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On the Feasible Region of Efficient Algorithms for Attributed Graph Alignment

Ziao Wang, Ning Zhang, Weina Wang, Lele Wang

TL;DR

This paper studies exact vertex alignment for attributed graphs under the attributed Erdős–Rényi pair model $\mathcal{G}(n,p,s_u;m,q,s_a)$. It proposes two polynomial-time algorithms, AttrRich and AttrSparse, each tailored to a different attribute-information regime, and proves they achieve exact recovery w.h.p. in their respective regions, approaching information-theoretic limits. The work also connects these results to seeded and bipartite variants, showing that attributes can substantially widen the computable feasible region in sparse settings while maintaining tractable complexity. Through detailed probabilistic analyses and lemmas (e.g., tail bounds and KL-divergence estimates), the authors quantify how anchor-based strategies plus witness-based propagation enable reliable recovery. Overall, the findings suggest attributes can significantly boost computational feasibility for graph alignment, especially in the sparse regime, and provide a rigorous framework for comparing polynomial-time and information-theoretic limits.

Abstract

Graph alignment aims at finding the vertex correspondence between two correlated graphs, a task that frequently occurs in graph mining applications such as social network analysis. Attributed graph alignment is a variant of graph alignment, in which publicly available side information or attributes are exploited to assist graph alignment. Existing studies on attributed graph alignment focus on either theoretical performance without computational constraints or empirical performance of efficient algorithms. This motivates us to investigate efficient algorithms with theoretical performance guarantee. In this paper, we propose two polynomial-time algorithms that exactly recover the vertex correspondence with high probability. The feasible region of the proposed algorithms is near optimal compared to the information-theoretic limits. When specialized to the seeded graph alignment problem under the seeded Erdős--Rényi graph pair model, the proposed algorithms extends the best known feasible region for exact alignment by polynomial-time algorithms.

On the Feasible Region of Efficient Algorithms for Attributed Graph Alignment

TL;DR

This paper studies exact vertex alignment for attributed graphs under the attributed Erdős–Rényi pair model . It proposes two polynomial-time algorithms, AttrRich and AttrSparse, each tailored to a different attribute-information regime, and proves they achieve exact recovery w.h.p. in their respective regions, approaching information-theoretic limits. The work also connects these results to seeded and bipartite variants, showing that attributes can substantially widen the computable feasible region in sparse settings while maintaining tractable complexity. Through detailed probabilistic analyses and lemmas (e.g., tail bounds and KL-divergence estimates), the authors quantify how anchor-based strategies plus witness-based propagation enable reliable recovery. Overall, the findings suggest attributes can significantly boost computational feasibility for graph alignment, especially in the sparse regime, and provide a rigorous framework for comparing polynomial-time and information-theoretic limits.

Abstract

Graph alignment aims at finding the vertex correspondence between two correlated graphs, a task that frequently occurs in graph mining applications such as social network analysis. Attributed graph alignment is a variant of graph alignment, in which publicly available side information or attributes are exploited to assist graph alignment. Existing studies on attributed graph alignment focus on either theoretical performance without computational constraints or empirical performance of efficient algorithms. This motivates us to investigate efficient algorithms with theoretical performance guarantee. In this paper, we propose two polynomial-time algorithms that exactly recover the vertex correspondence with high probability. The feasible region of the proposed algorithms is near optimal compared to the information-theoretic limits. When specialized to the seeded graph alignment problem under the seeded Erdős--Rényi graph pair model, the proposed algorithms extends the best known feasible region for exact alignment by polynomial-time algorithms.
Paper Structure (25 sections, 19 theorems, 97 equations, 4 figures, 1 table)

This paper contains 25 sections, 19 theorems, 97 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Main results
  4. Algorithm AttrRich
  5. Algorithm AttrSparse
  6. Seeded alignment in the sparse regime Mossel2020
  7. Discussion
  8. Comparison to the information theoretic limits
  9. Specialization to the seeded Erdős--Rényi graph alignment
  10. Specialization to the bipartite alignment
  11. Proof of Theorem \ref{['thm:achievability']}
  12. Proof of Theorem \ref{['thm:achievability2']}
  13. Proof of Lemmas
  14. Proof of Lemma \ref{['lem:E1']}
  15. Proof of Lemma \ref{['lem:A']}
  16. ...and 10 more sections

Key Result

Theorem 1

Consider the attributed Erdős--Rényi pair $\mathcal{G}(n,p,s_\mathrm{u}; m, q,s_\mathrm{a})$ with $p=o(1)$, $q=o(1)$, $s_\mathrm{u}=\Theta(1)$, and $s_\mathrm{a}=\Theta(1)$. Assume that and that there exists some constant $\epsilon>0$ such that Then there exists a polynomial-time algorithm, namely, Algorithm AttrRich with the parameters chosen in eq:x and eq:y, that achieves exact alignment w.h.

Figures (4)

  • Figure 1: Comparison between the feasible regions of the proposed algorithms and the information-theoretic limits: the shaded area (++) represents the information-theoretically feasible region given in Zhang--Wang--Wang2021; area is the feasible region for Algorithm AttrRich and area is the feasible region for Algorithm AttrSparse; area is the information-theoretically infeasible region given in Zhang--Wang--Wang2021.
  • Figure 2: Comparison between feasible regions of polynomial-time algorithms when $mqs_\mathrm{a}^2=0$ and when $mqs_\mathrm{a}^2=0.1\log n$. Subgraph (a) captures the case when $mqs_\mathrm{a}^2=0$. The green region is known to be feasible by a polynomial-time algorithm in chandelier2022, while no polynomial-time algorithms are known to be feasible in the red region. Subgraph (b) captures the case when $mqs_\mathrm{a}^2=0.1\log n$. The green region is feasible by the proposed algorithm AttrRich.
  • Figure 3: An illustration of attributed Erdős--Rényi pair model. We first sample a base graph $G$. Then we get $G_1$ and $G_2$ through edge subsampling $G$. The anonymized graph $G_2'$ is obtained through apply the permutation $\Pi^*$ on the user vertex set of $G_2$.
  • Figure 4: Comparison of the feasible region of Corollaries \ref{['cor:alg1']} and \ref{['cor:alg2']} to the feasible region in Mossel2020 and Shirani2017. On the top-left corner and bottom-right corner, the two blue regions are feasible for the proposed algorithms but not for any existing works. The red region is feasible for existing works, but not for the proposed algorithms. The green region is the overlap of our feasible region with the feasible region in the existing works.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Remark 1: Complexity of Algorithm AttrRich
  • Remark 2: Complexity of Algorithm AttrSparse
  • Theorem 3: Theorem 1 in Zhang--Wang--Wang2021
  • Corollary 1: Feasible region of Algorithm AttrRich
  • Corollary 2: Feasible region of Algorithm AttrSparse
  • Remark 3
  • Theorem 4: Comparison of polynomial-time algorithms
  • Remark 4
  • ...and 14 more