# Improved Achievability and Converse Bounds for ErdHos-R'enyi Graph Matching

arXiv: https://arxiv.org/abs/1602.01042 · HTML: https://mlsanity.com/paper/1602.01042/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2016.

## Built on by (most cited first, 30 of 30)

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- [Matching recovery threshold for correlated random graphs](https://mlsanity.com/paper/2205.14650/) (2022)
- [Detection threshold for correlated ErdHos-R'enyi graphs via densest subgraphs](https://mlsanity.com/paper/2203.14573/) (2022)
- [A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation](https://mlsanity.com/paper/2306.00266/) (2023)
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- [A polynomial time iterative algorithm for matching Gaussian matrices with non-vanishing correlation](https://mlsanity.com/paper/2212.13677/) (2022)
- [Matching Correlated Inhomogeneous Random Graphs using the k-core Estimator](https://mlsanity.com/paper/2302.05407/) (2023)
- [The Umeyama algorithm for matching correlated Gaussian geometric models in the low-dimensional regime](https://mlsanity.com/paper/2402.15095/) (2024)
- [A computational transition for detecting correlated stochastic block models by low-degree polynomials](https://mlsanity.com/paper/2409.00966/) (2024)
- [Efficiently matching random inhomogeneous graphs via degree profiles](https://mlsanity.com/paper/2310.10441/) (2023)
- [Clustered Graph Matching for Label Recovery and Graph Classification](https://mlsanity.com/paper/2205.03486/) (2022)
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- [Efficient Algorithms for Exact Graph Matching on Correlated Stochastic Block Models with Constant Correlation](https://mlsanity.com/paper/2305.19666/) (2023)
- [Gotta match 'em all: Solution diversification in graph matching matched filters](https://mlsanity.com/paper/2308.13451/) (2023)
- [Relational Learning in Pre-Trained Models: A Theory from Hypergraph Recovery Perspective](https://mlsanity.com/paper/2406.11249/) (2024)
- [Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery](https://mlsanity.com/paper/2501.02851/) (2025)
- [An Efficient and Privacy-Preserving Architecture for Cross-Institutional Collaborative RAG](https://mlsanity.com/paper/2605.25716/) (2026)
- [Information-Theoretic Thresholds for the Alignments of Partially Correlated Graphs](https://mlsanity.com/paper/2406.05428/) (2024)
- [Testing Dependency of Weighted Random Graphs](https://mlsanity.com/paper/2409.14870/) (2024)
- [Robust Random Graph Matching in Dense Graphs via an Approximate Message Passing Type Algorithm](https://mlsanity.com/paper/2412.16457/) (2024)
- [Asymptotically perfect seeded graph matching without edge correlation (and applications to inference)](https://mlsanity.com/paper/2506.02825/) (2025)
- [Chaining 2-FWL GNNs for Combinatorial Graph Alignment](https://mlsanity.com/paper/2510.03086/) (2025)
- [Covariate-assisted graph matching](https://mlsanity.com/paper/2512.11761/) (2025)
- [Matching and mixing: Matchability of graphs under Markovian error](https://mlsanity.com/paper/2601.20020/) (2026)

## Abstract (arXiv)

We consider the problem of perfectly recovering the vertex correspondence between two correlated ErdHos-R'enyi (ER) graphs. For a pair of correlated graphs on the same vertex set, the correspondence between the vertices can be obscured by randomly permuting the vertex labels of one of the graphs. In some cases, the structural information in the graphs allow this correspondence to be recovered. We investigate the information-theoretic threshold for exact recovery, i.e. the conditions under which the entire vertex correspondence can be correctly recovered given unbounded computational resources. Pedarsani and Grossglauser provided an achievability result of this type. Their result establishes the scaling dependence of the threshold on the number of vertices. We improve on their achievability bound. We also provide a converse bound, establishing conditions under which exact recovery is impossible. Together, these establish the scaling dependence of the threshold on the level of correlation between the two graphs. The converse and achievability bounds differ by a factor of two for sparse, significantly correlated graphs.