In the \emph{graph matching} problem we observe two graphs $G,H$ and the goal is to find an assignment (or matching) between their vertices such that some measure of edge agreement is maximized. We assume in this work that the observed pair $G,H$ has been drawn from the correlated Wigner model -- a popular model for correlated weighted graphs -- where the entries of the adjacency matrices of $G$ and $H$ are independent Gaussians and each edge of $G$ is correlated with one edge of $H$ (determined by the unknown matching) with the edge correlation described by a parameter $\sigma\in [0,1)$. In this paper, we analyse the performance of the \emph{projected power method} (PPM) as a \emph{seeded} graph matching algorithm where we are given an initial partially correct matching (called the seed) as side information. We prove that if the seed is close enough to the ground-truth matching, then with high probability, PPM iteratively improves the seed and recovers the ground-truth matching (either partially or exactly) in $\mathcal{O}(\log n)$ iterations. Our results prove that PPM works even in regimes of constant $\sigma$, thus extending the analysis in \citep{MaoRud} for the sparse Erd\H{o}s-R\'enyi model to the (dense) Wigner model. As a byproduct of our analysis, we see that the PPM framework generalizes some of the state-of-art algorithms for seeded graph matching. We support and complement our theoretical findings with numerical experiments on synthetic data.

翻译：在图匹配问题中，我们观察两个图$G$和$H$，目标是找到它们顶点之间的一个分配（或匹配），使得某种边一致性度量最大化。本文假设观测对$G,H$来自相关Wigner模型——一种常用的相关加权图模型，其中$G$和$H$的邻接矩阵元素为独立高斯变量，且$G$的每条边与$H$的一条边（由未知匹配决定）相关，边相关性由参数$\sigma\in [0,1)$描述。本文分析了投影幂法（PPM）作为种子图匹配算法的性能，其中初始的部分正确匹配（称为种子）作为辅助信息提供。我们证明：若种子与真实匹配足够接近，则高概率下PPM能在$\mathcal{O}(\log n)$次迭代中逐步改进种子并恢复真实匹配（部分或完全）。我们的结果证明PPM即使在常数$\sigma$的设定下仍有效，从而将文献\citep{MaoRud}中关于稀疏Erdős-Rényi模型的分析推广到（稠密）Wigner模型。作为分析的副产品，我们发现PPM框架概括了当前若干最先进的种子图匹配算法。我们通过合成数据上的数值实验支持并补充了理论发现。