Graph-Based Semi-Supervised Learning

We develop a source-driven fractional heat-kernel framework for semi-super-vised graph learning that combines nonlocal propagation with sustained label information. A fixed nonzero label source compatible with the Laplacian null space prevents asymptotic collapse into that space, providing a mechanism for mitigating oversmoothing at long diffusion times. The fractional order controls the relative modal attenuation and the spectral weighting of the sustained response, while the diffusion time sets the propagation horizon. We characterize conservation laws and equilibria on normalized and disconnected graphs, develop a null-space deflation, and analyze the approximation of the propagators. On Two-Moon, fractional orders improve source-free propagation, while compatible source-driven diffusion exceeds $96\%$ mean accuracy with one training label per class at orders $0.8$ and $1$. On Cora and CiteSeer, the source-driven pipeline improves mean accuracy over GAT by $9.2$ and $8.0$ percentage points at one training label per class, with model selection on $500$ labeled validation nodes and closely comparable classical and fractional pipeline configurations; it matches GAT on PubMed and trails it at ten and twenty labels per class on Cora. Within GraphHeat, validation-based exponent selection at a common diffusion time yields a paired gain of $1.45$ percentage points at one label per class.

Standard deep learning architectures used for classification generate label predictions with a projection head and softmax activation function. Although successful, these methods fail to leverage the relational information between samples for generating label predictions. In recent works, graph-based learning techniques, namely Laplace learning, have been heuristically combined with neural networks for both supervised and semi-supervised learning (SSL) tasks. However, prior works approximate the gradient of the loss function with respect to the graph learning algorithm or decouple the processes; end-to-end integration with neural networks is not achieved. In this work, we derive backpropagation equations, via the adjoint method, for inclusion of a general family of graph learning layers into a neural network. The resulting method, distinct from graph neural networks, allows us to precisely integrate similarity graph construction and graph Laplacian-based label propagation into a neural network layer, replacing a projection head and softmax activation function for general classification task. Our experimental results demonstrate smooth label transitions across data, improved generalization and robustness to adversarial attacks, and improved training dynamics compared to a standard softmax-based approach.

This paper addresses domain adaptation in graph-structured data under chronological splits. We consider transductive semi-supervised node classification (SSNC), in which labels of historical nodes are used to predict labels of more recent nodes. Temporal dependencies in graph connectivity induce distribution shifts across time, often degrading the performance of models trained on past nodes. Despite its practical relevance, domain adaptation under chronological graph splits remains relatively underexplored. We propose Symmetric Temporal Averaging Message Passing (STAMP), a timestamp-balanced operator that replaces standard neighborhood aggregation with temporally aligned propagation. We call its specialized variant for the heterogeneous ogbn-mag dataset STAMP-mag, reserving STAMP for the general formulation. By balancing messages across timestamps, STAMP enforces first-moment invariance across temporal domains while remaining scalable and easy to implement. Unlike many existing domain adaptation methods, whose assumptions are difficult to verify on graphs, STAMP relies on transparent, practically checkable conditions and is supported by theoretical analysis. Empirically, STAMP and STAMP-mag improve accuracy across all evaluated temporal graph benchmarks. STAMP achieves state-of-the-art performance on ogbn-arxiv, while STAMP-mag does so on ogbn-mag. For temporal-split Pokec, where no established benchmark exists, applying STAMP to the state-of-the-art model for the original dataset consistently improves performance under our chronological split. These results demonstrate that timestamp-balanced propagation is an effective and practical approach to mitigating temporal distribution shift in graph learning.