Randomly weighted d-complexes: Minimal spanning acycles and Persistence diagrams
Randomly weighted d-complexes: Minimal spanning acycles and Persistence diagrams
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DOI：
10.37236/8679
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发表时间：
2020-04-17
影响因子：
0.7
通讯作者：
Yogeshwaran, D.
中科院分区：
文献类型：
--
作者：
Skraba, Primoz;Thoppe, Gugan;Yogeshwaran, D.
加权d-复形是维度为d的单纯复形，其中每个面被分配实值权重。我们得出三个关键的结果，在这里关于持久性图和最小生成无环（MSA）等复杂的。首先，我们建立了一个等价的MSA面权重和死亡时间的持久性图。接下来，我们展示了MSA面权重的一个新的稳定性结果，由于我们的第一个结果，它也分别适用于死亡和出生时间。我们的最终结果涉及随机加权d-复合物的平均场模型的扰动。这里的d面权重是一些i.i.d.的扰动。分布，而所有低维面的权重为0。如果扰动衰减得足够快，我们表明，适当规模的极值最近的面权，面权的d-MSA，和相关的死亡时间收敛到一个非齐次泊松点过程。这个结果完全刻画了持久图和MSA的极值点。点过程收敛性和三点过程的渐近等价性对于任何加权随机复模型都是新的，甚至包括非扰动情形。最后，作为我们的稳定性结果的后果，我们表明，弗里兹的zeta（3）限制随机最小生成树和最近的扩展随机MSA日野和金泽也持有在适当的噪声设置。
A weighted d-complex is a simplicial complex of dimension d in which each face is assigned a real-valued weight. We derive three key results here concerning persistence diagrams and minimal spanning acycles (MSAs) of such complexes. First, we establish an equivalence between the MSA face-weights and death times in the persistence diagram. Next, we show a novel stability result for the MSA face-weights which, due to our first result, also holds true for the death and birth times, separately. Our final result concerns a perturbation of a mean-field model of randomly weighted d-complexes. The d-face weights here are perturbations of some i.i.d. distribution while all the lower-dimensional faces have a weight of 0. If the perturbations decay sufficiently quickly, we show that suitably scaled extremal nearest face-weights, face-weights of the d-MSA, and the associated death times converge to an inhomogeneous Poisson point process. This result completely characterizes the extremal points of persistence diagrams and MSAs. The point process convergence and the asymptotic equivalence of three point processes are new for any weighted random complex model, including even the non-perturbed case. Lastly, as a consequence of our stability result, we show that Frieze's zeta(3) limit for random minimal spanning trees and the recent extension to random MSAs by Hino and Kanazawa also hold in suitable noisy settings.