Related papers: Homologies of path complexes and digraphs
We study hypergraph visualization via its topological simplification. We explore both vertex simplification and hyperedge simplification of hypergraphs using tools from topological data analysis. In particular, we transform a hypergraph to…
Series-parallel network topologies generally exhibit simplified dynamical behavior and avoid high combinatorial complexity. A comprehensive analysis of how flow complexity emerges with a graph's deviation from series-parallel topology is…
We study properties of contiguity distance between simplicial maps. In particular, we show that simplicial versions of $LS$-category and topological complexity are particular cases of this more general notion.
We provide first a categorical exploration of, and then completion of the mapping of the relationships among, three fundamental perspectives on binary relations: as the incidence matrices of hypergraphs, as the formal contexts of concept…
Higher dimensional automata, i.e. labelled precubical sets, model concurrent systems. We introduce the homology graph of an HDA, which is a directed graph whose nodes are the homology classes of the HDA. We show that the homology graph is…
We explore pseudometrics for directed graphs in order to better understand their topological properties. The directed flag complex associated to a directed graph provides a useful bridge between network science and topology. Indeed, it has…
We show that the spaces of holomorphic and continuous maps from a smooth complex projective variety to a projective space have the same homology in a range depending on the degree of the maps.
We review the gauge and ghost cyle graph complexes as defined by Kreimer, Sars and van Suijlekom in "Quantization of gauge fields, graph polynomials and graph homology" and compute their cohomology. These complexes are generated by…
In this paper we will introduce and give topological properties of a new concept named simplicial distance which is the simplicial analog of the homotopic distance (in the sense of Marcias-Virgos and Mosquera-Lois in their paper [6]).…
We give an overview of differential cohomology from the point of view of algebraic topology. This includes a survey of several different definitions of differential cohomology groups, a discussion of differential characteristic classes, an…
We develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate directed persistent homology to classical persistent homology, prove some stability results,…
This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their…
The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes -- a purely combinatorial one and two geometric ones. It is shown, that most…
This short note introduces a notion of directed homotopy equivalence and of "directed" topological complexity (which elaborates on the notion that can be found in e.g. Farber's book) which have a number of desirable joint properties. In…
It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles,…
We define a simpler notion of symmetric topological complexity more ad hoc to the motion planning problem which was the original motivation for the definition of topological complexity. This is a homotopy invariant that we call…
The purpose of this note is to extend to Brownian loops some homology and holonomy results obtained in the case of discrete loops on a graph
We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original…
Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles…
The paper is devoted to an approach to the bounded cohomology theory based on the theories of simplicial sets and Postnikov systems. In particular, the main results of the bounded cohomology theory of topological spaces are extended to…