# Impossibility of Partial Recovery in the Graph Alignment Problem

arXiv: https://arxiv.org/abs/2102.02685 · HTML: https://mlsanity.com/paper/2102.02685/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 4 Feb 2021.

## Builds on (most cited first, 12 of 12)

- [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)
- [Exact alignment recovery for correlated ErdHos-R'enyi graphs](https://mlsanity.com/paper/1711.06783/) (2017)
- [Seeded Graph Matching via Large Neighborhood Statistics](https://mlsanity.com/paper/1807.10262/) (2018)
- [Spectral Graph Matching and Regularized Quadratic Relaxations II: ErdHos-R'enyi Graphs and Universality](https://mlsanity.com/paper/1907.08883/) (2019)
- [Partial Recovery in the Graph Alignment Problem](https://mlsanity.com/paper/2007.00533/) (2020)
- [From tree matching to sparse graph alignment](https://mlsanity.com/paper/2002.01258/) (2020)
- [Testing correlation of unlabeled random graphs](https://mlsanity.com/paper/2008.10097/) (2020)
- [Partial Recovery of ErdHos-R'enyi Graph Alignment via k-Core Alignment](https://mlsanity.com/paper/1809.03553/) (2018)
- [Seeded graph matching for correlated ErdHos-R'enyi graphs](https://mlsanity.com/paper/1304.7844/) (2013)
- [Sharp threshold for alignment of graph databases with Gaussian weights](https://mlsanity.com/paper/2010.16295/) (2020)
- [Spectral Graph Matching and Regularized Quadratic Relaxations I: The Gaussian Model](https://mlsanity.com/paper/1907.08880/) (2019)

## Built on by (most cited first, 21 of 21)

- [Random graph matching at Otter's threshold via counting chandeliers](https://mlsanity.com/paper/2209.12313/) (2022)
- [Correlated Stochastic Block Models: Exact Graph Matching with Applications to Recovering Communities](https://mlsanity.com/paper/2107.06767/) (2021)
- [Correlation detection in trees for planted graph alignment](https://mlsanity.com/paper/2107.07623/) (2021)
- [Detection threshold for correlated ErdHos-R'enyi graphs via densest subgraphs](https://mlsanity.com/paper/2203.14573/) (2022)
- [Statistical limits of correlation detection in trees](https://mlsanity.com/paper/2209.13723/) (2022)
- [A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation](https://mlsanity.com/paper/2306.00266/) (2023)
- [Exact Community Recovery in Correlated Stochastic Block Models](https://mlsanity.com/paper/2203.15736/) (2022)
- [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)
- [Efficiently matching random inhomogeneous graphs via degree profiles](https://mlsanity.com/paper/2310.10441/) (2023)
- [Gaussian Database Alignment and Gaussian Planted Matching](https://mlsanity.com/paper/2307.02459/) (2023)
- [Graph Matching via convex relaxation to the simplex](https://mlsanity.com/paper/2310.20609/) (2023)
- [Interdependency Matters: Graph Alignment for Multivariate Time Series Anomaly Detection](https://mlsanity.com/paper/2410.08877/) (2024)
- [Computational Lower Bounds for Correlated Random Graphs via Algorithmic Contiguity](https://mlsanity.com/paper/2502.09832/) (2025)
- [The feasibility of multi-graph alignment: a Bayesian approach](https://mlsanity.com/paper/2502.17142/) (2025)
- [The graph alignment problem: fundamental limits and efficient algorithms](https://mlsanity.com/paper/2404.12418/) (2024)
- [Information-Theoretic Thresholds for the Alignments of Partially Correlated Graphs](https://mlsanity.com/paper/2406.05428/) (2024)
- [Robust Random Graph Matching in Dense Graphs via an Approximate Message Passing Type Algorithm](https://mlsanity.com/paper/2412.16457/) (2024)
- [Chaining 2-FWL GNNs for Combinatorial Graph Alignment](https://mlsanity.com/paper/2510.03086/) (2025)
- [Sharp Statistical Thresholds for Attributed Gaussian Graph Matching](https://mlsanity.com/paper/2603.23305/) (2026)

## Abstract (arXiv)

Random graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For the correlated Erd"os-R'enyi model, we prove an impossibility result for partial recovery in the sparse regime, with constant average degree and correlation, as well as a general bound on the maximal reachable overlap. Our bound is tight in the noiseless case (the graph isomorphism problem) and we conjecture that it is still tight with noise. Our proof technique relies on a careful application of the probabilistic method to build automorphisms between tree components of a subcritical Erd"os-R'enyi graph.