Topology-Adaptive Hyperbolic Graph Attention Networks Guided by the Hyperbolic Sombor Index

2026-09-26

AI Summary

This study addresses the limitations of existing hyperbolic graph neural networks, which rely on shared curvature and lack explicit modeling of local hierarchical topology. To this end, we propose HSO-GAT, a model that pioneers the use of the hyperbolic Sombor index as a structural prior to capture hierarchical degree distribution characteristics, thereby unifying geometric adaptation with message-passing mechanisms. Specifically, the method achieves local geometric adaptation through node-level curvature scaling and introduces a structure-aware gated attention mechanism to optimize feature aggregation. Extensive experiments on eight benchmark datasets demonstrate that HSO-GAT achieves state-of-the-art performance in both node classification and link prediction tasks, validating the effectiveness of the proposed topology-guided hyperbolic learning paradigm.

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.

Problem

• Hyperbolic Graph Neural Networks
• Hierarchical Graphs
• Topological Patterns
• Curvature Adaptation
• Graph Attention

Innovation

• Hyperbolic Sombor Index
• Topology-Adaptive
• Hyperbolic Graph Attention Networks
• Local Curvature Adaptation
• Structure-Aware Message Passing