A Generalized Tikhonov Layer for Interpretable-by-design Graph Neural Networks

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, $Q = \mathrm{diag}(q_1,\ldots,q_n)$ 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 explanations necessarily yield distinct propagation operators, preventing degenerate explanations by design. Additionally, the Tikhonov layer provides, in a single layer, a global receptive field under mild support-connectivity conditions, 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.