论文标题
随机集团复合物中的最大持久性
Maximal persistence in random clique complexes
论文作者
论文摘要
我们研究了$ n $顶点的erdős-rényi随机集中复合物的持续同源。在这里，每个边缘$ e $一次出现在[0,1] $中的$ p_e \ in [0,1] $中随机随机选择的统一，并且一个周期$σ$的\ emph {permistence}分别定义为$ p_2 / p_1 $，其中$ p_1 $和$ p_2 $是周期的出生和死亡时间。我们表明，对于固定的$ k \ ge 1 $，具有高概率的$ k $ cycle的最大持久性大约是$ n^{1/k（k+1）} $。这些结果与Bobrowski，Kahle和Skraba的早期工作的随机几何环境形成鲜明对比。对于随机čech和vietoris-RIPS滤过，$ k $ cycle的最大持久性小得多，$ \ weft（\ log n / \ log n / \ log n / \ log n / \ log n / \ log n / \ firs n \ right）
We study the persistent homology of an Erdős--Rényi random clique complex filtration on $n$ vertices. Here, each edge $e$ appears at a time $p_e \in [0,1]$ chosen uniform randomly in the interval, and the \emph{persistence} of a cycle $σ$ is defined as $p_2 / p_1$, where $p_1$ and $p_2$ are the birth and death times of the cycle respectively. We show that for fixed $k \ge 1$, with high probability the maximal persistence of a $k$-cycle is of order roughly $n^{1/k(k+1)}$. These results are in sharp contrast with the random geometric setting where earlier work by Bobrowski, Kahle, and Skraba shows that for random Čech and Vietoris--Rips filtrations, the maximal persistence of a $k$-cycle is much smaller, of order $\left(\log n / \log \log n \right)^{1/k}$.
