Related papers: Computing the Homology of Hypergraphs
Topological representations are rapidly becoming a popular way to capture and encode higher-order interactions in complex systems. They have found applications in disciplines as different as cancer genomics, brain function, and…
We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell…
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of…
A theory of sections of simplicial height functions is developed. At the core of this theory lies the section complex, which is assembled from higher section spaces. The latter encode flow lines along the height, as well as their…
Homology is a tool in topological data analysis which measures the shape of the data. In many cases, these measurements translate into new insights which are not available by other means. To compute homology, we rely on mathematical…
Graph embeddings have become a key and widely used technique within the field of graph mining, proving to be successful across a broad range of domains including social, citation, transportation and biological. Graph embedding techniques…
Arone and Turchin defined graph-complexes computing the rational homotopy of the spaces of long embeddings. The graph-complexes split into a direct sum by the number of loops in graphs. In this paper we compute the homology of its two-loop…
A matching complex of a simple graph $G$ is a simplicial complex with faces given by the matchings of $G$. The topology of matching complexes is mysterious; there are few graphs for which the homotopy type is known. Marietti and Testa…
Graph embeddings, wherein the nodes of the graph are represented by points in a continuous space, are used in a broad range of Graph ML applications. The quality of such embeddings crucially depends on whether the geometry of the space…
In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…
Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between…
Intersection homology is defined for simplicial, singular and PL chains and it is well known that the three versions are isomorphic for a full filtered simplicial complex. In the literature, the isomorphism, between the singular and the…
The hyperlink prediction task, that of proposing new links between webpages, can be used to improve search engines, expand the visibility of web pages, and increase the connectivity and navigability of the web. Hyperlink prediction is…
We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding…
Homology with values in a connection with possibly irregular singular points on an algebraic curve is defined, generalizing homology with values in the underlying local system for a connection with regular singular points. Integration…
Computation of persistent homology of simplicial representations such as the Rips and the C\v{e}ch complexes do not efficiently scale to large point clouds. It is, therefore, meaningful to devise approximate representations and evaluate the…
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…
A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph into a large network. We propose two complementary MCMC algorithms for…
We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the…
We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this…