相关论文: Mayer Path Homology
This work is part of a series of papers focusing on multipath cohomology of directed graphs. Multipath cohomology is defined as the (poset) homology of the path poset -- i.e., the poset of disjoint simple paths in a graph -- with respect to…
We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…
A graphical model provides a compact and efficient representation of the association structure of a multivariate distribution by means of a graph. Relevant features of the distribution are represented by vertices, edges and other…
In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud…
In this note, we give a self-contained account on a construction for a directed homology theory based on modules over algebras, linking it to both persistence homology and natural homology. We study its first properties, among which some…
This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding…
The symmetries of paths in a manifold $M$ are classified with respect to a given pointwise proper action of a Lie group $G$ on $M$. Here, paths are embeddings of a compact interval into $M$. There are at least two types of symmetries:…
Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected…
For any acyclic quiver $Q$ without multiple edges, we construct a monoidal category $\mathcal{R}_Q$ whose indecomposable objects are tensor products (over the base field) of finite-dimensional modules over the path algebra of $Q$. We show…
In this paper, various kinds of invariants of directed graphs are summarized. In the first topic, the invariant w(G) for a directed graph G is introduced, which is primarily defined by S. Chen and X.M. Chen to solve a problem of weak…
Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics -- where it allows for synthetic…
The theory of path homology for digraphs was developed by Alexander Grigor'yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. In this paper, we consider the differential algebras on digraphs and define the parametrized homology of digraphs as…
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…
In this paper we explore the algebraic structure and combinatorial properties of eulerian magnitude homology. First, we analyze the diagonality conditions of eulerian magnitude homology, providing a characterization of complete graphs.…
This paper presents and explores a theory of \emph{multiholomorphic maps}. This group of ideas generalizes the theory of pseudoholomorphic curves in a direction suggested by consideration of the kinds of compatible geometric structures that…
In this paper, we study long paths and Hamiltonian paths in inhomogenous random graphs. In the first part of the paper, we consider an inhomogenous Erd\H{o}s-R\'enyi random graph $G_E$ with average edge density $p_n.$ We prove that if…
In this paper we study and compare two homology theories for (simple and undirected) graphs. The first, which was developed by Barcelo, Caprano, and White, is based on graph maps from hypercubes to the graph. The second theory was developed…
While the disciplines of physics and engineering sciences in many cases have taken advantage from accurate time-series prediction of system behaviour by applying ordinary differential equation systems upon precise basic physical laws such…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method…