Hyperbolic Graph Neural Networks Under the Microscope: The Role of Geometry-Task Alignment

TL;DR

The paper investigates when Hyperbolic Graph Neural Networks (HGNNs) genuinely outperform Euclidean GNNs by introducing geometry–task alignment as a central criterion. It combines theoretical results on model distortion with controlled synthetic tasks and real-world benchmarks to show HGNNs excel when the downstream task requires preserving the input graph's metric structure, especially on tree-like graphs for link prediction and pairwise-distance-sensitive problems. A key finding is that hyperbolic bias is not universally beneficial; its advantage emerges only under alignment between the task's metric needs and the graph's hyperbolic geometry, with node classification showing little to no gain. The study provides practical guidance: use HGNNs for geometry-aligned LP on tree-like graphs and note the potential for node regression as a promising, underexplored setting, while emphasizing that alignment criteria should drive model choice in graph representation learning.

Abstract

Many complex networks exhibit hyperbolic structural properties, making hyperbolic space a natural candidate for representing hierarchical and tree-like graphs with low distortion. Based on this observation, Hyperbolic Graph Neural Networks (HGNNs) have been widely adopted as a principled choice for representation learning on tree-like graphs. In this work, we question this paradigm by proposing an additional condition of geometry-task alignment, i.e., whether the metric structure of the target follows that of the input graph. We theoretically and empirically demonstrate the capability of HGNNs to recover low-distortion representations on two synthetic regression problems, and show that their geometric inductive bias becomes helpful when the problem requires preserving metric structure. Additionally, we evaluate HGNNs on the tasks of link prediction and node classification by jointly analyzing predictive performance and embedding distortion, revealing that only link prediction is geometry-aligned. Overall, our findings shift the focus from only asking "Is the graph hyperbolic?" to also questioning "Is the task aligned with hyperbolic geometry?", showing that HGNNs consistently outperform Euclidean models under such alignment, while their advantage vanishes otherwise.

Figures (10)

• Figure 1: Geometry--Task Alignment. Different latent geometries induce different degrees of alignment between embeddings and the prediction task. The top row depicts a latent manifold with increasing curvature magnitude, while the bottom rows plot the predictions $\hat{y}$ against the targets $y$. The first task (T1) benefits from the non-flat latent geometry since the targets require such a geometric inductive bias. On the other hand, no benefits can be observed when solving T2 despite the input graph being the same. For graphs with hyperbolic structure, decreasing latent curvature can make a task that is poorly represented in flat space become progressively more solvable, provided that the targets are aligned with the underlying input data geometry.
• Figure 2: Test Stress Loss ($\downarrow$) as a function of the embedding dimension for the PDP task on the synthetic tree and grid graphs. Each marker denotes the mean performance over $10$ random seeds; shaded regions indicate one standard deviation.
• Figure 3: NR MAE ($\downarrow$) of HGCN vs. its fixed curvature magnitude $c$ on the tree for $d=128$. We can see a curvature–performance trade-off.
• Figure 4: Normalized Stress loss ($\downarrow$) for predicting pairwise shortest-path distances using embeddings from LP (\ref{['fig:lp_real_stress_loss_norm']}) and NC (\ref{['fig:nc_real_stress_loss_norm']}). Hyperbolic models perform best for LP at small $\delta$, while gains for NC are marginal.
• Figure 5: LP ROC AUC ($\uparrow$) (\ref{['fig:lp_noisy_disease_roc']}, \ref{['fig:lp_noisy_airport_roc']}) and Normalized Stress Loss ($\downarrow$) (\ref{['fig:lp_noisy_disease_stress_loss_norm']}, \ref{['fig:lp_noisy_airport_stress_loss_norm']}) over different noise levels added to the features of the Disease and Airport datasets. HyboNet outperforms all other methods in terms of ROC AUC and it does so by better preserving the geometry as measured by the Stress Loss.

Theorems & Definitions (16)

• Definition 3.1: Embedding distortion sarkar2011low
• Theorem 4.1
• Definition 4.2: Model distortion
• Proposition 4.3: Euclidean embeddings
• Proposition 4.4: Similarities and isometries
• Proposition 4.5: Poincaré ball embeddings
• Definition 2.1: Lorentzian scalar product
• proof
• proof
• proof