Related papers: Persistent Betti numbers of random \v{C}ech comple…

In this paper the homology stability for symplectic groups over a ring with finite stable rank is established. First we develop a `nerve theorem' on the homotopy type of a poset in terms of a cover by subposets, where the cover is itself…

This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…

This paper studies Bernoulli cell complexes from the perspective of persistent homology, Tutte polynomials, and random-cluster models. Following the previous work [9], we first show the asymptotic order of the expected lifetime sum of the…

We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic…

Let $M$ be a compact, unit volume, Riemannian manifold with boundary. In this paper we study the homology of a random \v{C}ech-complex generated by a homogeneous Poisson process in $M$. Our main results are two asymptotic threshold…

We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…

We compute numerically the homology of several graph complexes in low loop orders, extending previous results.

0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…

We explore the recently introduced persistent reachability homology (PRH) of digraph data, i.e. data in the form of directed graphs. In particular, we study the effectiveness of PRH in network classification task in a key neuroscience…

Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…

We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by…

Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the…

Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the…

In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and…

A geometric entropy is defined as the Riemannian volume of the parameter space of a statistical manifold associated with a given network. As such it can be a good candidate for measuring networks complexity. Here we investigate its ability…

We show that the graph property of having a (very) large $k$-th Betti number $\beta_k$ for constant $k$ is testable with a constant number of queries in the dense graph model. More specifically, we consider a clique complex defined by an…

In this paper, we introduce a novel persistence framework for Morse decompositions in Markov chains using combinatorial multivector fields. Our approach provides a structured method to analyze recurrence and stability in finite-state…

We find the asymptotic number of connected graphs with $k$ vertices and $k-1+l$ edges when $k,l$ approach infinity, reproving a result of Bender, Canfield and McKay. We use the {\em probabilistic method}, analyzing breadth-first search on…

Permutations of correlated sequences of random variables appear naturally in a variety of applications such as graph matching and asynchronous communications. In this paper, the asymptotic statistical behavior of such permuted sequences is…

The machine learning technique of persistent homology classifies complex systems or datasets by computing their topological features over a range of characteristic scales. There is growing interest in applying persistent homology to…