Graph Alignment via Birkhoff Relaxation

Abstract: We consider the graph alignment problem, wherein the objective is to find a vertex correspondence between two graphs that maximizes the edge overlap. The graph alignment problem is an instance of the quadratic assignment problem (QAP), known to be NP-hard in the worst case even to approximately solve. In this paper, we analyze Birkhoff relaxation, a tight convex relaxation of QAP, and present theoretical guarantees on its performance when the inputs follow the Gaussian Wigner Model. More specifically, the weighted adjacency matrices are correlated Gaussian Orthogonal Ensemble with correlation $1/\sqrt{1+\sigma^2}$. Denote the optimal solutions of the QAP and Birkhoff relaxation by $\Pi^\star$ and $X^\star$ respectively. We show that $|X^\star-\Pi^\star|_F^2 = o(n)$ when $\sigma = o(n^{-1.25})$ and $|X^\star-\Pi^\star|_F^2 = \Omega(n)$ when $\sigma = \Omega(n^{-0.5})$. Thus, the optimal solution $X^\star$ transitions from a small perturbation of $\Pi^\star$ for small $\sigma$ to being well separated from $\Pi^\star$ as $\sigma$ becomes larger than $n^{-0.5}$. This result allows us to guarantee that simple rounding procedures on $X^\star$ align $1-o(1)$ fraction of vertices correctly whenever $\sigma = o(n^{-1.25})$. This condition on $\sigma$ to ensure the success of the Birkhoff relaxation is state-of-the-art.

• The paper introduces Birkhoff relaxation, a convex approach that replaces permutation matrices with doubly stochastic matrices to tackle graph alignment.
• The study establishes convergence conditions where, for σ = o(n^-1.25), rounding the relaxed solution recovers near-perfect vertex alignment.
• Results reveal a phase transition in performance with increasing noise, prompting further research to extend applicability to noisier settings.

Graph Alignment via Birkhoff Relaxation

Introduction

The paper "Graph Alignment via Birkhoff Relaxation" investigates the graph alignment problem, an instance of the Quadratic Assignment Problem (QAP), which involves maximizing edge overlap between two graphs. The problem, recognized as NP-hard, is approached with Birkhoff relaxation—analyzing its performance under the Gaussian Wigner Model where the adjacency matrices belong to a correlated Gaussian Orthogonal Ensemble.

Methodology

The core of the paper lies in the formulation of the Birkhoff relaxation to solve QAP efficiently. This convex relaxation replaces permutation matrices with the Birkhoff polytope (doubly stochastic matrices), turning the otherwise complex problem into a more tractable form. The Birkhoff relaxation is defined as minimizing the Frobenius norm squared of the difference between the products of adjacency matrices and the relaxed permutation matrix.

The relaxation transitions smoothly to better approximate the true permutation matrix under certain conditions on the noise level, defined by parameter $\sigma$. The paper establishes bounds on the Frobenius distance between the optimal solutions of the QAP ($\Pi^\star$) and the relaxation ($X^\star$), revealing the transition point where the relaxation remains a close approximation.

Theoretical Analysis

The presentation of theoretical guarantees is a substantial contribution:

1. Convergence Conditions: For $\sigma = o(n^{-1.25})$, $X^\star$ closely approximates $\Pi^\star$, ensuring that straightforward rounding of $X^\star$ leads to correct alignment for $1-o(1)$ of the vertices. This condition is posited as the current best for assuring successful graph alignment via Birkhoff relaxation.
2. Separation Condition: Conversely, when $\sigma$ grows to $\Omega(n^{-0.5})$, $X^\star$ becomes significantly separated from $\Pi^\star$, though potential recovery of $\Pi^\star$ is still possible via projection of $X^\star$.

Results and Implications

The results indicate a phase transition in the success of Birkhoff relaxation correlated with $\sigma$. This transition is theoretically aligned but nuanced with practical performance, where even in cases where $X^\star$ is separated from $\Pi^\star$, the possibility of post-processing recovery exists. The paper suggests these bounds are conservative and anticipates future research could relax these $\sigma$ thresholds closer to $n^{-0.5}$, enhancing the practical appeal and applicability of this relaxation technique.

Future Research Directions

The findings underscore the complexity and potential of Birkhoff relaxation for graph alignment, urging further exploration into:

• Lowering the condition on $\sigma$ to enable applicability in noisier settings.
• Investigating the structural properties and post-processing techniques for enhanced recovery of permutation matrices even when $X^\star$ is distanced from $\Pi^\star$.
• Extending theoretical and empirical analysis to broaden application scenarios where Birkhoff relaxation is effective, integrating insights from spectral and convex optimization domains.

Conclusion

This paper presents a detailed theoretical treatment of Birkhoff relaxation's capability in addressing graph alignment, providing promising guidelines for achieving near-optimal vertex correspondence under certain conditions. As a foundational step, it sets the stage for robust application of convex relaxation techniques in network analysis and related fields, where efficient and precise graph alignment is crucial.