Limit theorems for Betti numbers of random simplicial complexes

Kahle, Matthew; Meckes, Elizabeth

doi:10.4310/hha.2013.v15.n1.a17

Cited by 107 publications

(161 citation statements)

References 19 publications

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“…Alternatively we could only provide separate limit theorems for each individual region. Similar phenomena have been pointed out in a series of works [8,11,17,24,35], in which various limit theorems for topological invariants in different regimes were derived (though they are not directly related to the layered structure described above).…”

Section: Introductionsupporting

confidence: 74%

Convergence of persistence diagrams for topological crackle

Owada¹,

Bobrowski²

2020

Bernoulli

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In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of homological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to homological cycles generated by "noisy" samples where the support is unbounded. We aim to establish weak convergence results for persistence diagramsa point process representation for persistent homology, where each homological cycle is represented by its (birth, death) coordinates. In this work we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. In this framework we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a short lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.2010 Mathematics Subject Classification. Primary 60G70. Secondary 55U10, 60F05, 60G55.

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Section: Introductionsupporting

confidence: 74%

Convergence of persistence diagrams for topological crackle

Owada¹,

Bobrowski²

2020

Bernoulli

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“…Recently, as a higher-dimensional analogue of a random geometric graph, there has been growing interest in the asymptotics of the so-called random Cěch complex. See, for example, [18], [19], and [28], while [10] provides an elegant review of that direction.…”

Section: Introductionmentioning

confidence: 99%

Functional central limit theorem for subgraph counting processes

Owada¹

2017

Electron. J. Probab.

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The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region -called a weak core -in which the random points are highly densely scattered and form a giant geometric graph.2010 Mathematics Subject Classification. Primary 60G70, 60D05. Secondary 60G15, 60G18.

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“…On the contrary, in Y r (·), the unique dynamic point y 2 meets only y 3 periodically. been studied in various contexts and with various names such as Betti curve, feature counting function, etc, [2,28,34,35,42,62]. The rank function rk(M X ) of M X has also been extensively considered [17,19,51,58,59].…”

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confidence: 99%

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

Kim

Mémoli

2020

Discrete Comput Geom

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Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS).In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter "spatiotemporal" filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov-Hausdorff distance which permits metrizing the collection of all DMSs.On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well known interleaving distance. In either case, the metric d can be computed in polynomial time. IntroductionStability and tractability of TDA for studying metric spaces. Finite point clouds or finite metric spaces are amongst the most common data representations considered in topological data analysis (TDA) [13,29,33]. In particular, the stability of the Single Linkage Hierarchical Clustering (SLHC) method [16] or the stability of the persistent homology of filtered Rips complexes built on metric spaces [22,23] motivates adopting these constructions when studying metric spaces arising in applications.Whereas there has been extensive applications of TDA to static metric data (thanks to the aforementioned theoretical underpinnings), there is not much study of dynamic metric data from the TDA perspective. Our motivation for considering dynamic metric data stems from the study and characterization of flocking/swarming behaviors of animals [5,36,37,39,53,57,63,69], convoys [41], moving clusters [43], or mobile groups [40,70]. In this paper, by extending ideas from [16,22,23,47,46], we aim at establishing a TDA framework for the study of dynamic metric spaces (DMSs) which comes together with stability theorems. We begin by describing and comparing relevant work with ours. Lack of an adequate metric for DMSs. In [55], Munch considers vineyards -a certain notion of time-varying persistence diagrams introduced by Cohen-Steiner et al. [25] -as signatures for dynamic point clouds. Munch, in particular, shows that vineyards are stable 1 [24] under perturbations of the input dynamic point cloud ...

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Limit theorems for Betti numbers of random simplicial complexes

Cited by 107 publications

References 19 publications

Convergence of persistence diagrams for topological crackle

Convergence of persistence diagrams for topological crackle

Functional central limit theorem for subgraph counting processes

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

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