cs.LGMar 20, 2026

Geometric Mixture-of-Experts with Curvature-Guided Adaptive Routing for Graph Representation Learning

Authors: Haifang Cao, Yu Wang, Timing Li, Xinjie Yao, Pengfei Zhu

Organizations: College of Intelligence and Computing, Tianjin University, Tianjin, China

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Abstract

Graph-structured data typically exhibits complex topological heterogeneity, making it difficult to model accurately within a single Riemannian manifold. While emerging mixed-curvature methods attempt to capture such diversity, they often rely on implicit, task-driven routing that lacks fundamental geometric grounding. To address this challenge, we propose a Geometric Mixture-of-Experts framework (GeoMoE) that adaptively fuses node representations across diverse Riemannian spaces to better accommodate multi-scale topological structures. At its core, GeoMoE leverages Ollivier-Ricci Curvature (ORC) as an intrinsic geometric prior to orchestrate the collaboration of specialized experts. Specifically, we design a graph-aware gating network that assigns node-specific fusion weights, regularized by a curvature-guided alignment loss to ensure interpretable and geometry-consistent routing. Additionally, we introduce a curvature-aware contrastive objective that promotes geometric discriminability by constructing positive and negative pairs according to curvature consistency. Extensive experiments on six benchmark datasets demonstrate that GeoMoE outperforms state-of-the-art baselines across diverse graph types.

Figures & tables

Figure 1: (Left) Topological visualization of real-world graphs (IV), combining grid (I), hierarchy (II), and cycle (III) structures. (Right) Performance of single-geometry models vs. our GeoMoE across these structures.
Figure 2: An overview of GeoMoE. We first encode the input graph via Euclidean, hyperbolic, and spherical geometric experts, then dynamically fuse their outputs through a curvature-guided gating alignment mechanism. The curvature-aware contrastive learning objective further refines the fused representations to enhance geometric discriminability.
MethodCoraCiteSeerPubMedPhotoWikiCSAirport
GCN ( Kipf, 2016 )82.02±0.4582.02_{\pm\text{\scriptsize 0.45}}70.24±0.8570.24_{\pm\text{\scriptsize 0.85}}78.45±0.3678.45_{\pm\text{\scriptsize 0.36}}92.84±0.2592.84_{\pm\text{\scriptsize 0.25}}74.29±1.0674.29_{\pm\text{\scriptsize 1.06}}83.19±1.7683.19_{\pm\text{\scriptsize 1.76}}
GAT ( Veličković et al., 2017 )82.72±0.6982.72_{\pm\text{\scriptsize 0.69}}71.45±1.1571.45_{\pm\text{\scriptsize 1.15}}77.23±0.5177.23_{\pm\text{\scriptsize 0.51}}92.70±0.5592.70_{\pm\text{\scriptsize 0.55}}76.55±0.8876.55_{\pm\text{\scriptsize 0.88}}86.10±1.3986.10_{\pm\text{\scriptsize 1.39}}
SGC ( Wu et al., 2019 )81.91±0.6381.91_{\pm\text{\scriptsize 0.63}}71.95±0.5571.95_{\pm\text{\scriptsize 0.55}}78.60±0.2478.60_{\pm\text{\scriptsize 0.24}}91.96±1.2391.96_{\pm\text{\scriptsize 1.23}}75.54±1.2575.54_{\pm\text{\scriptsize 1.25}}84.07±1.9184.07_{\pm\text{\scriptsize 1.91}}
GraphCON ( Rusch et al., 2022 )82.54±1.2582.54_{\pm\text{\scriptsize 1.25}}71.29±0.9671.29_{\pm\text{\scriptsize 0.96}}78.66±1.2378.66_{\pm\text{\scriptsize 1.23}}91.94±1.1291.94_{\pm\text{\scriptsize 1.12}}76.64±0.5976.64_{\pm\text{\scriptsize 0.59}}79.25±2.5179.25_{\pm\text{\scriptsize 2.51}}
NodeFormer ( Wu et al., 2022 )82.34±0.8582.34_{\pm\text{\scriptsize 0.85}}72.47±1.2772.47_{\pm\text{\scriptsize 1.27}}79.13±1.1079.13_{\pm\text{\scriptsize 1.10}}93.46±0.6293.46_{\pm\text{\scriptsize 0.62}}73.27±0.8773.27_{\pm\text{\scriptsize 0.87}}82.66±0.6982.66_{\pm\text{\scriptsize 0.69}}
ACMP ( Wang et al., 2022 )82.65±0.7882.65_{\pm\text{\scriptsize 0.78}}73.01±1.1473.01_{\pm\text{\scriptsize 1.14}}79.12±0.5479.12_{\pm\text{\scriptsize 0.54}}92.24±1.7192.24_{\pm\text{\scriptsize 1.71}}77.12±0.5877.12_{\pm\text{\scriptsize 0.58}}88.96±1.7488.96_{\pm\text{\scriptsize 1.74}}
Table 1: Accuracy ( %±\%\pm standard deviation) for Cora, CiteSeer, PubMed, Photo and WikiCS and F1 score ( %±\%\pm standard deviation) for Airport for node classification. Scores are averaged over ten runs. The best results are boldfaced while the runner-ups are underlined.
 ExpertsLossesDatasets
VariantEuc.Hyp.Sph.Lalign\mathcal{L}_{\text{align}}Lcontr\mathcal{L}_{\text{contr}}CoraCiteSeerPubMedPhotoWikiCSAirport
(a)✓    82.0270.2478.4592.8474.2983.19
(b) ✓   79.7169.4175.5693.3877.6289.25
(c)  ✓  76.6870.5776.8591.6175.8585.63
(d)✓✓ ✓✓82.9771.1278.7393.8878.1391.97
(e)✓ ✓✓✓82.7571.3778.8493.4275.7687.28
Table 2: Ablation study. In the table, “✓” denotes GeoMoE with the component. The best results are boldfaced.
Figure 3: Consistency between gating weights and ORC on Photo.
Figure 4: Sensitivity to curvature threshold θ\theta .
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset#Nodes#Edges#Classes#Features
Cora2,7085,42971,433
CiteSeer3,3274,73263,703
PubMed19,71744,3383500
Photo7,650119,0818745
WikiCS11,701216,12310300
Airport3,18818,63144
Appendix
Table 3: Statistics of the experimental datasets.
Figure 5: Impact of negative sample count KK .
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