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

The gist The authors develop a source-driven fractional heat-kernel framework for semi-supervised graph learning that combines nonlocal propagation with sustained label information to mitigate oversmoothing at long diffusion times.

How it works

The method introduces a fixed nonzero label source compatible with the Laplacian null space to prevent asymptotic collapse into that space, which provides 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.

The framework builds upon personalized propagation of neural predictions and graph diffusion convolution, adopting heat-kernel filtering to realize spectral diffusion over a graph. It introduces fractional heat-kernel diffusion with a source-driven propagation term, enabling a tunable nonlocal smoothing effect under sparse supervision. The compatible-source dynamics have the limit Π0U0 + (Lˢ)†F, which solves the fractional Poisson system.

The source term F is chosen so that it injects supervision only through the labeled nodes and, at the same time, remains orthogonal to the null space of the Laplacian used in the diffusion. This condition preserves the corresponding conserved label functionals and prevents secular drift in the source-driven solution.

Spectral Analysis and Approximation

The spectral decomposition of the heat kernel matrix e−tL yields e−tL = Ue−tΛU−1 = Xn k=1 e−tλk ϕkϕ⊤k, where modes with large eigenvalue decay strongly while modes with small eigenvalue are comparatively preserved. The kernel induces a diffusion distance on the vertices defined by Dt(i, j)2:= Xl∈V Ht(i, l) − Ht(j, l)2, which is related to the spectral decomposition.

Approximation techniques include:

1. Truncated Spectral Decomposition, where the error is controlled by the omitted signal energy and the cutoff eigenvalue λm+1.

2. Chebyshev Polynomial Approximation, which avoids eigendecomposition while maintaining accuracy for moderate t. For non-integer s, this approximation has an algebraic decay rate of O(m−2s).

Fractional Heat Kernel Dynamics

The fractional Laplacian Lˢ, s > 0, provides a tunable spectral response and nonlocal spatial coupling. The action of e−tLs on mode ϕk is multiplication by the scalar filter gs(λk) = e−tλs k.

Proposition 3.1 characterizes attenuation for a fixed Laplacian normalization and a common diffusion time t, showing that decreasing the fractional order increases attenuation on positive-frequency modes below λ = 1 and decreases it on modes above λ = 1. The ratio of the fractional filter to the classical one at a fixed eigenvalue is Rs(λ, t) = gs(λ)g1 (λ) = exp−t(λs − λ), which decays exponentially in t for 0 1.

Integration into GNN Architectures

The fractional heat kernel can be used as a graph propagation operator inside standard neural architectures, where the layer becomes H(l+1) = σ e−tlLs H(l)W(l). The parameter tl controls the diffusion scale at layer l, while s controls the spectral profile of the propagation.

Multi-scale heat kernel aggregation is achieved by H(l+1) = σ X R r=1 α(l)r e−t(l)r Ls H(l)W(l)r!, where the weights are parametrized by unconstrained logits η(l)r.

Source-Driven Framework and Deflation

The source term F is constructed so that it injects supervision only through the labeled nodes and, at the same time, remains orthogonal to the null space of the Laplacian used in the diffusion. The exact solution for a time-independent F is U(t) = e−tLsU0 + t · h(tLs)F.

To remove null-space components, deflation is developed by shifting the entire null space, defining Lˢγ:= Lˢ + γΠ0. The solution to the deflated flow converges to Uγ,∞ = (Lˢγ)−1F = (Lˢ)†F + γ−1Π0F.

Experimental Evaluation

Experiments on Two-Moon show that 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.

The framework provides both a mathematical account of sustained supervision under diffusion and a practical means of selecting its spectral weighting. The experiments identify the roles of these components under sparse supervision.

Conclusion

The source-driven fractional heat-kernel framework provides a precise mechanism for mitigating oversmoothing at long diffusion times by retaining label-dependent variation outside the null space at arbitrarily long diffusion times. The order tunes the strength of this preference, and the diffusion time sets the propagation horizon.

The framework thus provides both a mathematical account of sustained supervision under diffusion and a practical means of selecting its spectral weighting. The experiments identify the roles of these components under sparse supervision. The source-driven pipeline improves mean accuracy over GAT by 9.2 and 8.0 percentage points at one training label per class. The framework thus provides both a mathematical account of sustained supervision under diffusion and a practical means of selecting its spectral weighting. The experiments identify the roles of these components under sparse supervision.

The results illustrate classification with long propagation horizons, while the order sets the relative equilibrium weights λ−s k. The source sustains the non-null response, while the order tunes its spectral composition.

The source-driven pipeline improves mean accuracy over GAT by 9.2 and 8.0 percentage points at one training label per class. The framework thus provides both a mathematical account of sustained supervision under diffusion and a practical means of selecting its spectral weighting. The experiments identify the roles of these components under sparse supervision.

The framework thus provides both a mathematical account of sustained supervision under diffusion and a practical means of selecting its spectral weighting <ref:2510.

1. The system can mitigate oversmoothing at long diffusion times by implementing a fixed nonzero label source compatible with the Laplacian null space, which prevents asymptotic collapse into that space.

2. The system can achieve significant performance gains in semi-supervised settings, as demonstrated by the claim that source-driven pipeline improves mean accuracy over GAT by 9.2 and 8.0 percentage points at one training label per class on Cora and CiteSeer.

3. The system can dynamically adjust its spectral filtering capabilities by controlling the fractional order, as the fractional order controls the relative modal attenuation and the spectral weighting of the sustained response.

4. The system can utilize a multi-scale diffusion approach by using a layer form like "H(l+1) = σ X R r=1 α(l)r e−t(l)r Ls H(l)W(l)r !, where t(l)r > 0 are diffusion times."

5. The system can incorporate learned graph refinement by employing a GAT-refined graph in the diffusion stage, which involves building a new adjacency matrix A′ from embeddings using A′ = 1/2 Wk + 1/2 W.

6. The system can perform unsupervised node classification and label propagation by utilizing the long-time limit of the source response, which is given by (L s)†F, as this represents a precise mechanism for mitigating oversmoothing at long diffusion times.

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.