Hierarchical graphs embed in hyperbolic space with lower distortion than in Euclidean space owing to its negative curvature. However, their gradient-based learning is hampered at large radii, where the Poincaré ball and the Lorentz hyperboloid models fail numerically. Polar coordinates avoid this problem, but the hyperbolic metric scales the angular step by the hyperbolic sine of the radius, freezing angular motion. We observe that this factor is a choice, silently fixed by existing implementations: the Euclidean tangent parametrization, for instance, uses the radius itself. We show that other choices are not only possible but preferable. They are endpoints of a one-parameter family of optimization preconditioners with curvatures from −1 to 0, while the embedding remains at curvature −1. We show that since the Euclidean preconditioner rearranges a layout but refines it poorly, while an intermediate one refines far better once a layout is in place, combining them in two stages reduces the loss on real-world trees by 46-74% over the best single curvature.
Figures & tables
Caterp.
Marsupialia
Fabaceae
Cichlidae
Falconif.
PowerLaw
Europe
∣V∣
200
837
1067
3134
281
200
399
∣E∣
199
836
1066
3133
280
780
5993
diameter
41
36
44
64
27
4
5
radial span s
19.9
17.7
21.8
27.9
13.2
1.5
2.4
Hydra init.
1763
6406
13545
460985
363
17.9
51.2
Single curvature
Table 1: Upper part: graphs used in the experiments and some statistics. Lower part: loss (stress) for each optimization method ( ×103 ). Single curvature : one c throughout, from the Hydra initialization. Two-stage : c=10−6 up to the handover (stage 1), then the c in each row for stage 2. Gains are relative changes in loss (negative is better): curvature gain compares the best of c∈{0.2,0.3} with c=10−6 during stage 2; schedule gain compares the best two-stage run with the best single-curvature run.
Figure 1: Stress along the iterations when a single curvature c is used for the whole optimization, starting from the Hydra initialization. Left: Fabaceae ( s=21.8 ); right: Falconiformes ( s=13.2 ). Graphs with larger radial span s are more sensitive to the pre-conditioner curvature.
Figure 2: Stress along the iterations when the two-phase scheme is used (first c→0 and then the other curvatures), starting from the Hydra initialization. Left: Caterpillar; right: Marsupialia. Even in real graphs the gain of a second phase that uses an intermediate value of c is significant.
Hyperbolic embeddings provide compact geometric representations of complex networks in hyperbolic spaces, but systematic comparisons of methods developed in machine learning, network science, and algorithmics remain rare. We benchmark 13 unsupervised hyperbolic graph embedders under a unified protocol for link prediction and topology reconstruction on synthetic and empirical networks. The protocol captures both missing-link recovery and the preservation of local and global network structure. Maximum-likelihood and representation-learning-based approaches, including hybrid variants, achieve the strongest overall performance, although no method dominates across all tasks and structural regimes. Performance is more strongly associated with embedding paradigm than with disciplinary origin. We identify the network regimes in which different paradigms succeed or fail and provide practical guidance for method selection in downstream applications.
Robert Jankowski, Maksim Kitsak, Dorota Celińska-Kopczyńska
TU Delft · Indiana University · Institute of Informatics, University of Warsaw
Hyperbolic geometry has emerged as an effective latent space for representing complex networks, owing to its ability to capture hierarchical organization and heterogeneous connectivity patterns using low-dimensional embeddings. As a result, numerous hyperbolic graph representation learning methods have been proposed in recent years. However, their practical adoption and systematic comparison remain challenging, as implementations are fragmented and shared tools for reproducible and fair evaluation are lacking. In this work, we introduce a unified open-source framework for hyperbolic graph representation learning that integrates several widely used embedding methods under a common optimization interface. The novel framework enables consistent training, visualization, and evaluation of hyperbolic embeddings, and interfaces seamlessly with standard network analysis tools. Leveraging this unified setup, we conduct an experimental study of hyperbolic embedding methods on real-world networks, focusing on two canonical downstream tasks: link prediction and node classification. Beyond predictive accuracy, the study offers practical insights into the strengths and limitations of existing approaches, thereby facilitating informed method selection and fostering reproducible research in hyperbolic graph representation learning.
Facultad de Ingenier´ıa, Universidad de la Rep´ublica, Uruguay · Centro Interdisciplinario en Ciencia de Datos y Aprendizaje Autom´atico (CICADA), Universidad de la Rep´ublica, Uruguay
Biomedical knowledge graphs combine ontology-derived hierarchies with transversal associations among heterogeneous entities such as phenotypes, diseases, genes, proteins, and patients. This hybrid structure raises the question of whether hyperbolic embeddings, which naturally capture tree-like organization, remain useful beyond purely hierarchical graphs. We present a preliminary study of hyperbolic graph representation learning for Mendelian-disease differential diagnosis on a patient-integrated biomedical graph. Experiments on isolated ontology subgraphs show that hyperbolic models achieve strong performance in substantially lower dimensions than Euclidean baselines. We then evaluate the models on a link-prediction task that ranks candidate diseases for each patient. Results suggest that hyperbolic embeddings can exploit biomedical hierarchical structure while supporting diagnostic reasoning over heterogeneous patient-level graphs.
Pietro Miotto, Lucia Mellini, Tommaso Marzi +5
Department of Computer Science, Università degli Studi di Milano, Milan, Italy · Università della Svizzera italiana (USI), Lugano, Switzerland · Politecnico di Milano, Milan, Italy +2