Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion

To combat oversmoothing in Graph Convolutional Networks, Sheaf Neural Networks (SNNs) were proposed as a generalization by equipping the graph with a sheaf structure and replacing the graph Laplacian with a sheaf Laplacian $\mathcal{L}$. Existing analyses connect sheaf diffusion to oversmoothing via the harmonic space ($\ker\mathcal{L}$), taking its absolute dimension as an indicator of anti-oversmoothing capacity. However, absolute dimension alone is not a reliable measure: certain sheaf configurations inflate $\dim \ker \mathcal{L}$ while their harmonic sections remain entirely constant, without enriching discriminative capacity. We instead introduce the first relative, geometric approach, yielding a precise characterisation of anti-oversmoothing capacity. Under natural conditions on stalk transportation and global sheaf structure, we establish an index-theoretic comparison criterion showing that one sheaf's harmonic space genuinely contains another's beyond trivial inflation. We illustrate this with a concrete instance and further introduce \textit{GyroSheaf}, a sheaf with curved gyrovector-space stalks, extending the criterion to the non-linear setting via local tangent-space linearization. Experiments across ten models confirm the theoretical criterion: sheaf models violating the criterion collapse despite possessing index jumps, while compliant models maintain depth-stable representations.

Temporal Sheaf Neural Networks with Dynamic Orthogonal Transport

We introduce Temporal Sheaf Neural Networks (TSNN), a temporal link prediction framework that equips each node with a time-varying orthogonal frame and compares node states only after explicit transport between local coordinate systems. In contrast to existing continuous-time graph models that operate in a shared global embedding space, TSNN models node-specific and evolving interaction semantics through dynamic local frames. The model parameterizes per-node frames via efficient low-rank Householder products, preserves stored hidden states exactly under frame updates, and uses a geometric-residual decoder that anchors predictions on transported distances while learning residual corrections. All computations are strictly causal and use only the pre-event history. We show that the symmetric degree-normalized sheaf Laplacian is orthogonally similar to the symmetric normalized graph Laplacian, with the random-walk normalized form similar in the corresponding degree metric; the full-active, feature-scaled diffusion used by TSNN is exactly a metric-gradient step on the combinatorial sheaf Dirichlet energy, with a degree-free monotone-descent and non-expansiveness guarantee. Frame drift perturbs updates only linearly. Across TGB v2 link-prediction and temporal-heterogeneous leaderboards, together with the DGB benchmark suite, TSNN matches or surpasses the strongest prior methods on most benchmarks, with the largest improvements on graphs exhibiting strong node-role heterogeneity. Ablations confirm the distinct benefit of dynamic frames, orthogonal transport, and geometric-residual decoding.