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 Gaussian Wigner (CGW) 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 (Mao et al. 2023) for the sparse Correlated Erdos-Renyi(CER) model to the (dense) CGW 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\emph{图匹配}问题中，我们观测到两个图$G$和$H$，目标是找到它们顶点之间的一个分配（或匹配），使得某种边一致性的度量最大化。本文假设观测对$G,H$来自相关高斯Wigner（CGW）模型——一种流行的相关加权图模型——其中$G$和$H$的邻接矩阵元素是独立的高斯变量，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\emph{投影幂方法}（PPM）作为一种\emph{种子}图匹配算法的性能，其中我们获得一个初始部分正确匹配（称为种子）作为辅助信息。我们证明，如果种子足够接近真实匹配，那么在高概率下，PPM在$\mathcal{O}(\log n)$次迭代内逐步改进种子并恢复真实匹配（部分或完全）。我们的结果证明了PPM即使在常数$\sigma$的区间内也有效，从而将(Mao等人，2023)对稀疏相关Erdos-Renyi（CER）模型的分析推广到（稠密）CGW模型。作为分析的副产品，我们看到PPM框架推广了种子图匹配的一些最新算法。我们通过合成数据的数值实验支持并补充了理论发现。