Related papers: Strongly Hyperbolic Unit Disk Graphs
Hyperbolic geometry plays an important role within function theory of the disk. For example, via the Schwarz-Pick Lemma, the isometries of the unit disk $\mathbb D$ with respect to this geometry are the conformal self-maps of $\mathbb D$.…
Recently there was an increasing interest in applications of graph neural networks in non-Euclidean geometry; however, are non-Euclidean representations always useful for graph learning tasks? For different problems such as node…
In this draft paper, we introduce a novel architecture for graph networks which is equivariant to the Euclidean group in $n$-dimensions. The model is designed to work with graph networks in their general form and can be shown to include…
Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks or social…
Suppose that $X$ is an infinite, connected, locally finite, quasi-transitive graph with the property that every bi-infinite quasi-geodesic uniformly coarsely separates $X$ into exactly two deep pieces. We show that such an $X$ is…
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers…
Hyperbolic deep learning leverages the metric properties of hyperbolic spaces to develop efficient and informative embeddings of hierarchical data. Here, we focus on the solvable group structure of hyperbolic spaces, which follows naturally…
Neural embeddings have been used with great success in Natural Language Processing (NLP). They provide compact representations that encapsulate word similarity and attain state-of-the-art performance in a range of linguistic tasks. The…
Due to its geometric properties, hyperbolic space can support high-fidelity embeddings of tree- and graph-structured data, upon which various hyperbolic networks have been developed. Existing hyperbolic networks encode geometric priors not…
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…
Hyperbolic uniform disk graphs (HUDGs) are intersection graphs of disks with some radius $r$ in the hyperbolic plane, where $r$ may be constant or depend on the number of vertices in a family of HUDGs. We show that HUDGs with constant…
Diffusion generative models (DMs) have achieved promising results in image and graph generation. However, real-world graphs, such as social networks, molecular graphs, and traffic graphs, generally share non-Euclidean topologies and hidden…
A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius…
Weak unit disk contact graphs are graphs that admit representing nodes as a collection of internally disjoint unit disks whose boundaries touch if there is an edge between the corresponding nodes. In this work we focus on graphs without…
Connectomes are spatially embedded networks whose architecture has been shaped by physical constraints and communication needs throughout evolution. Using a decentralized navigation protocol, we investigate the relationship between the…
A family of graphs optimized as the topologies for supercomputer interconnection networks is proposed. The special needs of such network topologies, minimal diameter and mean path length, are met by special constructions of the weight…
Clustering, as an unsupervised technique, plays a pivotal role in various data analysis applications. Among clustering algorithms, Spectral Clustering on Euclidean Spaces has been extensively studied. However, with the rapid evolution of…
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
Learning good image representations that are beneficial to downstream tasks is a challenging task in computer vision. As such, a wide variety of self-supervised learning approaches have been proposed. Among them, contrastive learning has…
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links,…