We introduce several geometric notions, including the width of a homology class, to theory persistent homology. These ideas provide interpretations persistence diagrams. Indeed, we give quantitative and descriptions “life span” or “persistence” class. As case study, analyze power filtration on unweighted graphs explicit bounds for life spans classes in diagrams all dimensions.

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from topological viewpoint. A prevalent technique such analysis involves computation of homology groups and their persistence. These concepts are well suited spaces that not directed. As result, one needs concept accommodates orientations in input space. Path-homology developed graphs by Grigoryan e...

A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose framework that optimises the choice for dataset graphs, such their diagrams capture features graphs are best suited to given data science problem. Since graph is derived from its Laplacian, our encodes geometric properties in can be applied without priori...

Persistent homology typically studies the evolution of homology groups Hp(X) (with coefficients in a field) along a filtration of topological spaces. A∞-persistence extends this theory by analysing the evolution of subspaces such as V := Ker ∆n|Hp(X) ⊆ Hp(X), where {∆m}m≥1 denotes a structure of A∞-coalgebra on H∗(X). In this paper we illustrate how A∞-persistence can be useful beyond persisten...

The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of R-valued functions, the result was later cast in a more general algebraic form, in the language of persistence modules and interleavings. In this paper, we establish an analogue of this algebraic stability theorem for z...

We relate the machinery of persistence modules to Legendrian contact homology theory and Poisson bracket invariants, use it show existence connecting trajectories symplectic Hamiltonian flows.

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in R. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in R persistence has been extended to a pairing...

Abstract. Persistent homology was shown by Zomorodian and Carlsson [35] to be homology of graded chain complexes with coefficients in the graded ring k[t]. As such, the behavior of persistence modules — graded modules over k[t] — is an important part in the analysis and computation of persistent homology. In this paper we present a number of facts about persistence modules; ranging from the wel...