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Attributed Network Alignment: Statistical Limits and Efficient Algorithm

Dong Huang, Chenyang Tian, Pengkun Yang

Abstract

This paper studies the problem of recovering a hidden vertex correspondence between two correlated graphs when both edge weights and node features are observed. While most existing work on graph alignment relies primarily on edge information, many real-world applications provide informative node features in addition to graph topology. To capture this setting, we introduce the featured correlated Gaussian Wigner model, where two graphs are coupled through an unknown vertex permutation, and the node features are correlated under the same permutation. We characterize the optimal information-theoretic thresholds for exact recovery and partial recovery of the latent mapping. On the algorithmic side, we propose QPAlign, an algorithm based on a quadratic programming relaxation, and demonstrate its strong empirical performance on both synthetic and real datasets. Moreover, we also derive theoretical guarantees for the proposed procedure, supporting its reliability and providing convergence guarantees.

Attributed Network Alignment: Statistical Limits and Efficient Algorithm

Abstract

This paper studies the problem of recovering a hidden vertex correspondence between two correlated graphs when both edge weights and node features are observed. While most existing work on graph alignment relies primarily on edge information, many real-world applications provide informative node features in addition to graph topology. To capture this setting, we introduce the featured correlated Gaussian Wigner model, where two graphs are coupled through an unknown vertex permutation, and the node features are correlated under the same permutation. We characterize the optimal information-theoretic thresholds for exact recovery and partial recovery of the latent mapping. On the algorithmic side, we propose QPAlign, an algorithm based on a quadratic programming relaxation, and demonstrate its strong empirical performance on both synthetic and real datasets. Moreover, we also derive theoretical guarantees for the proposed procedure, supporting its reliability and providing convergence guarantees.

Paper Structure

This paper contains 46 sections, 19 theorems, 187 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Results
  3. Related Work
  4. Attributed graph alignment
  5. Other graph models
  6. Our Contribution
  7. Information-theoretic Thresholds
  8. Possibility Results
  9. Impossibility Results
  10. QPAlign: Quadratic Programming relaxation for attributed graph Alignment
  11. Time complexity
  12. Numerical Results
  13. Simulation Studies
  14. Real Data Analysis
  15. ACM-DBLP dataset
  16. ...and 31 more sections

Key Result

Theorem 1

Under featured correlated Gaussian Wigner model, if $d=\omega(\log n)$ and $n\log (\tfrac{1}{1-\rho^2})+2d\log (\tfrac{1}{1-r^2})\ge (4+\epsilon)\log n$ for some constant $\epsilon>0$, then there exists an estimator $\hat{\pi}$ such that, for any fixed constant $0<\delta<1$ and $\pi^*\in \mathcal{S} Conversely, for any constant $0<\delta<1$, if $n\log (\frac{1}{1-\rho^2})+2d\log (\frac{1}{1-r^2})\

Figures (8)

  • Figure 1: Overlap between the estimator $\hat{\pi}$ in Algorithm \ref{['alg:qap-relax']} and the ground truth $\pi^*$ in two models with $n=3000$ and $d=512$, evaluated across varying correlations $\rho\in [0,1]$ and $r\in [0,1]$.
  • Figure 2: Phase-transition boundaries of QPAlign under different regularization parameters $\lambda$, together with the information-theoretic exact recovery limit.
  • Figure 3: Overlap vs. $\lambda$ on ACM-DBLP and Douban datasets.
  • Figure 4: Overlap between the estimator $\hat{\pi}$ in Algorithm \ref{['alg:qap-relax']} and the ground truth $\pi^*$ in two models with $n=100$ and $d=16$, evaluated across varying correlations $\rho\in [0,1]$ and $r\in [0,1]$.
  • Figure 5: Overlap between the estimator $\hat{\pi}$ in Algorithm \ref{['alg:qap-relax']} and the ground truth $\pi^*$ evaluated by different algorithms across varying correlations $\sigma\in [0,0.5]$ with different $\lambda$.
  • ...and 3 more figures

Theorems & Definitions (27)

  • Definition 1: Featured correlated Gaussian Wigner model
  • Theorem 1: Partial Recovery
  • Theorem 2: Exact Recovery
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Proposition 3: Impossibility result, partial recovery
  • Proposition 4: Impossibility result, exact recovery
  • Proposition 5
  • Proposition 6
  • ...and 17 more