Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

Hiraoka, Yasuaki; Shirai, Tomoyuki

doi:10.1002/rsa.20718

Cited by 25 publications

(22 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The main results include a rigorous proof of the convergence of L k (K (d) n )/n d−1 to I d−1 and a determination of the growth exponent of E[L k (C n )]. These results solve problems posed in [10] and are summarized in the following theorems. In particular, E[L d−1 (K (d) n )]/n d−1 converges to I d−1 as n → ∞.…”

Section: Introductionmentioning

confidence: 69%

“…In [10], the asymptotic behavior of E[L d−1 (K (d) n )]/n d−1 as n → ∞ is also discussed, and a possible limiting constant I d−1 is found by a heuristic argument. For the exact value of I d−1 , see (4.10) and (4.11) and Section 4.4.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.5 is a special case of these theorems, and Theorem 1.2 also follows from them. The proof of Theorem 1.2 in [10] is based on the monotonicity of β d−1 (K (d) n (t)) with respect to t. Our approach is different and is applicable to more general random filtrations. For the proof of Theorem 1.4 and an extension (Theorem 4.11), we additionally make use of the result by Linial and Peled [15] on the convergence of β d (K (d) n (c/n))/n d as n → ∞ for each c ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic behavior of lifetime sums for random simplicial complex processes

Hino¹,

Kanazawa²

2019

J. Math. Soc. Japan

View full text Add to dashboard Cite

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 higherdimensional counterpart of Frieze's ζ(3)-limit theorem for the Erdős-Rényi 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.2010 Mathematics Subject Classification. Primary 05C80, 60D05; Secondary 55U10, 05E45, 60C05. Key Words and Phrases. Linial-Meshulam complex process, random clique complex process, multiparameter random simplicial complex, lifetime sum, Betti number. K n (t) decreases, and β 0 (K n (t)) denotes the zeroth (reduced) Betti number of K n (t), that is, the number of connected components of K n (t) minus one. This type of relation holds for a general increasing family of graphs. Applying this formula and analyzing β 0 (K n (t)) in detail, Frieze [7] obtained the following significant result about the behavior of W n .and for any ε > 0,Recently, there has been a growing interest in studying random simplicial complexes as a higher-dimensional generalization of random graphs. Since an Erdős-Rényi graph can be regarded as a one-dimensional random simplicial complex, and graph connectivity can be equivalently described as the vanishing of the zeroth (reduced) homology, it is natural to seek a higher-dimensional analogue to the theory of Erdős-Rényi's G(n, p) model. The d-Linial-Meshulam model [14] and the random clique complex model [12] are typical models of this type. The d-Linial-Meshulam model Y d (n, p) is defined as the distribution of d-dimensional random simplicial complexes with n vertices and the complete (d−1)-dimensional skeleton such that each d-simplex is placed with independent probability p. The random clique complex model C(n, p) is defined as the distribution of the clique complex of the Erdős-Rényi graph that follows G(n, p). Here, given a graph G, its clique complex Cl(G) is defined as the maximal simplicial complex among those for which the one-dimensional skeletons are equal to G. Linial, Meshulam, and Wallach [14,16] exhibited the threshold for the vanishing of the (d − 1)-th homology for the d-Linial-Meshulam model, which is analogous to the connectivity threshold of the Erdős-Rényi graph. Later, Kahle [13] obtained similar results for the random clique complex model.Along another line, Hiraoka and Shirai [10] obtained a higher-dimensional analogue of (1.1) in the context of the theory of persistent homology. Persistent homologies can describe the topological features of a filtration (i.e., an increasing family of simplicial complexes; see, e.g., [3,17]). In par...

show abstract

Section: Introductionmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic behavior of lifetime sums for random simplicial complex processes

Hino¹,

Kanazawa²

2019

J. Math. Soc. Japan

View full text Add to dashboard Cite

show abstract

“…We now discuss the definition in more detail. Apart from being more restrictive than that in [HS17,Kal83], our definition differs from that of [HS17] in its use of field coefficients over integer coefficients. Clearly, in the case of d = 1, S is a minimal spanning tree on the graph K 1 .…”

Section: Spanning Acyclesmentioning

confidence: 99%

Randomly Weighted $d$-Complexes: Minimal Spanning Acycles and Persistence Diagrams

Škraba¹,

Thoppe

Yogeshwaran³

2020

Electron. J. Combin.

View full text Add to dashboard Cite

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.

show abstract

“…The persistent homology originated in algebraic topology, and thus, there are only a limited number of probabilistic and statistical studies that have treated it. The present paper contains some of the earliest results from a pure probabilistic viewpoint, other papers being [30], [11], and [21]. In particular, [11] investigated probabilistic features of bars of the maximum size, from which they tried to capture "extremal" behavior of bars.…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

Owada¹

2018

Ann. Appl. Probab.

View full text Add to dashboard Cite

We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on R d by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology. It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, i.e., the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes. We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology.

show abstract

Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

Cited by 25 publications

References 30 publications

Asymptotic behavior of lifetime sums for random simplicial complex processes

Asymptotic behavior of lifetime sums for random simplicial complex processes

Randomly Weighted $d$-Complexes: Minimal Spanning Acycles and Persistence Diagrams

Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

Contact Info

Product

Resources

About