解决随机图匹配的尖锐重构问题
Settling the Sharp Reconstruction Thresholds of Random Graph Matching
摘要：
本文研究了两个边相关随机图的隐点对应的恢复问题。我们专注于高斯模型，其中两个图是具有相关高斯权重的完全图，以及Erdens-Rényi模型，其中两个图是从共同的父Erdens-Rényi图$\mathcal{G}(n,p)$子采样的。对于稠密图与$p=n^{-o(1)}$，我们证明了存在一个尖锐的阈值，在此之上，可以正确地匹配所有，但一个消失的分数顶点和低于正确匹配任何正分数是不可能的，一种现象被称为“全有或全无”相变。更惊人的是，在高斯设置中，在阈值以上，所有顶点都可以以高概率精确匹配。与此相反，稀疏的Erdens-Rényi图与$p=n^{-Θ(1)}$，我们表明，全有或全无的现象不再成立，我们确定的阈值为一个常数因子。在此过程中，我们还推导出了精确恢复的尖锐阈值，锐化了Erdens-Rényi图中的现有结果。否定结果的证明建立在基于截断二阶矩计算的互信息的紧密表征和将互信息与重构误差的积分相关联的“面积定理”之上。积极的结果如下紧密分析的最大似然估计，考虑到循环结构的诱导置换的边缘。
英文摘要：
This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the Erdős-Rényi model where the two graphs are subsampled from a common parent Erdős-Rényi graph $\mathcal{G}(n,p)$. For dense graphs with $p=n^{-o(1)}$, we prove that there exists a sharp threshold, above which one can correctly match all but a vanishing fraction of vertices and below which correctly matching any positive fraction is impossible, a phenomenon known as the "all-or-nothing" phase transition. Even more strikingly, in the Gaussian setting, above the threshold all vertices can be exactly matched with high probability. In contrast, for sparse Erdős-Rényi graphs with $p=n^{-Θ(1)}$, we show that the all-or-nothing phenomenon no longer holds and we determine the thresholds up to a constant factor. Along the way, we also derive the sharp threshold for exact recovery, sharpening the existing results in Erdős-Rényi graphs. The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation and an "area theorem" that relates the mutual information to the integral of the reconstruction error. The positive results follows from a tight analysis of the maximum likelihood estimator that takes into account the cycle structure of the induced permutation on the edges.
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