AI 中文总结
该研究针对紧致黎曼流形上的随机几何图，提出仅需 O(n) 条边的方法，在热力学超临界 regime 下可高概率推断其一阶同调群，确定了正确尺度并证明结果最优。
AI 中文摘要
考虑紧致黎曼流形 M 上的随机几何图 $G_M(n;r)$，其顶点为 n 个独立采样点构成的点云，边连接距离不超过 r 的顶点。我们证明，在热力学 regime（即有界期望平均度）下，若 $G_M(n;r)$ 在连续渗流意义下是超临界的，则当 $n \to \infty$ 时，可高概率从 $G_M(n;r)$ 正确推断出 M 的一阶同调群 $H_1(M)$。具体而言，可通过取 $G_M(n;r)$ 的循环空间，再商去所有度量直径为 $O(r|\log r|)$（或图直径为 $O(|\log r|)$）的循环，得到 $H_1(M)$。我们估计 $H_1(M)$ 的方法利用了超临界渗流的粗拓扑事实，而非通常考察点云邻域拓扑的方法。以往方法使用需要 $O(n \log n)$ 条边的组合模型，而我们的方法仅需 $O(n)$ 条边。我们还证明，在热力学 regime 的所有相态中，若改为商去度量直径为 $o(r|\log r|)$ 的循环，则高概率无法恢复 $H_1(M)$，因此 $\Theta(r|\log r|)$ 是“正确尺度”。在此过程中，我们证明任意紧致 d 维黎曼流形都具有 Bobrowski 和 Skraba \cite{BS2020} 定义的“一阶同调渗流阈值”，且该阈值与 $\mathbb{R}^d$ 上的连续渗流阈值重合，这一结果此前仅对平坦环面已知。这强烈表明我们的结果是最优的，即无法在亚临界热力学 regime 下从 $G_M(n;r)$ 推断出 $H_1(M)$。所有结果对任意系数的同调均成立。
英文摘要
Consider a random geometric graph $G_M(n;r)$ on a compact Riemannian manifold $M$, whose vertices are a cloud of $n$ independently sampled points, and whose edges connect vertices at distance $\le r$. We show that, in the thermodynamic (i.e. bounded expected average degree) regime, if $G_M(n;r)$ is supercritical in the sense of continuum percolation, then the first homology group $H_1(M)$ of $M$ can be correctly inferred from $G_M(n;r)$ with high probability as $n \to \infty$. Specifically, one can obtain $H_1(M)$ by taking the cycle space of $G_M(n;r)$ and quotienting out all the cycles of metric diameter $O(r|\log r|)$ (or of graph diameter $O(|\log r|)$). Our method of estimating $H_1(M)$ exploits a coarse-topological fact about supercritical percolation, as opposed to usual methods, which examine the topology of neighborhoods of the point cloud. Whereas previous methods use combinatorial models which require $O(n \log n)$ edges, our method only requires $O(n)$ edges. We also show that, in all phases of the thermodynamic regime, if one instead takes the quotient by cycles of metric diameter $o(r|\log r|)$, with high probability, one will not recover $H_1(M)$. Thus $Θ(r|\log r|)$ is the ``right scale.'' On the way, we show that an arbitrary compact $d$-dimensional Riemannian manifold has a \emph{first homological percolation threshold} in the sense of Bobrowski and Skraba \cite{BS2020} which coincides with the continuum percolation threshold on $\R^d$, a result previously only known for the flat torus. This strongly suggests that our results are optimal, in the sense that $H_1(M)$ cannot be inferred from $G_M(n;r)$ in the subcritical thermodynamic regime. All results hold for homology with arbitrary coefficients.
Comments34 pages, 4 figures
