随机加权$d-$复合体：最小生成无环和持久图
Randomly Weighted $d-$complexes: Minimal Spanning Acycles and Persistence Diagrams
摘要：
加权的$d-$复合体是维度$d$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，我们建立了持久性图中的MSA面权重和\emph{death times}之间的等价性。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，而所有低维面具有$0$的权重。如果扰动衰减足够快，我们表明，适当缩放的极值最近的面权，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。对于任何加权随机复模型，甚至在无扰动情况下，三点过程的点过程收敛性和渐近等价性都是新的。最后，作为我们稳定性结果的结果，我们表明Frieze对随机最小生成树的$ζ(3)$限制以及Hino和Kanazawa最近对随机MSA的扩展也在适当的噪音环境中成立。
英文摘要：
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 \emph{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 perturbation 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 $ζ(3)$ limit for random minimal spanning trees and the recent extension to random MSAs by Hino and Kanazawa also hold in suitable noisy settings.
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