Random Geometric Complexes
Random Geometric Complexes
复制标题
随机几何复合体
DOI：
--
复制
发表时间：
2009
影响因子：
0.8
通讯作者：
Matthew Kahle
中科院分区：
文献类型：
--
作者：
Matthew Kahle
我们研究了基于 ℝd 中随机点构建的 Čech 和 Vietoris-Rips 复合体的预期拓扑性质。我们找到了随机几何图的连通性和组件计数的已知结果的高维类似物。然而，当 k>0 时，更高的同源性 Hk 并不是单调的。特别是，对于每个 k>0，我们表现出两个阈值，一个是同源性从消失到非消失的阈值，另一个是同源性返回到消失的阈值。我们给出了稀疏区域中 Betti 数的期望和密集区域中的界限的渐近公式。
We study the expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝd. We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology Hk is not monotone when k>0.In particular, for every k>0, we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes.