arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2610.16675math.PRcs.DSmath.STstat.TH
收藏

随机几何图中的图匹配I：高效算法

Graph Matching in Random Geometric Graphs I: Efficient Algorithms

Tsinghua University（清华大学）；Peking University（北京大学）；Yale University（耶鲁大学）

Dong Huang, Ankang Jiao, Shuangping Li, Zhangsong Li

备注：
62 pages, 4 figures
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

该研究针对两个相关稠密随机几何图的隐藏顶点对应恢复问题，提出一种随机多项式时间算法，可在$d$满足特定条件时精确恢复置换$\pi$，还揭示了相关系数与计算成本的权衡关系。

摘要：

我们研究两个相关的稠密随机几何图之间隐藏顶点对应关系的恢复问题。每个图由$\mathbb{R}^d$中的$n$个潜在单位向量生成；对应顶点的向量$u_i^A$和$u_i^B$满足内积$\langle u_i^A,u_i^B\rangle=\rho$，且第二个图的顶点标签被未知置换$\pi$置换。对于每个固定的相关系数$\rho\in(0,1]$以及每个介于0和1之间的边密度，我们给出一种随机多项式时间算法，当$d\ge(\log n)^L$（其中$L$仅依赖于相关系数和密度界）时，该算法能以趋近于1的概率精确恢复$\pi$。该算法通过计数带有固定数量种子的装饰路径，为每个顶点分配一个签名。此类带种子模板的数量随路径长度呈指数增长，且底数为任意大的常数，因此所得统计量可放大任意小的常数相关系数。我们通过两种方式控制潜在几何结构产生的依赖关系：当$d$大于$n/(\log n)^{L_0}$（$L_0$为合适的常数）时，通过将局部矩与相关的Erdős–Rényi图的局部矩进行比较；当$d$较小时，通过潜在几何结构的谱估计结合谐波校正。我们还提供了低阶证据，证明相关系数与计算成本之间存在权衡关系。

英文摘要：

We study the recovery of the hidden vertex correspondence between two correlated dense random geometric graphs. Each graph is generated from $n$ latent unit vectors in $\mathbb R^d$; the vectors $u_i^A$ and $u_i^B$ of corresponding vertices satisfy $\langle u_i^A,u_i^B\rangle=ρ$, and the vertex labels of the second graph are permuted by an unknown permutation $π$. For every fixed correlation $ρ\in(0,1]$ and every edge density bounded away from zero and one, we give a randomized polynomial time algorithm that recovers $π$ exactly with probability tending to one whenever $d\ge(\log n)^L$, where $L$ depends only on the correlation and the density bounds. The algorithm assigns to each vertex a signature formed by counting decorated paths with a constant number of seeds. The number of such seeded templates grows exponentially with the path length, with an arbitrarily large constant base, so the resulting statistics amplify an arbitrarily small constant correlation. We control the dependencies created by the latent geometry in two ways: when $d$ is larger than $n/(\log n)^{L_0}$ for a suitable constant $L_0$, by comparing local moments with those of correlated Erdős--Rényi graphs; for smaller $d$, by spectral estimation of the latent geometry combined with a harmonic correction. We also provide low-degree evidence for a tradeoff between the correlation and the computational cost.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑