# Learning Parametrised Graph Shift Operators

arXiv: https://arxiv.org/abs/2101.10050 · HTML: https://mlsanity.com/paper/2101.10050/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 51 references that are in the map.
- First posted on arXiv 25 Jan 2021.

## Builds on (most cited first, 30 of 31)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Representation Learning on Graphs: Methods and Applications](https://mlsanity.com/paper/1709.05584/) (2017)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Towards Deeper Graph Neural Networks](https://mlsanity.com/paper/2007.09296/) (2020)
- [Position-aware Graph Neural Networks](https://mlsanity.com/paper/1906.04817/) (2019)
- [What graph neural networks cannot learn: depth vs width](https://mlsanity.com/paper/1907.03199/) (2019)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Spectral redemption: clustering sparse networks](https://mlsanity.com/paper/1306.5550/) (2013)
- [Coloring graph neural networks for node disambiguation](https://mlsanity.com/paper/1912.06058/) (2019)
- [Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel](https://mlsanity.com/paper/1309.4111/) (2013)
- [Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs](https://mlsanity.com/paper/1501.06087/) (2015)
- [Spectral Clustering of Graphs with the Bethe Hessian](https://mlsanity.com/paper/1406.1880/) (2014)
- [Stability of Graph Neural Networks to Relative Perturbations](https://mlsanity.com/paper/1910.09655/) (2019)
- [Optimal Laplacian regularization for sparse spectral community detection](https://mlsanity.com/paper/1912.01419/) (2019)

## Built on by (most cited first, 16 of 16)

- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [Modularity-Aware Graph Autoencoders for Joint Community Detection and Link Prediction](https://mlsanity.com/paper/2202.00961/) (2022)
- [Adaptive Kernel Graph Neural Network](https://mlsanity.com/paper/2112.04575/) (2021)
- [Weisfeiler and Leman go Hyperbolic: Learning Distance Preserving Node Representations](https://mlsanity.com/paper/2211.02501/) (2022)
- [How Does Message Passing Improve Collaborative Filtering?](https://mlsanity.com/paper/2404.08660/) (2024)
- [Towards Better Graph Representation Learning with Parameterized Decomposition & Filtering](https://mlsanity.com/paper/2305.06102/) (2023)
- [Permute Me Softly: Learning Soft Permutations for Graph Representations](https://mlsanity.com/paper/2110.01872/) (2021)
- [Edge-Varying Fourier Graph Networks for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2210.03093/) (2022)
- [Flexible Diffusion Scopes with Parameterized Laplacian for Heterophilic Graph Learning](https://mlsanity.com/paper/2409.09888/) (2024)
- [Unveiling the Sampling Density in Non-Uniform Geometric Graphs](https://mlsanity.com/paper/2210.08219/) (2022)
- [Demystifying Topological Message-Passing with Relational Structures: A Case Study on Oversquashing in Simplicial Message-Passing](https://mlsanity.com/paper/2506.06582/) (2025)
- [Which Graph Shift Operator? A Spectral Answer to an Empirical Question](https://mlsanity.com/paper/2602.06557/) (2026)
- [Graph Enabled Cross-Domain Knowledge Transfer](https://mlsanity.com/paper/2304.03452/) (2023)
- [Supervised Attention Using Homophily in Graph Neural Networks](https://mlsanity.com/paper/2307.05217/) (2023)
- [Topological Neural Networks: Mitigating the Bottlenecks of Graph Neural Networks via Higher-Order Interactions](https://mlsanity.com/paper/2402.06908/) (2024)
- [Local-Global Geometric Insights for Graph Neural Networks via Entropic Curvature](https://mlsanity.com/paper/2607.22381/) (2026)

## Abstract (arXiv)

In many domains data is currently represented as graphs and therefore, the graph representation of this data becomes increasingly important in machine learning. Network data is, implicitly or explicitly, always represented using a graph shift operator (GSO) with the most common choices being the adjacency, Laplacian matrices and their normalisations. In this paper, a novel parametrised GSO (PGSO) is proposed, where specific parameter values result in the most commonly used GSOs and message-passing operators in graph neural network (GNN) frameworks. The PGSO is suggested as a replacement of the standard GSOs that are used in state-of-the-art GNN architectures and the optimisation of the PGSO parameters is seamlessly included in the model training. It is proved that the PGSO has real eigenvalues and a set of real eigenvectors independent of the parameter values and spectral bounds on the PGSO are derived. PGSO parameters are shown to adapt to the sparsity of the graph structure in a study on stochastic blockmodel networks, where they are found to automatically replicate the GSO regularisation found in the literature. On several real-world datasets the accuracy of state-of-the-art GNN architectures is improved by the inclusion of the PGSO in both node- and graph-classification tasks.