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Asymmetric graph alignment and the phase transition for asymmetric tree correlation testing

Jakob Maier, Laurent Massoulié

TL;DR

This work extends graph alignment to asymmetric correlated Erdős–Rényi graphs with varying node numbers and edge densities by introducing a tree-based framework that reduces global alignment to local tree correlation testing. It proposes MPAlign, a polynomial-time algorithm that leverages one-sided tree tests on dangling subtrees to recover a substantial portion of the planted node correspondences in the sparse regime. A rigorous information-theoretic phase transition is established for asymmetric tree testing: the feasibility threshold is governed by Otter's constant $oldsymbol{eta} \,\approx\,0.338$, with testing possible when $ss'>\alpha$ and $\ u$ large enough, and impossible when $ss'\le\alpha$; this yields practical implications for random subgraph isomorphism. The findings connect local Galton–Watson tree behavior to global graph alignment, provide a formal diagonalization of the likelihood ratio, and resolve an open problem about polynomial-time solutions to certain random SIP instances in the asymmetric setting, with potential extensions to broader network models.

Abstract

Graph alignment - identifying node correspondences between two graphs - is a fundamental problem with applications in network analysis, biology, and privacy research. While substantial progress has been made in aligning correlated Erdős-Rényi graphs under symmetric settings, real-world networks often exhibit asymmetry in both node numbers and edge densities. In this work, we introduce a novel framework for asymmetric correlated Erdős-Rényi graphs, generalizing existing models to account for these asymmetries. We conduct a rigorous theoretical analysis of graph alignment in the sparse regime, where local neighborhoods exhibit tree-like structures. Our approach leverages tree correlation testing as the central tool in our polynomial-time algorithm, MPAlign, which achieves one-sided partial alignment under certain conditions. A key contribution of our work is characterizing these conditions under which asymmetric tree correlation testing is feasible: If two correlated graphs $G$ and $G'$ have average degrees $λs$ and $λs'$ respectively, where $λ$ is their common density and $s,s'$ are marginal correlation parameters, their tree neighborhoods can be aligned if $ss' > α$, where $α$ denotes Otter's constant and $λ$ is supposed large enough. The feasibility of this tree comparison problem undergoes a sharp phase transition since $ss' \leq α$ implies its impossibility. These new results on tree correlation testing allow us to solve a class of random subgraph isomorphism problems, resolving an open problem in the field.

Asymmetric graph alignment and the phase transition for asymmetric tree correlation testing

TL;DR

This work extends graph alignment to asymmetric correlated Erdős–Rényi graphs with varying node numbers and edge densities by introducing a tree-based framework that reduces global alignment to local tree correlation testing. It proposes MPAlign, a polynomial-time algorithm that leverages one-sided tree tests on dangling subtrees to recover a substantial portion of the planted node correspondences in the sparse regime. A rigorous information-theoretic phase transition is established for asymmetric tree testing: the feasibility threshold is governed by Otter's constant , with testing possible when and large enough, and impossible when ; this yields practical implications for random subgraph isomorphism. The findings connect local Galton–Watson tree behavior to global graph alignment, provide a formal diagonalization of the likelihood ratio, and resolve an open problem about polynomial-time solutions to certain random SIP instances in the asymmetric setting, with potential extensions to broader network models.

Abstract

Graph alignment - identifying node correspondences between two graphs - is a fundamental problem with applications in network analysis, biology, and privacy research. While substantial progress has been made in aligning correlated Erdős-Rényi graphs under symmetric settings, real-world networks often exhibit asymmetry in both node numbers and edge densities. In this work, we introduce a novel framework for asymmetric correlated Erdős-Rényi graphs, generalizing existing models to account for these asymmetries. We conduct a rigorous theoretical analysis of graph alignment in the sparse regime, where local neighborhoods exhibit tree-like structures. Our approach leverages tree correlation testing as the central tool in our polynomial-time algorithm, MPAlign, which achieves one-sided partial alignment under certain conditions. A key contribution of our work is characterizing these conditions under which asymmetric tree correlation testing is feasible: If two correlated graphs and have average degrees and respectively, where is their common density and are marginal correlation parameters, their tree neighborhoods can be aligned if , where denotes Otter's constant and is supposed large enough. The feasibility of this tree comparison problem undergoes a sharp phase transition since implies its impossibility. These new results on tree correlation testing allow us to solve a class of random subgraph isomorphism problems, resolving an open problem in the field.

Paper Structure

This paper contains 51 sections, 32 theorems, 265 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Document structure.
  4. Related work
  5. Notations
  6. Setting and random graph model
  7. Asymmetric correlated Erdős--Rényi graphs
  8. True node correspondences
  9. One-sided partial alignment
  10. From graphs to trees and back
  11. Tree notations.
  12. Tree neighborhoods
  13. Correlated Galton Watson trees
  14. One-sided hypothesis testing
  15. Dangling tree trick
  16. ...and 36 more sections

Key Result

Theorem 1.1

For a pair of sparse asymmetric correlated Erdős--Rényi graphs $(G, G')$ with global density $\lambda > 0$ and correlation parameters $s, s' \in [0,1]$, the following statements hold:

Figures (3)

  • Figure 1: Sampling process of $(G, G') \sim \mathrm{CER}(N, q, q', r, r')$. Nodes and edges added during the augmentations are red and blue, respectively.
  • Figure 2: Sampling process of $(t, t') \sim\mathbb{P}^{\,\text{corr}}$ up to depth 3. The blue and red components are independently sampled and attached to the grey intersection tree.
  • Figure 3: Two examples of unlabeled trees and their representation as sequences of natural numbers indexed by trees.

Theorems & Definitions (72)

  • Theorem 1.1: main results --- informal statement
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.1: overlap
  • Definition 2.2: error fraction
  • Definition 2.3
  • Remark 2.5
  • Lemma 3.1: Uniform bound on the neighborhood sizes
  • ...and 62 more