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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.28578 |
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Table of Contents:
- We propose the Tikhonov layer, a graph neural network layer that is interpretable by design: once trained, its learned parameters directly reveal which node features and which aspects of the graph topology were leveraged for prediction. In practice, the layer's propagation matrix takes the closed-form $R = (p(L)+Q)^{-1} Q$, where $L$ is the normalized graph Laplacian, $Q = diag(q_1,...,q_n)$ a learnable diagonal matrix of positive node-importance scores, and $p(\cdot)$ a learnable polynomial. For any input feature $x$, the layer output $Rx$ is the exact minimizer of a generalized graph Tikhonov problem that trades off node-level data fidelity against a topology-driven regularization penalty. The learned pair $\{\{q_i\},p\}$ constitutes a built-in explanation: large $q_i$ indicates that node $i$'s own features drive the prediction, while small $q_i$ signals reliance on the local graph topology; the shape of $p$ reveals whether homophily, heterophily, or a band-pass response is exploited. Expressivity is preserved by routing complexity through a dedicated, arbitrarily deep Q-network that produces the importance scores, while the Tikhonov layer itself remains transparent. We prove that distinct node-importance matrices yield distinct propagation operators, structurally coupling the explanation to the computation. Additionally, the Tikhonov layer provides, in a single layer, a global receptive field, mitigating both oversmoothing and oversquashing. Experiments on standard graph classification benchmarks confirm that the model matches (and sometimes outperforms) opaque baselines while producing interpretable and faithful explanations.