# Perfect Recovery for Random Geometric Graph Matching with Shallow Graph Neural Networks

arXiv: https://arxiv.org/abs/2402.07340 · HTML: https://mlsanity.com/paper/2402.07340/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 37 of its 60 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 30 of 37)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [What graph neural networks cannot learn: depth vs width](https://mlsanity.com/paper/1907.03199/) (2019)
- [Convergence and Stability of Graph Convolutional Networks on Large Random Graphs](https://mlsanity.com/paper/2006.01868/) (2020)
- [Graph Convolution for Semi-Supervised Classification: Improved Linear Separability and Out-of-Distribution Generalization](https://mlsanity.com/paper/2102.06966/) (2021)
- [Generalization Analysis of Message Passing Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2202.00645/) (2022)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [Learning Theory Can (Sometimes) Explain Generalisation in Graph Neural Networks](https://mlsanity.com/paper/2112.03968/) (2021)
- [Efficient random graph matching via degree profiles](https://mlsanity.com/paper/1811.07821/) (2018)
- [Settling the Sharp Reconstruction Thresholds of Random Graph Matching](https://mlsanity.com/paper/2102.00082/) (2021)
- [Transferability Properties of Graph Neural Networks](https://mlsanity.com/paper/2112.04629/) (2021)
- [Spectral Graph Matching and Regularized Quadratic Relaxations II: ErdHos-R'enyi Graphs and Universality](https://mlsanity.com/paper/1907.08883/) (2019)
- [Partial Recovery of ErdHos-R'enyi Graph Alignment via k-Core Alignment](https://mlsanity.com/paper/1809.03553/) (2018)
- [Correlated Stochastic Block Models: Exact Graph Matching with Applications to Recovering Communities](https://mlsanity.com/paper/2107.06767/) (2021)
- [Random Graph Matching in Geometric Models: the Case of Complete Graphs](https://mlsanity.com/paper/2202.10662/) (2022)
- [On the Non-asymptotic and Sharp Lower Tail Bounds of Random Variables](https://mlsanity.com/paper/1810.09006/) (2018)
- [Mean-field theory of graph neural networks in graph partitioning](https://mlsanity.com/paper/1810.11908/) (2018)
- [Random Graph Matching with Improved Noise Robustness](https://mlsanity.com/paper/2101.11783/) (2021)
- [Matching recovery threshold for correlated random graphs](https://mlsanity.com/paper/2205.14650/) (2022)
- [Graph and graphon neural network stability](https://mlsanity.com/paper/2010.12529/) (2020)
- [Strong recovery of geometric planted matchings](https://mlsanity.com/paper/2107.05567/) (2021)
- [Optimal Inference in Contextual Stochastic Block Models](https://mlsanity.com/paper/2306.07948/) (2023)

## Abstract (arXiv)

We study the graph matching problem in the presence of vertex feature information using shallow graph neural networks. Specifically, given two graphs that are independent perturbations of a single random geometric graph with sparse binary features, the task is to recover an unknown one-to-one mapping between the vertices of the two graphs. We show under certain conditions on the sparsity and noise level of the feature vectors, a carefully designed two-layer graph neural network can, with high probability, recover the correct mapping between the vertices with the help of the graph structure. Additionally, we prove that our condition on the noise parameter is tight up to logarithmic factors. Finally, we compare the performance of the graph neural network to directly solving an assignment problem using the noisy vertex features and demonstrate that when the noise level is at least constant, this direct matching fails to achieve perfect recovery, whereas the graph neural network can tolerate noise levels growing as fast as a power of the size of the graph. Our theoretical findings are further supported by numerical studies as well as real-world data experiments.