Fractional Heat Kernel for Semi-Supervised Graph Learning with Small Training Sample Size

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.