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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…
We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either…
Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a ubiquitous tool in TDA. Basing our study on this, we…
We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the…
To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the tail of these noise distributions. This study advances…
We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to…
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…
We develop a general framework for the probabilistic analysis of random finite point clouds in the context of topological data analysis. We extend the notion of a barcode of a finite point cloud to compact metric spaces. Such a barcode…
We investigate the homology of an unbounded noisy sample on $\mathbb{R}^d$, under various assumptions on the sampling density. This investigation is based on previous results by Adler, Bobrowski, and Weinberger (\cite{crackle}), and Owada…
A limit theorem for a sequence of diffusion processes on graphs is proved in a case when vary both parameters of the processes (the drift and diffusion coefficients on every edge and the asymmetry coefficients in every vertex), and…
This paper develops new limit theory for data that are generated by networks or more generally display cross-sectional dependence structures that are governed by observable and unobservable characteristics. Strategic network formation…
Let $P$ be a simple,stationary point process having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $P_n:= P \cap W_n$ be its…
Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration. Most filtrations used for persistent homology depend…
We study persistent Betti numbers and persistence diagrams obtained from time series and random fields. It is well known that the persistent Betti function is an efficient descriptor of the topology of a point cloud. So far, convergence…
A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate…
For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…
We consider random filtered complexes built over marked point processes on Euclidean spaces. Examples of our filtered complexes include a filtration of $\check{\textrm{C}}$ech complexes of a family of sets with various sizes, growths, and…
This paper is second in the series, following Pranav et al. (2019), focused on the characterization of geometric and topological properties of 3D Gaussian random fields. We focus on the formalism of persistent homology, the mainstay of…
Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity…
This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable…