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This paper introduces HSO-GAT, a novel graph neural network architecture that leverages hyperbolic geometry and the Hyperbolic Sombor Index to better capture hierarchical topological patterns in graphs. By adapting local curvature and gating message propagation based on structural priors, the model achieves state-of-the-art results in node classification and link prediction across eight benchmark datasets. The work advances the theoretical understanding of how geometric scaling properties can be unified with feature-driven attention for improved representation learning.
Topology-Adaptive Hyperbolic Graph Attention Networks Guided by the Hyperbolic Sombor Index
arXiv:2609.32275v1 Announce Type: new Abstract: 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 \textbf{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.
Abstract: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 \textbf{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.
Machine Learning (cs.LG)
arXiv:2609.32275 [cs.LG]
(or arXiv:2609.32275v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.32275 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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