Related papers: A limit theorem for persistence diagrams of random…
Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature…
We analyze site percolation on directed and undirected graphs with site-dependent open-site probabilities. We construct upper bounds on cluster susceptibilities, vertex connectivity functions, and the expected number of simple open cycles…
We consider graphs with vertices of degree 1 or 2 and prove that the numbers of components of sizes 2 to q have a limit normal distribution for any q > 1. The result is also extended to multigraphs.
We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving…
We prove a strong law of large numbers for a class of strongly mixing processes. Our result rests on recent advances in understanding of concentration of measure. It is simple to apply and gives finite-sample (as opposed to asymptotic)…
This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of solutions snapshots, what…
We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.
In this paper we extend two limit theorems which were recently obtained for fragmentation processes to such processes with immigration. More precisely, in the setting with immigration we consider a limit theorem for the process counted with…
Persistent homology studies the birth and death of cycles in a parameterized family of spaces. In this paper, we study the birth and death of cycles in a multifiltration of a chain complex with the goal of producing a persistence diagram…
We construct a flow of continuous time and discrete state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching superprocesses over the positive…
This paper introduces a reformulation of the classical convergence theorem for spectral sequences of filtered complexes which provides an algorithm to effectively compute the induced filtration on the total (co)homology, as soon as the…
We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett, who showed that for arbitrarily small infection parameter $\lambda$, the survival time of…
We extend Berge's Maximum Theorem to allow for incomplete preferences. We first provide a simple version of the Maximum Theorem for convex feasible sets and a fixed preference. Then, we show that if, in addition to the traditional…
The paper develops a general framework for constrained clustering which is based on the close connection of geometric clustering and diagrams. Various new structural and algorithmic results are proved (and known results generalized and…
We consider perturbations of interval maps with indifferent fixed points, which we refer to as wobbly interval intermittent maps, for which stable laws for general H\"older observables fail. We obtain limit laws for such maps and H\"older…
We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the…
Population structure can be modelled by evolutionary graphs, which can have a substantial, but very subtle influence on the fate of the arising mutants. Individuals are located on the nodes of these graphs, competing with each other to…
We consider a population with non-overlapping generations, whose size goes to infinity. It is described by a discrete genealogy which may be time non-homogeneous and we pay special attention to branching trees in varying environments. A…
In this article, we consider a sequence $(N_n)_{n \geq 1}$ of point processes, whose points lie in a subset $E$ of $\bR \verb2\2 \{0\}$, and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…