Related papers: Computing singular simplicial homologies of digrap…
Directed graphs can be studied by their associated directed flag complex. The homology of this complex has been successful in applications as a topological invariant for digraphs. Through comparison with path homology theory, we derive a…
The singular cubical homology theory for the category of quivers or digraphs can be constructed similarly to the classical singular homology theory for topological spaces. The case of digraphs and quivers differs from the topological case…
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…
In this paper, we extend the notion of directed clique complex to quivers and introduce an associated homology theory. By applying this construction to biquandle coloring quivers, we obtain new invariants of links. We then introduce a…
Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs -- specifically simplicial, relative, and chain…
Directed graphs are ubiquitous models for networks, and topological spaces they generate, such as the directed flag complex, have become useful objects in applied topology. The simplices are formed from directed cliques. We extend Atkin's…
Persistent homology enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to…
We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent…
We study the homology of simplicial and cubical sets with symmetries. These are simplicial and cubical sets with additional maps expressing the symmetries of simplices and cubes. We consider the chain complex computing the homology groups…
We present a fast algorithm for computing discrete cubical homology of graphs over finite fields with an appropriate characteristic. This algorithm improves on several computational steps compared to constructions in the existing…
Discrete cubical homology arose as the homology theory associated with discrete cubical homotopy theory. Despite the combinatorial nature of this homology, its computation has posed a significant challenge to the researchers in the field.…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
This survey describes some useful properties of the local homology of abstract simplicial complexes. Although the existing literature on local homology is somewhat dispersed, it is largely dedicated to the study of manifolds, submanifolds,…
We provide a characterization of two types of directed homology for fully-connected, feedforward neural network architectures. These exact characterizations of the directed homology structure of a neural network architecture are the first…
In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call "strong…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain…
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…
The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove…
The neighborhood complex of a graph was introduced by Lov\'asz to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such `Hom complexes' are also…
Persistent homology theory is a relatively new but powerful method in data analysis. Using simplicial complexes, classical persistent homology is able to reveal high dimensional geometric structures of datasets, and represent them as…