# Hyperbolic Graph Convolutional Neural Networks

@article{Chami2019HyperbolicGC, title={Hyperbolic Graph Convolutional Neural Networks}, author={Ines Chami and Rex Ying and Christopher R{\'e} and Jure Leskovec}, journal={Advances in neural information processing systems}, year={2019}, volume={32}, pages={ 4869-4880 } }

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 an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation…

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## References

SHOWING 1-10 OF 52 REFERENCES

Neural Embeddings of Graphs in Hyperbolic Space

- Computer ScienceArXiv
- 2017

A new concept that exploits recent insights and proposes learning neural embeddings of graphs in hyperbolic space is presented and experimental evidence that embedding graphs in their natural geometry significantly improves performance on downstream tasks for several real-world public datasets is provided.

Hyperbolic Graph Neural Networks

- Computer ScienceNeurIPS
- 2019

A novel GNN architecture for learning representations on Riemannian manifolds with differentiable exponential and logarithmic maps is proposed and a scalable algorithm for modeling the structural properties of graphs is developed, comparing Euclidean and hyperbolic geometry.

Hyperbolic Attention Networks

- Computer ScienceICLR
- 2019

This work introduces hyperbolic attention networks to endow neural networks with enough capacity to match the complexity of data with hierarchical and power-law structure and re-expressing the ubiquitous mechanism of soft attention in terms of operations defined for hyperboloid and Klein models.

Poincaré Embeddings for Learning Hierarchical Representations

- Computer ScienceNIPS
- 2017

This work introduces a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n-dimensional Poincare ball -- and introduces an efficient algorithm to learn the embeddings based on Riemannian optimization.

Hyperbolic Entailment Cones for Learning Hierarchical Embeddings

- Computer ScienceICML
- 2018

This work presents a novel method to embed directed acyclic graphs through hierarchical relations as partial orders defined using a family of nested geodesically convex cones and proves that these entailment cones admit an optimal shape with a closed form expression both in the Euclidean and hyperbolic spaces.

Representation Tradeoffs for Hyperbolic Embeddings

- Computer ScienceICML
- 2018

A hyperbolic generalization of multidimensional scaling (h-MDS), which offers consistently low distortion even with few dimensions across several datasets, is proposed and a PyTorch-based implementation is designed that can handle incomplete information and is scalable.

Poincaré GloVe: Hyperbolic Word Embeddings

- Computer ScienceICLR
- 2019

Empirically, based on extensive experiments, it is proved that the embeddings, trained unsupervised, are the first to simultaneously outperform strong and popular baselines on the tasks of similarity, analogy and hypernymy detection.

How Powerful are Graph Neural Networks?

- Computer ScienceICLR
- 2019

This work characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures, and develops a simple architecture that is provably the most expressive among the class of GNNs.

Hyperbolic Geometry of Complex Networks

- Computer Science, MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

It is shown that targeted transport processes without global topology knowledge are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.

Inductive Representation Learning on Large Graphs

- Computer ScienceNIPS
- 2017

GraphSAGE is presented, a general, inductive framework that leverages node feature information (e.g., text attributes) to efficiently generate node embeddings for previously unseen data and outperforms strong baselines on three inductive node-classification benchmarks.