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Exact Graph Matching in Correlated Gaussian-Attributed Erdős-Rényi Model

Joonhyuk Yang, Hye Won Chung

TL;DR

This work studies exact graph matching in the correlated Gaussian-attributed Erdős-Rényi model, where both edge structure and correlated Gaussian node features are available. It proposes a two-step algorithm that first uses edge information via a $k$-core matching to obtain a large correct partial alignment and then completes the matching using Gaussian attributes, and it derives information-theoretic achievability and impossibility thresholds showing the regimes where features provide a gain. The main contributions are (i) a formal CGD model with parameters $(n,\mathbf{p},d,\rho)$, (ii) thresholds that couple edge and feature information through terms like $np_{11} + \frac{d}{4}\log\frac{1}{1-\rho^2}$, and (iii) an extension of the $k$-core analysis to a broader edge regime and its integration with MAP-based feature matching. The results quantify the benefit of jointly using edge and node information for exact graph matching and point to directions for designing efficient algorithms that achieve the information-theoretic limits.

Abstract

Graph matching problem aims to identify node correspondence between two or more correlated graphs. Previous studies have primarily focused on models where only edge information is provided. However, in many social networks, not only the relationships between users, represented by edges, but also their personal information, represented by features, are present. In this paper, we address the challenge of identifying node correspondence in correlated graphs, where additional node features exist, as in many real-world settings. We propose a two-step procedure, where we initially match a subset of nodes only using edge information, and then match the remaining nodes using node features. We derive information-theoretic limits for exact graph matching on this model. Our approach provides a comprehensive solution to the real-world graph matching problem by providing systematic ways to utilize both edge and node information for exact matching of the graphs.

Exact Graph Matching in Correlated Gaussian-Attributed Erdős-Rényi Model

TL;DR

This work studies exact graph matching in the correlated Gaussian-attributed Erdős-Rényi model, where both edge structure and correlated Gaussian node features are available. It proposes a two-step algorithm that first uses edge information via a -core matching to obtain a large correct partial alignment and then completes the matching using Gaussian attributes, and it derives information-theoretic achievability and impossibility thresholds showing the regimes where features provide a gain. The main contributions are (i) a formal CGD model with parameters , (ii) thresholds that couple edge and feature information through terms like , and (iii) an extension of the -core analysis to a broader edge regime and its integration with MAP-based feature matching. The results quantify the benefit of jointly using edge and node information for exact graph matching and point to directions for designing efficient algorithms that achieve the information-theoretic limits.

Abstract

Graph matching problem aims to identify node correspondence between two or more correlated graphs. Previous studies have primarily focused on models where only edge information is provided. However, in many social networks, not only the relationships between users, represented by edges, but also their personal information, represented by features, are present. In this paper, we address the challenge of identifying node correspondence in correlated graphs, where additional node features exist, as in many real-world settings. We propose a two-step procedure, where we initially match a subset of nodes only using edge information, and then match the remaining nodes using node features. We derive information-theoretic limits for exact graph matching on this model. Our approach provides a comprehensive solution to the real-world graph matching problem by providing systematic ways to utilize both edge and node information for exact matching of the graphs.
Paper Structure (16 sections, 15 theorems, 51 equations, 1 algorithm)

This paper contains 16 sections, 15 theorems, 51 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Correlated Gaussian-attributed Erdős-Rényi model
  3. Related work and our contribution
  4. Notation
  5. Main Result
  6. Proof of Theorem \ref{['thm:achievability']}
  7. Algorithm and outline of proof
  8. $k$-core matching
  9. Database matching
  10. Proof of Theorem \ref{['thm:impossibbility']}
  11. Analysis of $k$-core matching
  12. Conclusion and open problems
  13. Proof of Lemmas on Erdős-Rényi model
  14. Proof of Lemma \ref{['lem:k-core']}
  15. Notation
  16. ...and 1 more sections

Key Result

Theorem 2.1

Consider the correlated Gaussian-attributed ER model $(G_1,G_2)\sim \mathcal{CGD}(n,\mathbf{p},d,\rho)$. For any arbitrary small constant $\epsilon >0$, if then there exists an estimator $\hat{\pi}$ such that $\hat{\pi}=\pi_*$ with probability $1-o(1)$.

Theorems & Definitions (23)

  • Theorem 2.1: Achievability
  • Remark 2.1
  • Theorem 2.2: Impossibility
  • Remark 2.2
  • Definition 3.1: Matching
  • Definition 3.2: $k$-core matching and $k$-core estimator
  • Theorem 3.1: Partial matching from $k$-core estimator
  • Theorem 3.2: Exact matching from $k$-core estimator
  • Theorem 3.3: Theorem 1 in DCK19
  • Lemma 4.1
  • ...and 13 more