Related papers: Barcode Embeddings for Metric Graphs
A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in…
Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the…
Graph homologies are powerful tools to compute the rational homotopy group of the space of long embeddings. Two graph homologies have been invented from two approaches to study the space of long embeddings: the hairy graph homology from…
In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren, and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3)(2019) 479-500] and study the…
Consider an ergodic unimodular random one-ended planar graph $\G$ of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By "locally…
This paper introduces new invariants of rigid vertex graph embeddings by using non-local combinatorial information that is available at each graphical node. The new non-local information that we use in this paper involves parity in the…
Node2vec is a graph embedding method that learns a vector representation for each node of a weighted graph while seeking to preserve relative proximity and global structure. Numerical experiments suggest Node2vec struggles to recreate the…
Theoretical results from discrete geometry suggest that normed spaces can abstractly embed finite metric spaces with surprisingly low theoretical bounds on distortion in low dimensions. In this paper, inspired by this theoretical insight,…
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…
We study the topological persistence of the (path) configuration spaces and the (path) independence complexes for digraphs as well as their underlying graphs. We construct some canonical embeddings from the (path) independence complexes of…
Let $G$ be a topological group, let $\phi$ be a continuous endomorphism of $G$ and let $H$ be a closed $\phi$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is,…
In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some…
We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…
We discuss various aspects of a local-to-global embedding technique and the metric geometry of stable metric spaces and of two of its important subclasses: locally finite spaces and proper spaces. We explain how the barycentric gluing…
Magnitude homology is an emerging framework that captures the intrinsic topological and geometric features of metric spaces, demonstrating significant potential for topoplogical data analysis and geometric data analysis. This work…
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
Embedding static graphs in low-dimensional vector spaces plays a key role in network analytics and inference, supporting applications like node classification, link prediction, and graph visualization. However, many real-world networks…
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…
We consider the following properties of compact oriented irreducible graph-manifolds: to contain a $\pi_1$-injective surface (immersed, virtually embedded or embedded), be (virtually) fibered over $S^1$, and to carry a metric of nonpositive…