Related papers: Homology of weighted path complexes and directed h…
Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [3, 4, 12]). In this…
Let $A$ be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded $A$-modules, providing weighted versions of…
This paper provides an overview of results, concerning longest or heaviest paths, in the area of random directed graphs on the integers along with some extensions. We study first-order asymptotics of heaviest paths allowing weights both on…
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph ${\tt G}$ is the simplicial complex whose faces are the multipaths of ${\tt G}$. We compute the Euler characteristic, and associated…
We study some homological invariants of a given generalized bound path algebra in terms of those of the algebras used in its construction. We discuss the particular case where the algebra is a generalized path algebra and give conditions…
Hypergraphs, as a generalization of simplicial complexes, have long been a subject of interest in their geometric interpretation. The subdivision of simplicial complexes can, to some extent, provide insights into the geometry of simplicial…
We study the homotopy theory of diagrams of chain complexes over a field indexed by a finite poset, and show that it can be completely described in terms of appropriate diagrams of graded vector spaces.
The last decade has seen the development of path homology and magnitude homology -- two homology theories of directed graphs, each satisfying classic properties such as Kunneth and Mayer-Vietoris theorems. Recent work of Asao has shown that…
Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than…
We show that for a complete complex algebraic variety the pure component of homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce…
Weighted hypergraphs are generalizations of weighted simplicial complexes. In recent years, weighted Laplacians of weighted simplicial complexes have been studied. In 2016, as a generalization of the homology of simplicial complexes, the…
We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality…
This paper studies a notion of directed homology for preordered spaces, called the homology digraph. We show that the homology digraph is a directed homotopy invariant and establish variants of the main results of ordinary singular homology…
We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of…
We propose a novel method for topological analysis of unweighted graphs which is based on \textit{persistent homology}. The proposed method maps the input graph to a complete weighted graph where the weighting function maps each edge to a…
We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse…
We state the graph-theoretic computational problem of finding tight paths in a directed, edge-weighted graph, as well as its simplification of finding tight pairs. These problems are motivated by the need of algorithms that find so-called…
In some applications of matching, the structural or hierarchical properties of the two graphs being aligned must be maintained. The hierarchical properties are induced by the direction of the edges in the two directed graphs. These…
In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in…
This paper defines an invariant associated to Whitehead's certain exact sequence of a simply connected CW-complex which is much more elementary - and less powerful - than the boundary invariant of Baues. Nevertheless, in good cases, it…