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

2026-05-27

AI Summary

This work addresses the limited interpretability of graph neural networks (GNNs), which often obscures how predictions depend on node features and graph structure. To this end, the authors propose the Tikhonov layer—a novel GNN layer whose output is the closed-form solution to a generalized graph Tikhonov regularization problem. By incorporating learnable node importance parameters and polynomial-based regularizers, the layer intrinsically embeds interpretability without sacrificing expressive power. It uniquely combines a global receptive field with robustness against over-smoothing and over-squashing, while guaranteeing strict alignment between the model’s explanations and its prediction logic. Experiments demonstrate that the model matches or exceeds the performance of non-interpretable baselines on standard graph classification benchmarks. Moreover, it yields intuitive interpretations: the importance parameter \( q_i \) indicates reliance on a node’s own features (large \( q_i \)) versus local topology (small \( q_i \)), and the learned polynomial shape reveals homophilic, heterophilic, or band-pass spectral response mechanisms.

Abstract

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.

Problem

• interpretable graph neural networks
• Tikhonov regularization
• graph topology
• node importance
• model transparency

Innovation

• Tikhonov regularization
• interpretable-by-design
• graph neural networks
• global receptive field
• node importance