# Graph Matching with Partially-Correct Seeds

arXiv: https://arxiv.org/abs/2004.03816 · HTML: https://mlsanity.com/paper/2004.03816/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Apr 2020.

## Built on by (most cited first, 10 of 10)

- [Exact Matching of Random Graphs with Constant Correlation](https://mlsanity.com/paper/2110.05000/) (2021)
- [Random Graph Matching with Improved Noise Robustness](https://mlsanity.com/paper/2101.11783/) (2021)
- [A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation](https://mlsanity.com/paper/2306.00266/) (2023)
- [The Power of D-hops in Matching Power-Law Graphs](https://mlsanity.com/paper/2102.12975/) (2021)
- [Seeded graph matching for the correlated Gaussian Wigner model via the projected power method](https://mlsanity.com/paper/2204.04099/) (2022)
- [SeedGNN: Graph Neural Networks for Supervised Seeded Graph Matching](https://mlsanity.com/paper/2205.13679/) (2022)
- [Graph Matching via convex relaxation to the simplex](https://mlsanity.com/paper/2310.20609/) (2023)
- [The graph alignment problem: fundamental limits and efficient algorithms](https://mlsanity.com/paper/2404.12418/) (2024)
- [Efficient Algorithms for Exact Graph Matching on Correlated Stochastic Block Models with Constant Correlation](https://mlsanity.com/paper/2305.19666/) (2023)
- [Statistical Limits for Testing Correlation of Hypergraphs](https://mlsanity.com/paper/2202.05888/) (2022)

## Abstract (arXiv)

Graph matching aims to find the latent vertex correspondence between two edge-correlated graphs and has found numerous applications across different fields. In this paper, we study a seeded graph matching problem, which assumes that a set of seeds, i.e., pre-mapped vertex-pairs, is given in advance. While most previous work requires all seeds to be correct, we focus on the setting where the seeds are partially correct. Specifically, consider two correlated graphs whose edges are sampled independently from a parent ER graph G(n,p). A mapping between the vertices of the two graphs is provided as seeds, of which an unknown β fraction is correct. We first analyze a simple algorithm that matches vertices based on the number of common seeds in the 1-hop neighborhoods, and then further propose a new algorithm that uses seeds in the 2-hop neighborhoods. We establish non-asymptotic performance guarantees of perfect matching for both 1-hop and 2-hop algorithms, showing that our new 2-hop algorithm requires substantially fewer correct seeds than the 1-hop algorithm when graphs are sparse. Moreover, by combining our new performance guarantees for the 1-hop and 2-hop algorithms, we attain the best-known results (in terms of the required fraction of correct seeds) across the entire range of graph sparsity and significantly improve the previous results in cite10.14778/2794367.2794371,lubars2018correcting when p≥ n-5/6. For instance, when p is a constant or p=n-3/4, we show that only Omega(√nlog n) correct seeds suffice for perfect matching, while the previously best-known results demand Omega(n) and Omega(n3/4log n) correct seeds, respectively. Numerical experiments corroborate our theoretical findings, demonstrating the superiority of our 2-hop algorithm on a variety of synthetic and real graphs.