Related papers: Persistent hypergraph homology and its application…
Modeling higher-order interactions (HOI) has emerged as a crucial challenge in complex systems analysis, as many phenomena cannot be fully captured by pairwise relationships alone. Hypergraphs, which generalize graphs by allowing…
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and…
Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior…
Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in a finite dimensional vector space…
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…
This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level-set persistence is presented using sheaves and cosheaves.
\"Uberhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of \"uberhomology, is related to dominating sets in graphs. To…
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence…
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under…
Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the \v{C}ech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers.…
The field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis.…
Motivated by applications to a wide range of assemble-to-order systems, operations scheduling, healthcare systems and collaborative economy applications, we introduce a stochastic matching model on hypergraphs, extending the model in [15]…
This work presents a new method to quantify connectivity in transportation networks. Inspired by the field of topological data analysis, we propose a novel approach to explore the robustness of road network connectivity in the presence of…
Persistent homology (PH) is a popular tool for topological data analysis that has found applications across diverse areas of research. It provides a rigorous method to compute robust topological features in discrete experimental…
In topological data analysis, persistent homology characterizes robust topological features in data and it has a summary representation, called a persistence diagram. Statistical research for persistence diagrams have been actively…
The persistence barcode (equivalently, the persistence diagram), which can be obtained from the interval decomposition of a persistence module, plays a pivotal role in applications of persistent homology. For multi-parameter persistent…
We apply persistent homology, the main method in topological data analysis, to the study of demographic data. Persistence diagrams efficiently summarize information about clusters or peaks in a region's demographic data. To illustrate how…
Persistent Homology (PH) offers stable, multi-scale descriptors of intrinsic shape structure by capturing connected components, loops, and voids that persist across scales, providing invariants that complement purely geometric…
We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics…
This paper introduces a novel hypergraph classification algorithm. The use of hypergraphs in this framework has been widely studied. In previous work, hypergraph models are typically constructed using distance or attribute based methods.…