Related papers: Homological aspects of oriented hypergraphs
Networks are often studied as graphs, where the vertices stand for entities in the world and the edges stand for connections between them. While relatively easy to study, graphs are often inadequate for modeling real-world situations,…
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…
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point…
This paper extends graphic statics by describing the forces and moments in any 3D rigid-jointed frame structure in terms of cell complexes using homology theory of algebraic topology. Graphic statics provides a highly geometric way to…
The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…
Recent developments in applied algebraic topology can simplify and extend results in graphic statics - the analysis of equilibrium forces, dual diagrams, and more. The techniques introduced here are inspired by recent developments in…
As complex networks find applications in a growing range of disciplines, the diversity of naturally occurring and model networks being studied is exploding. The adoption of a well-developed collection of network taxonomies is a natural…
Hypergraphs provide a natural way to represent polyadic relationships in network data. For large hypergraphs, it is often difficult to visually detect structures within the data. Recently, a scalable polygon-based visualization approach was…
We investigate structural implications arising from the condition that a given directed graph does not interpret, in the sense of primitive positive interpretation with parameters or orbits, every finite structure. Our results generalize…
We define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory, as well as for structures on sets of isomorphism types…
We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy…
The study of hypergraphs has received a lot of attention over the past few years, however up until recently there has been no interest in systems where higher order interactions are not undirected. In this article we introduce the notion of…
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…
Based on a simple example, it is explained how the homological analysis may be applied for modeling of the electric circuits. The homological branch, mesh and nodal analyses are presented. Geometrical interpretations are given.
We propose to study homomorphisms of connectome graphs. Homomorphisms can be studied as sequences of elementary homomorphisms - folds, which identify pairs of vertices. Several fold types are defined. Initial computation results for some…
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…
We determine all possible triples of depth, dimension, and regularity of edge ideals of weighted oriented graphs with a fixed number of vertices. Also, we compute all the possible Betti table sizes of edge ideals of weighted oriented trees…
We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our…
We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads, respectively. Algebras for these models are homotopy versions of the corresponding structures.
The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of…