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

arXiv: https://arxiv.org/abs/2605.28578 · HTML: https://mlsanity.com/paper/2605.28578/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 32 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 22 of 22)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [GNNExplainer: Generating Explanations for Graph Neural Networks](https://mlsanity.com/paper/1903.03894/) (2019)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Parameterized Explainer for Graph Neural Network](https://mlsanity.com/paper/2011.04573/) (2020)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Graph Neural Networks with convolutional ARMA filters](https://mlsanity.com/paper/1901.01343/) (2019)
- [Understanding Attention and Generalization in Graph Neural Networks](https://mlsanity.com/paper/1905.02850/) (2019)
- [Contextual Stochastic Block Models](https://mlsanity.com/paper/1807.09596/) (2018)
- [ProtGNN: Towards Self-Explaining Graph Neural Networks](https://mlsanity.com/paper/2112.00911/) (2021)
- [Towards Self-Explainable Graph Neural Network](https://mlsanity.com/paper/2108.12055/) (2021)
- [Community detection thresholds and the weak Ramanujan property](https://mlsanity.com/paper/1311.3085/) (2013)
- [Explaining the Explainers in Graph Neural Networks: a Comparative Study](https://mlsanity.com/paper/2210.15304/) (2022)
- [Design of graph filters and filterbanks](https://mlsanity.com/paper/1711.02046/) (2017)
- [Reconsidering Faithfulness in Regular, Self-Explainable and Domain Invariant GNNs](https://mlsanity.com/paper/2406.15156/) (2024)
- [Can You Hear Me Now? A Benchmark for Long-Range Graph Propagation](https://mlsanity.com/paper/2512.17762/) (2025)
- [GNN Explanations that do not Explain and How to find Them](https://mlsanity.com/paper/2601.20815/) (2026)
- [Bermuda Triangles: GNNs Fail to Detect Simple Topological Structures](https://mlsanity.com/paper/2105.00134/) (2021)

## Built on by (most cited first, 1 of 1)

- [Reachability is not enough: Diagnosing long-range behavior in GNNs](https://mlsanity.com/paper/2609.33674/) (2026)

## Abstract (arXiv)

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(q1,...,qn) a learnable diagonal matrix of positive node-importance scores, and p(·) 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 qi,p constitutes a built-in explanation: large qi indicates that node i's own features drive the prediction, while small qi 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.