Organizations: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China · Beijing Key Lab of Intelligent Telecommunications Software and Multimedia, Beijing University of Posts and Telecommunications, Beijing 100876, China
Hyperbolic graph representation learning has emerged as a powerful paradigm for modeling hierarchical data. Unlike Euclidean space, which exhibits polynomial volume growth, hyperbolic space grows exponentially with radius, naturally accommodating tree-like structures without the geometric distortion known as Euclidean crowding. Despite strong empirical performance, the theoretical mechanisms underlying hyperbolic embeddings remain poorly understood. In this work, we provide a geometric quantifcation research for Euclidean crowding, a generalization-theoretic explanation for the superiority of hyperbolic space on hierarchical data, and further develop an adaptive Euclidean-hyperbolic fusion learning framework toward bridging this gap. Specifically, we first establish a geometric quantification of Euclidean crowding by analyzing layer-wise minimum separation in node embeddings, proving that fixed-dimensional Euclidean embeddings of tree-like graphs inevitably exhibit layer-wise crowding that leads to weaker margin-based generalization bounds. Second, we provide a generalization-bound for the superiority of hyperbolic geometry on hierarchical data. Third, we propose EH-Fusion, a geometry-adaptive framework that lifts Euclidean GNN representations into hyperbolic space via exponential mapping and adaptively fuses both geometries based on node-level neighborhood expansion patterns. Extensive experiments validate our theoretical insights and demonstrate the effectiveness of EH-Fusion on benchmark datasets.
Figures & tables
Fig. 1: Euclidean crowding and its connection to generalization. In hierarchical data, the exponential growth of nodes with depth compresses pairwise distances in Euclidean space, reducing the geometric separation Δ and the induced functional margin γ . This margin reduction leads to a looser margin-based generalization bound, whereas hyperbolic space can better preserve hierarchical distances with lower distortion.
Fig. 2: The overall framework of EH-Fusion: Upper part: A shared Euclidean graph encoder yields node representations, which are then mapped onto multiple geometric manifolds for metric learning. Lower part: High-order degree growth information is extracted to capture local geometric characteristics and guide the adaptive aggregation of manifold-specific prediction probabilities.
Model
Disease ( δ=0 )
Airport ( δ=1 )
Citeseer ( δ=5 )
Cora ( δ=11 )
8
16
32
64
8
16
32
64
8
16
32
64
8
16
32
64
poinEmb
70.5
71.8
75.2
78.2
96.2
96.5
96.7
96.7
81.4
80.7
80.3
80.9
86.1
86.4
87.8
87.8
HNN
94.2
94.0
94.2
94.1
93.8
94.1
94.3
94.3
95.5
96.2
96.1
96.3
92.6
92.8
93.7
93.5
GCN
85.8
82.4
85.5
87.1
93.1
94.1
93.9
95.1
92.2
93.7
95.2
95.8
90.6
92.6
92.9
93.8
GAT
81.9
82.8
82.2
89.9
94.3
93.3
95.3
94.1
94.8
95.8
96.6
97.2
92.3
93.8
94.5
95.2
HGCN
89.3
89.3
89.9
89.8
94.4
95.0
95.3
96.2
95.4
96.1
96.4
96.6
93.8
94.6
95.0
95.0
TABLE I: AUC(%) for link prediction task.
Ele-Fashion
Model
Image Encoder
dim=8
dim=16
dim=32
dim=64
GCN
ImageBind
79.45
82.43
82.46
83.52
CLIP
78.73
79.53
80.91
80.57
DINOv2
79.50
80.13
80.07
81.84
ViT
78.35
80.46
79.80
80.81
GCN2L
ImageBind
80.34
83.08
83.30
84.29
TABLE II: ACC(%) performance comparison for image-attributed graph.
Disease
Airport
Citeseer
Cora
HGCN
89.9
96.2
96.6
95.0
GCN
87.1
95.1
95.8
93.8
+Exp2P
94.6
96.9
93.9
93.9
+Exp2L
94.9
97.2
92.0
92.4
+Exp+EH-Fusion
95.4
97.3
95.9
94.1
FHNN
96.0
96.7
95.0
94.9
TABLE III: Ablation study of the hyperbolic mapping layer and EH-Fusion module.
Fig. 3: Effect of the exponential mapping layer and EH-Fusion on strongly hyperbolic datasets. A single hyperbolic mapping significantly improves performance, and adaptive fusion provides further gains across embedding dimensions.
Fig. 4: AUC comparison of GAT in Lorentz hyperbolic space and Euclidean space on datasets with varying δ -hyperbolicity levels. Smaller δ indicates stronger hyperbolicity.
Fig. 5: EH-Fusion analysis. Adaptive manifold selection mitigates geometric mismatch and stabilizes performance across datasets.
Fig. 6: Effect of bacebone model on Exponential Mapping and EH-Fusion moudule.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Fig. 7: Ramp Loss vs. 0–1 Loss. The ramp loss serves as a 1/γ -Lipschitz surrogate that upper-bounds the 0 – 1 loss. The parameter γ controls the slope of the loss, while γ0 denotes the functional margin.
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.
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
Hyperbolic geometry has emerged as a principled space for representing hierarchical graphs. However, existing hyperbolic graph neural networks typically rely on shared curvature configurations and feature-driven attention, failing to explicitly exploit local hierarchical topological patterns. To bridge this gap, we introduce the Hyperbolic Sombor Index (HSO) as a lightweight structural prior for capturing hierarchy-indicative degree stratification. Building on this, we propose HSO-GAT, a topology-adaptive hyperbolic graph attention network that unifies geometric adaptation and message propagation. Specifically, it comprises two complementary modules: HSO-Guided Local Curvature Adaptation, which performs adaptive node-wise geometric scaling from aggregated node-level HSO signals, and HSO-Gated Hyperbolic Graph Attention, which enables structure-aware message passing through feature-conditioned gating. Theoretically, we establish the monotonic sensitivity of edge-level HSO to degree imbalance and analyze the validity and radial scaling properties of node-adaptive hyperbolic mappings. Extensive experiments on eight benchmark datasets demonstrate that HSO-GAT consistently achieves state-of-the-art performance in both node classification and link prediction tasks.