-aware hierarchical pooling scheme. Built upon local spectral coarsening, HiSP projects fine-grained stalk features onto low-frequency modes of the cluster-wise sheaf Laplacian and preserves sheaf energy consistency via Galerkin operators. The method integrates cochain-level prolongation maps with a lifting-based sheaf Laplacian, enabling efficient batch processing within PyTorch Geometric. HiSP not only explicitly distinguishes and quantifies truncation and realization losses incurred during coarsening but also significantly enhances multiscale representation capacity while preserving the underlying sheaf structure.

This study addresses the over-smoothing problem in deep neural networks and the unreliability of existing harmonic space dimension metrics by proposing a relative geometric criterion grounded in index theory to precisely characterize over-smoothing resistance. Pioneering a relative geometric analysis perspective, this work integrates local tangent space linearization with the stalk structure of curved gyrovector spaces to construct a nonlinear gyrosheaf diffusion model, thereby overcoming the limitations of absolute dimensionality. Extensive experiments across ten architectures validate the effectiveness of the proposed criterion. Notably, the gyrosheaf diffusion model successfully maintains representation stability and prevents collapse in deep networks, confirming the theoretical validity of this approach. These findings establish a robust framework for analyzing and mitigating over-smoothing through rigorous geometric principles rather than conventional spectral methods.

This work addresses the limitation of conventional graph neural networks (GNNs) in handling node features represented as Gaussian distributions, which often disregard their intrinsic geometric and algebraic structures, leading to information loss. To overcome this, the authors propose Gaussian Simplex Neural Networks (GSNNs), the first GNN framework that explicitly incorporates the geometry of the Gaussian manifold. Leveraging cellular sheaf theory, they design a message-passing mechanism tailored for Gaussian-distributed features and extend the graph Laplacian operator to the Gaussian space while rigorously preserving its essential mathematical properties. Experimental results on both synthetic and real-world datasets demonstrate that GSNNs significantly outperform baseline approaches that naively concatenate Gaussian parameters, thereby validating their effectiveness in capturing and leveraging the structural information inherent in Gaussian representations.

This study addresses the failure of long-range information propagation and the over-smoothing bottleneck in graph neural networks by proposing the ONDA framework. To the best of our knowledge, this work is the first to couple wave dynamics with matrix-valued transport, leveraging operator-valued information waves to facilitate long-range graph learning. By introducing second-order oscillatory dynamics alongside layered transport operators, ONDA ensures that cross-node influence remains undiminished during propagation, thereby overcoming the fundamental limitations of conventional diffusion models. Extensive experiments demonstrate that ONDA significantly outperforms existing state-of-the-art methods across diverse benchmark tasks, including long-range propagation, graph transfer learning, and heterogeneous graph modeling.