[Paper Review] A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation

Jian Ding, Zhangsong Li|arXiv (Cornell University)|Jun 1, 2023

Random Matrices and Applications4 citations

TL;DR

This paper presents a polynomial-time iterative algorithm for matching two correlated Erdős–Rényi random graphs with non-vanishing edge correlation, even when edge density decays polynomially as $ n^{-\alpha+o(1)} $ for $ \alpha \in [0,1) $. The algorithm succeeds in recovering the latent vertex correspondence with high probability, marking the first such result for non-vanishing correlation across all edge density regimes, including below the square root of Otter’s constant (~0.338).

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 $\approx 0.338$).

Motivation & Objective

• Address the computational gap in random graph matching by developing an efficient algorithm that works under non-vanishing correlation, regardless of edge density regime.
• Overcome limitations of prior algorithms that required correlation above the square root of Otter’s constant (~0.338) for polynomial-time success.
• Extend the applicability of iterative matching algorithms beyond Gaussian Wigner matrices to sparse random graphs with correlated edges.
• Demonstrate robustness of the iterative algorithm framework under varying graph sparsity and correlation levels, suggesting universality in algorithmic design.
• Provide a theoretical foundation for algorithmic universality in random graph matching, encouraging future work on structured models like stochastic block or geometric graphs.

Proposed method

• Propose an iterative algorithm inspired by prior work on correlated Gaussian Wigner matrices, adapting it to sparse Erdős–Rényi graphs.
• Use a tilted measure and Gaussian approximation to analyze the behavior of edge correlation signals in the matching process.
• Define key events such as $ \mathcal{E}_\diamond $, $ \mathcal{B}_{t^*} $, and $ \mathtt{U} $ to control error propagation and vertex misalignment.
• Employ a finishing step that processes unmatched vertices in a chain-based matching structure, ensuring high-probability recovery.
• Apply union bounds and concentration inequalities to bound the probability of large sets of failed matches, leveraging $ \Delta \ll \varepsilon $ and $ \Delta \leq e^{-(\log\log n)^8} $.
• Use spectral-like components $ g_{t^* - 1}Y(k,v) $ and $ \check{Y}(k,v) $ to model local neighborhood correlations and derive signal strength bounds.

Experimental results

Research questions

• RQ1 Can a polynomial-time algorithm recover the latent vertex correspondence in correlated Erdős–Rényi graphs when edge correlation is non-vanishing, even at low edge densities?
• RQ2 Does the iterative algorithm framework from Gaussian matrix matching extend to sparse random graphs with correlated edges?
• RQ3 What is the minimal correlation threshold below which existing polynomial-time algorithms fail, and can this work break that barrier?
• RQ4 How does the algorithm maintain high success probability as edge density decays as $ n^{-\alpha+o(1)} $ with $ \alpha < 1 $?
• RQ5 To what extent is the iterative matching algorithm robust to sparsity and weak correlation, suggesting universality across random graph models?

Key findings

• The algorithm achieves polynomial time complexity $ O(n^C) $ for some constant $ C = C(\alpha, \rho) $, independent of the edge density decay rate as long as $ \alpha < 1 $.
• The algorithm succeeds in recovering the latent matching $ \pi $ with probability $ 1 - o(1) $ when edge correlation $ \rho \in (0,1] $ is non-vanishing.
• The running time exponent $ C $ tends to infinity as $ \rho \to 0 $, suggesting this may be the best possible dependence under the current framework.
• The algorithm handles edge densities as low as $ q = n^{-\alpha+o(1)} $ with $ \alpha \in [0,1) $, including sparse regimes where average degree grows slowly.
• The proof establishes that the number of failed matches is bounded by $ O(n / \log n) $, and the finishing step recovers a matching of size at least $ m/4 - n/(\log n)^2 $, which is sublinear.
• The method relies on Gaussian approximation in high-probability events, with error terms controlled via $ \Delta^{10} $ and $ \exp\{-(\log n)^{1.8}\} $ tail bounds.