16 research outputs found
Randomly weighted d-complexes: Minimal spanning acycles and Persistence diagrams
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
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
Asymptotic behavior of lifetime sums for random simplicial complex processes
We study the homological properties of random simplicial complexes. In
particular, we obtain the asymptotic behavior of lifetime sums for a class of
increasing random simplicial complexes; this result is a higher-dimensional
counterpart of Frieze's -limit theorem for the Erd\H{o}s-R\'{e}nyi
graph process. The main results include solutions to questions posed in an
earlier study by Hiraoka and Shirai about the Linial-Meshulam complex process
and the random clique complex process. One of the key elements of the arguments
is a new upper bound on the Betti numbers of general simplicial complexes in
terms of the number of small eigenvalues of Laplacians on links. This bound can
be regarded as a quantitative version of the cohomology vanishing theorem.Comment: 39 pages, minor correction
ランダム単体複体のトポロジーとパーシステントホモロジーの生存時間
Tohoku University水藤博課
Refinement of Interval Approximations for Fully Commutative Quivers
A fundamental challenge in multiparameter persistent homology is the absence
of a complete and discrete invariant. To address this issue, we propose an
enhanced framework that realizes a holistic understanding of a fully
commutative quiver's representation via synthesizing interpretations obtained
from intervals. Additionally, it provides a mechanism to tune the balance
between approximation resolution and computational complexity. This framework
is evaluated on commutative ladders of both finite-type and infinite-type. For
the former, we discover an efficient method for the indecomposable
decomposition leveraging solely one-parameter persistent homology. For the
latter, we introduce a new invariant that reveals persistence in the second
parameter by connecting two standard persistence diagrams using interval
approximations. We subsequently present several models for constructing
commutative ladder filtrations, offering fresh insights into random filtrations
and demonstrating our toolkit's effectiveness in analyzing the topology of
materials
A higher-dimensional homologically persistent skeleton
Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data — for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when data points split into good clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as holes and voids that persist for a large interval in a 1-parameter family of shapes associated to a cloud. These features can be visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimum spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale