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

Haifang Cao; Boan Tao; Xiyuan Gao; Timing Li; Yu Wang; Pengfei Zhu

cs.LGSep 26, 2026

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

Authors: Haifang Cao, Boan Tao, Xiyuan Gao, Timing Li, Yu Wang, Pengfei Zhu

Organizations: Tianjin University Tainjin, China

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 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.

Figures & tables

Figure 1. Topological sensitivity of the edge-level Hyperbolic Sombor Index (HSO). (Left) Three representative graph structures: (a) Tree-Like, (b) Grid-Like, and (c) Cycle-Like. (Right) The distribution of edge-level HSO values and their proportions across different structures, where labels (e.g., 3.162 / 66.67%) denote the specific HSO value and the corresponding percentage of edges. Tree-like graphs exhibit higher and broader distributions due to degree imbalances.

Figure 2. Overview of the proposed HSO-GAT framework. Edge-level HSO $b_{ij}$ captures local degree imbalance, and its neighborhood aggregation yields the node-level HSO $b_{i}$ . The HLCA module transforms $b_{i}$ into node-wise curvature $c_{i}$ to guide adaptive hyperbolic mappings while keeping representations in a shared Poincaré ball. The HSGA module performs multi-head attention in the tangent space, combining feature compatibility with a gated edge-level HSO bias. By coupling HLCA and HSGA, HSO-GAT enables topology-adaptive geometric modeling and structure-aware message propagation.

Dataset	Pubmed	Computer	Photo	WikiCS	Airport	Disease	CS	ogbn-arxiv
Nodes	19,717	13,752	7,650	11,701	3,188	1,044	18,333	169,343
Edges	44,338	245,861	119,081	216,123	18,631	1,043	81,894	1,166,243
Classes	3	10	8	10	4	2	15	40
Features	500	767	745	300	4	1,000	6,805	128
$\delta$	3.5	1.5	2.5	1.5	1.0	0.0	2.0	2.5

Table 1. Statistics of the experimental datasets.

Method	PubMed	Computer	Photo	WikiCS	Airport	Disease	CS	ogbn-arxiv
Method	( $\delta=3.5$ )	( $\delta=1.5$ )	( $\delta=2.5$ )	( $\delta=1.5$ )	( $\delta=1.0$ )	( $\delta=0.0$ )	( $\delta=2.0$ )	( $\delta=2.5$ )
GCN (2017)	$78.45_{\pm\text{\scriptsize 0.36}}$	$90.17_{\pm\text{\scriptsize 0.69}}$	$92.84_{\pm\text{\scriptsize 0.25}}$	$74.29_{\pm\text{\scriptsize 1.06}}$	$83.19_{\pm\text{\scriptsize 1.76}}$	$79.73_{\pm\text{\scriptsize 1.88}}$	$93.10_{\pm\text{\scriptsize 0.14}}$	$68.36_{\pm\text{\scriptsize 0.28}}$
GraphSAGE (2017)	$76.88_{\pm\text{\scriptsize 0.56}}$	$89.38_{\pm\text{\scriptsize 0.98}}$	$91.27_{\pm\text{\scriptsize 2.44}}$	$74.22_{\pm\text{\scriptsize 1.36}}$	$88.55_{\pm\text{\scriptsize 0.84}}$	$75.29_{\pm\text{\scriptsize 2.76}}$	$92.55_{\pm\text{\scriptsize 0.19}}$	$66.86_{\pm\text{\scriptsize 0.50}}$
GAT (2018)	$77.23_{\pm\text{\scriptsize 0.51}}$	$91.16_{\pm\text{\scriptsize 0.55}}$	$92.70_{\pm\text{\scriptsize 0.51}}$	$76.55_{\pm\text{\scriptsize 0.88}}$	$86.10_{\pm\text{\scriptsize 1.39}}$	$80.25_{\pm\text{\scriptsize 0.54}}$	$93.12_{\pm\text{\scriptsize 0.15}}$	$68.05_{\pm\text{\scriptsize 0.30}}$
HGNN (2019)	$76.94_{\pm\text{\scriptsize 1.12}}$	$91.35_{\pm\text{\scriptsize 0.63}}$	$91.82_{\pm\text{\scriptsize 0.35}}$	$74.89_{\pm\text{\scriptsize 0.42}}$	$84.87_{\pm\text{\scriptsize 2.09}}$	$82.16_{\pm\text{\scriptsize 1.41}}$	$94.63_{\pm\text{\scriptsize 0.09}}$	$67.34_{\pm\text{\scriptsize 0.40}}$
HGCN (2019)	$77.03_{\pm\text{\scriptsize 0.49}}$	$90.58_{\pm\text{\scriptsize 0.21}}$	$93.19_{\pm\text{\scriptsize 0.29}}$	$76.67_{\pm\text{\scriptsize 0.44}}$	$88.82_{\pm\text{\scriptsize 1.66}}$	$89.36_{\pm\text{\scriptsize 1.02}}$	$94.24_{\pm\text{\scriptsize 0.17}}$	$65.32_{\pm\text{\scriptsize 0.25}}$

Table 2. Node classification results. We report accuracy (

\%\pm

standard deviation) on PubMed, Computer, Photo, WikiCS, CS and ogbn-arxiv, and Macro-F1 (

\%\pm

standard deviation) on Airport and Disease. All scores are averaged over ten runs. The best result in each column is in bold and the runner-up is underlined. OOM denotes out of memory.

Method	PubMed	Computer	Photo	WikiCS	Airport	Disease	CS	ogbn-arxiv
Method	( $\delta=3.5$ )	( $\delta=1.5$ )	( $\delta=2.5$ )	( $\delta=1.5$ )	( $\delta=1.0$ )	( $\delta=0.0$ )	( $\delta=2.0$ )	( $\delta=2.5$ )
GCN (2017)	$93.59_{\pm\text{\scriptsize 0.07}}$	$93.38_{\pm\text{\scriptsize 0.21}}$	$94.60_{\pm\text{\scriptsize 0.37}}$	$94.19_{\pm\text{\scriptsize 0.04}}$	$91.99_{\pm\text{\scriptsize 0.20}}$	$72.20_{\pm\text{\scriptsize 0.81}}$	$94.67_{\pm\text{\scriptsize 0.06}}$	$93.27_{\pm\text{\scriptsize 0.06}}$
GraphSAGE (2017)	$94.18_{\pm\text{\scriptsize 0.07}}$	$93.60_{\pm\text{\scriptsize 0.13}}$	$93.27_{\pm\text{\scriptsize 0.44}}$	$94.66_{\pm\text{\scriptsize 0.57}}$	$94.62_{\pm\text{\scriptsize 0.29}}$	$70.76_{\pm\text{\scriptsize 1.38}}$	$93.50_{\pm\text{\scriptsize 0.03}}$	$95.35_{\pm\text{\scriptsize 0.07}}$
GAT (2018)	$92.46_{\pm\text{\scriptsize 0.18}}$	$94.66_{\pm\text{\scriptsize 0.24}}$	$92.05_{\pm\text{\scriptsize 1.22}}$	$95.03_{\pm\text{\scriptsize 0.22}}$	$94.38_{\pm\text{\scriptsize 0.21}}$	$79.18_{\pm\text{\scriptsize 1.36}}$	$94.34_{\pm\text{\scriptsize 0.08}}$	$91.25_{\pm\text{\scriptsize 0.44}}$
HGNN (2019)	$93.47_{\pm\text{\scriptsize 0.28}}$	$96.93_{\pm\text{\scriptsize 0.27}}$	$97.70_{\pm\text{\scriptsize 0.14}}$	$96.97_{\pm\text{\scriptsize 0.10}}$	$96.78_{\pm\text{\scriptsize 0.28}}$	$75.08_{\pm\text{\scriptsize 1.43}}$	$96.21_{\pm\text{\scriptsize 0.07}}$	$96.20_{\pm\text{\scriptsize 0.18}}$
HGCN (2019)	$95.51_{\pm\text{\scriptsize 0.03}}$	$97.72_{\pm\text{\scriptsize 0.10}}$	$97.39_{\pm\text{\scriptsize 0.02}}$	$98.16_{\pm\text{\scriptsize 0.05}}$	$97.95_{\pm\text{\scriptsize 0.11}}$	$89.59_{\pm\text{\scriptsize 0.88}}$	$97.08_{\pm\text{\scriptsize 0.07}}$	$97.02_{\pm\text{\scriptsize 0.05}}$

Table 3. Link prediction results. ROC-AUC (

\%\pm

standard deviation) for all datasets. All scores are averaged over ten runs. The best result in each column is in bold and the runner-up is underlined. OOM denotes out of memory.

Method	PubMed		Disease
Method	NC	LP	NC	LP
Base (HAT)	75.56	96.55	90.43	88.91
w/o HLCA	78.54	97.96	96.01	97.39
w/o HSGA	77.66	98.27	94.89	97.64
w/o $\mathcal{L}_{\mathrm{curv}}$	78.83	98.39	96.57	98.11
w/ Deg. Ratio	79.38	98.35	97.37	97.96

Table 4. Ablation study on PubMed and Disease datasets. w/ Deg. Ratio replaces the HSO prior with the degree ratio of incident edges. The best result in each column is in bold.

Figure 3. Node classification accuracy (%) of HSO-GAT across different node-level HSO levels. Nodes are grouped from Level 1 to Level 4 according to increasing HSO values.

Graph Neural Networks

Graph Neural Networks learn from graph-structured data by combining information across connected nodes, edges, and larger structures.

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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.

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