We analyze a new spectral graph matching algorithm, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), for recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. Extending the exact recovery guarantees established in a companion paper for Gaussian weights, in this work, we prove the universality of these guarantees for a general correlated Wigner model. In particular, for two Erdős-Rényi graphs with edge correlation coefficient \(1-\sigma ^2\) and average degree at least \({\text {polylog}}(n)\), we show that GRAMPA exactly recovers the latent vertex correspondence with high probability when \(\sigma \lesssim 1/{\text {polylog}}(n)\). Moreover, we establish a similar guarantee for a variant of GRAMPA, corresponding to a tighter quadratic programming relaxation of the quadratic assignment problem. Our analysis exploits a resolvent representation of the GRAMPA similarity matrix and local laws for the resolvents of sparse Wigner matrices.

我们分析了一种新的谱图匹配算法，即 GRAph Matching by Pairwise eigen-Alignments (GRAMPA)，用于恢复两个未标记的边相关加权图之间的潜在顶点对应关系。扩展在高斯权重的配套论文中建立的精确恢复保证，在这项工作中，我们证明了这些保证对于一般相关 Wigner 模型的普遍性。特别是，对于具有边相关系数\(1-\sigma ^2\)和平均度至少\({\text {polylog}}(n)\)的两个 Erdős-Rényi 图，我们证明 GRAMPA 准确地恢复了当\(\sigma \lesssim 1/{\text {polylog}}(n)\)时，潜在顶点对应的概率很高. 此外，我们为 GRAMPA 的变体建立了类似的保证，对应于二次分配问题的更严格的二次规划松弛。我们的分析利用 GRAMPA 相似矩阵的解析表示和稀疏 Wigner 矩阵的解析的局部定律。