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A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation

Jian Ding, Zhangsong Li

TL;DR

The paper develops a polynomial-time iterative algorithm to recover the latent vertex permutation between two correlated Erdős–Rényi graphs with edge density q= n^{−α+o(1)} and non-vanishing correlation ρ. It extends a Gaussian Wigner-matrix framework to a graph setting by constructing iterative signal-bearing sets and leveraging Gaussian smoothing, density comparison, and projection techniques. The analysis shows that the algorithm achieves an almost exact matching and then, via seeded graph matching, exact recovery with probability 1−o(1); the running time is polynomial in n with a constant depending on α and ρ. This work demonstrates algorithmic universality across random matrix and random graph models and provides a robust approach for graph matching in regimes where edge density vanishes polynomially in n. The contributed methodology combines iterative signal amplification, careful probabilistic control of “bad” events, and a Gaussian-dominance strategy to achieve exact recovery under non-vanishing correlation even when q is sparse."

Abstract

We propose an efficient algorithm for matching two correlated Erdős--Rényi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- α+o(1)}$ for a constant $α\in [0,1)$, we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of $q$) when the edge correlation is below the square root of the Otter's constant (which is $\approx 0.338$).

A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation

TL;DR

The paper develops a polynomial-time iterative algorithm to recover the latent vertex permutation between two correlated Erdős–Rényi graphs with edge density q= n^{−α+o(1)} and non-vanishing correlation ρ. It extends a Gaussian Wigner-matrix framework to a graph setting by constructing iterative signal-bearing sets and leveraging Gaussian smoothing, density comparison, and projection techniques. The analysis shows that the algorithm achieves an almost exact matching and then, via seeded graph matching, exact recovery with probability 1−o(1); the running time is polynomial in n with a constant depending on α and ρ. This work demonstrates algorithmic universality across random matrix and random graph models and provides a robust approach for graph matching in regimes where edge density vanishes polynomially in n. The contributed methodology combines iterative signal amplification, careful probabilistic control of “bad” events, and a Gaussian-dominance strategy to achieve exact recovery under non-vanishing correlation even when q is sparse."

Abstract

We propose an efficient algorithm for matching two correlated Erdős--Rényi graphs with vertices whose edges are correlated through a latent vertex correspondence. When the edge density for a constant , we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of ) when the edge correlation is below the square root of the Otter's constant (which is ).
Paper Structure (27 sections, 40 theorems, 278 equations, 1 figure)

This paper contains 27 sections, 40 theorems, 278 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Backgrounds and related works
  3. Our contributions
  4. Notations
  5. An iterative matching algorithm
  6. Preprocessing
  7. Initialization
  8. Iteration
  9. Almost exact matching
  10. From almost exact matching to exact matching
  11. Formal description of the algorithm
  12. Running time analysis
  13. Analysis of the matching algorithm
  14. Outline of the proof
  15. Preliminaries on probability and linear algebra
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose that $q = n^{-\alpha+o(1)} \leq 1/2$ for some constant $\alpha\in [0, 1)$ and that $\rho\in (0, 1]$ is a constant. Then there exist a constant $C=C(\alpha,\rho)$ and an algorithm (see Algorithm algo:matching below) with time-complexity $O(n^{C})$ that recovers the latent matching with probab

Figures (1)

  • Figure 1: Logic of the definition

Theorems & Definitions (79)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Definition 3.1
  • Proposition 3.2
  • ...and 69 more