Fractional Heat Kernel for Semi-Supervised Graph Learning with Small Training Sample Size
Farid Bozorgnia, Vyacheslav Kungurtsev, Shirali Kadyrov, Mohsen Yousefnezhad
TL;DR
The paper tackles semi-supervised graph learning in regimes with very few labeled nodes by introducing a fractional heat-kernel diffusion framework. It develops spectral and approximation analyses for the heat kernel, defines a fractional Laplacian-based diffusion, and integrates these operators into GNNs (GIN, GCN, GAT) with multi-scale diffusion and continuous supervision. A self-training protocol leveraging fractional diffusion is proposed, alongside theoretical guarantees such as mass conservation and long-time behavior. Empirical results on synthetic and real datasets (Two-Moon, Cora) demonstrate improved performance in ultra-sparse labeling scenarios and illustrate the method's scalability and robustness, with significant gains particularly when labels are scarce.
Abstract
In this work, we introduce novel algorithms for label propagation and self-training using fractional heat kernel dynamics with a source term. We motivate the methodology through the classical correspondence of information theory with the physics of parabolic evolution equations. We integrate the fractional heat kernel into Graph Neural Network architectures such as Graph Convolutional Networks and Graph Attention, enhancing their expressiveness through adaptive, multi-hop diffusion. By applying Chebyshev polynomial approximations, large graphs become computationally feasible. Motivating variational formulations demonstrate that by extending the classical diffusion model to fractional powers of the Laplacian, nonlocal interactions deliver more globally diffusing labels. The particular balance between supervision of known labels and diffusion across the graph is particularly advantageous in the case where only a small number of labeled training examples are present. We demonstrate the effectiveness of this approach on standard datasets.