Laplacian Features for Learning with Hyperbolic Space

Due to its geometric properties, hyperbolic space can support high-fidelity embeddings of tree- and graph-structured data. As a result, various hyperbolic networks have been developed which outperform Euclidean networks on many tasks: e.g. hyperbolic graph convolutional networks (GCN) can outperform vanilla GCN on some graph learning tasks. However, most existing hyperbolic networks are complicated, computationally expensive, and numerically unstable -- and they cannot scale to large graphs due to these shortcomings. With more and more hyperbolic networks proposed, it is becoming less and less clear what key component is necessary to make the model behave. In this paper, we propose HyLa, a simple and minimal approach to using hyperbolic space in networks: HyLa maps once from a hyperbolic-space embedding to Euclidean space via the eigenfunctions of the Laplacian operator in the hyperbolic space. We evaluate HyLa on graph learning tasks including node classification and text classification, where HyLa can be used together with any graph neural networks. When used with a linear model, HyLa shows significant improvements over hyperbolic networks and other baselines

Hyperbolic Graph Representation Learning: A Tutorial

Graph-structured data are widespread in real-world applications, such as social networks, recommender systems, knowledge graphs, chemical molecules etc. Despite the success of Euclidean space for graph-related learning tasks, its ability to model complex patterns is essentially constrained by its polynomially growing capacity. Recently, hyperbolic spaces have emerged as a promising alternative for processing graph data with tree-like structure or power-law distribution, owing to the exponential growth property. Different from Euclidean space, which expands polynomially, the hyperbolic space grows exponentially which makes it gains natural advantages in abstracting tree-like or scale-free graphs with hierarchical organizations. In this tutorial, we aim to give an introduction to this emerging field of graph representation learning with the express purpose of being accessible to all audiences. We first give a brief introduction to graph representation learning as well as some preliminary Riemannian and hyperbolic geometry. We then comprehensively revisit the hyperbolic embedding techniques, including hyperbolic shallow models and hyperbolic neural networks. In addition, we introduce the technical details of the current hyperbolic graph neural networks by unifying them into a general framework and summarizing the variants of each component. Moreover, we further introduce a series of related applications in a variety of fields. In the last part, we discuss several advanced topics about hyperbolic geometry for graph representation learning, which potentially serve as guidelines for further flourishing the non-Euclidean graph learning community.Comment: Accepted as ECML-PKDD 2022 Tutoria

Discrete-time Temporal Network Embedding via Implicit Hierarchical Learning in Hyperbolic Space

Representation learning over temporal networks has drawn considerable attention in recent years. Efforts are mainly focused on modeling structural dependencies and temporal evolving regularities in Euclidean space which, however, underestimates the inherent complex and hierarchical properties in many real-world temporal networks, leading to sub-optimal embeddings. To explore these properties of a complex temporal network, we propose a hyperbolic temporal graph network (HTGN) that fully takes advantage of the exponential capacity and hierarchical awareness of hyperbolic geometry. More specially, HTGN maps the temporal graph into hyperbolic space, and incorporates hyperbolic graph neural network and hyperbolic gated recurrent neural network, to capture the evolving behaviors and implicitly preserve hierarchical information simultaneously. Furthermore, in the hyperbolic space, we propose two important modules that enable HTGN to successfully model temporal networks: (1) hyperbolic temporal contextual self-attention (HTA) module to attend to historical states and (2) hyperbolic temporal consistency (HTC) module to ensure stability and generalization. Experimental results on multiple real-world datasets demonstrate the superiority of HTGN for temporal graph embedding, as it consistently outperforms competing methods by significant margins in various temporal link prediction tasks. Specifically, HTGN achieves AUC improvement up to 9.98% for link prediction and 11.4% for new link prediction. Moreover, the ablation study further validates the representational ability of hyperbolic geometry and the effectiveness of the proposed HTA and HTC modules.Comment: KDD202

Neural Embeddings of Graphs in Hyperbolic Space

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 success of neural embeddings has prompted significant amounts of research into applications in domains other than language. One such domain is graph-structured data, where embeddings of vertices can be learned that encapsulate vertex similarity and improve performance on tasks including edge prediction and vertex labelling. For both NLP and graph based tasks, embeddings have been learned in high-dimensional Euclidean spaces. However, recent work has shown that the appropriate isometric space for embedding complex networks is not the flat Euclidean space, but negatively curved, hyperbolic space. We present a new concept that exploits these recent insights and propose learning neural embeddings of graphs in hyperbolic space. We provide experimental evidence that embedding graphs in their natural geometry significantly improves performance on downstream tasks for several real-world public datasets.Comment: 7 pages, 5 figure

A Unification Framework for Euclidean and Hyperbolic Graph Neural Networks

Hyperbolic neural networks are able to capture the inherent hierarchy of graph datasets, and consequently a powerful choice of GNNs. However, they entangle multiple incongruent (gyro-)vector spaces within a layer, which makes them limited in terms of generalization and scalability. In this work, we propose to use Poincar\'e disk model as our search space, and apply all approximations on the disk (as if the disk is a tangent space derived from the origin), and thus getting rid of all inter-space transformations. Such an approach enables us to propose a hyperbolic normalization layer, and to further simplify the entire hyperbolic model to a Euclidean model cascaded with our hyperbolic normalization layer. We applied our proposed nonlinear hyperbolic normalization to the current state-of-the-art homogeneous and multi-relational graph networks. We demonstrate that not only does the model leverage the power of Euclidean networks such as interpretability and efficient execution of various model components, but also it outperforms both Euclidean and hyperbolic counterparts in our benchmarks