# The Phantom Alignment Strength Conjecture: Practical use of graph matching alignment strength to indicate a meaningful graph match

arXiv: https://arxiv.org/abs/2103.00624 · HTML: https://mlsanity.com/paper/2103.00624/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2021.

## Built on by (most cited first, 4 of 4)

- [Lost in the Shuffle: Testing Power in the Presence of Errorful Network Vertex Labels](https://mlsanity.com/paper/2208.08638/) (2022)
- [Clustered Graph Matching for Label Recovery and Graph Classification](https://mlsanity.com/paper/2205.03486/) (2022)
- [Detection of Model-based Planted Pseudo-cliques in Random Dot Product Graphs by the Adjacency Spectral Embedding and the Graph Encoder Embedding](https://mlsanity.com/paper/2312.11054/) (2023)
- [Asymptotically perfect seeded graph matching without edge correlation (and applications to inference)](https://mlsanity.com/paper/2506.02825/) (2025)

## Abstract (arXiv)

The alignment strength of a graph matching is a quantity that gives the practitioner a measure of the correlation of the two graphs, and it can also give the practitioner a sense for whether the graph matching algorithm found the true matching. Unfortunately, when a graph matching algorithm fails to find the truth because of weak signal, there may be "phantom alignment strength" from meaningless matchings that, by random noise, have fewer disagreements than average (sometimes substantially fewer); this alignment strength may give the misleading appearance of significance. A practitioner needs to know what level of alignment strength may be phantom alignment strength and what level indicates that the graph matching algorithm obtained the true matching and is a meaningful measure of the graph correlation. The it Phantom Alignment Strength Conjecture introduced here provides a principled and practical means to approach this issue. We provide empirical evidence for the conjecture, and explore its consequences.