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

Haifang Cao; Yu Wang; Timing Li; Xinjie Yao; Pengfei Zhu

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

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.

Method	Cora	CiteSeer	PubMed	Photo	WikiCS	Airport
GCN ( Kipf, 2016 )	$82.02_{\pm\text{\scriptsize 0.45}}$	$70.24_{\pm\text{\scriptsize 0.85}}$	$78.45_{\pm\text{\scriptsize 0.36}}$	$92.84_{\pm\text{\scriptsize 0.25}}$	$74.29_{\pm\text{\scriptsize 1.06}}$	$83.19_{\pm\text{\scriptsize 1.76}}$
GAT ( Veličković et al., 2017 )	$82.72_{\pm\text{\scriptsize 0.69}}$	$71.45_{\pm\text{\scriptsize 1.15}}$	$77.23_{\pm\text{\scriptsize 0.51}}$	$92.70_{\pm\text{\scriptsize 0.55}}$	$76.55_{\pm\text{\scriptsize 0.88}}$	$86.10_{\pm\text{\scriptsize 1.39}}$
SGC ( Wu et al., 2019 )	$81.91_{\pm\text{\scriptsize 0.63}}$	$71.95_{\pm\text{\scriptsize 0.55}}$	$78.60_{\pm\text{\scriptsize 0.24}}$	$91.96_{\pm\text{\scriptsize 1.23}}$	$75.54_{\pm\text{\scriptsize 1.25}}$	$84.07_{\pm\text{\scriptsize 1.91}}$
GraphCON ( Rusch et al., 2022 )	$82.54_{\pm\text{\scriptsize 1.25}}$	$71.29_{\pm\text{\scriptsize 0.96}}$	$78.66_{\pm\text{\scriptsize 1.23}}$	$91.94_{\pm\text{\scriptsize 1.12}}$	$76.64_{\pm\text{\scriptsize 0.59}}$	$79.25_{\pm\text{\scriptsize 2.51}}$
NodeFormer ( Wu et al., 2022 )	$82.34_{\pm\text{\scriptsize 0.85}}$	$72.47_{\pm\text{\scriptsize 1.27}}$	$79.13_{\pm\text{\scriptsize 1.10}}$	$93.46_{\pm\text{\scriptsize 0.62}}$	$73.27_{\pm\text{\scriptsize 0.87}}$	$82.66_{\pm\text{\scriptsize 0.69}}$
ACMP ( Wang et al., 2022 )	$82.65_{\pm\text{\scriptsize 0.78}}$	$73.01_{\pm\text{\scriptsize 1.14}}$	$79.12_{\pm\text{\scriptsize 0.54}}$	$92.24_{\pm\text{\scriptsize 1.71}}$	$77.12_{\pm\text{\scriptsize 0.58}}$	$88.96_{\pm\text{\scriptsize 1.74}}$

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.

	Experts			Losses		Datasets
Variant	Euc.	Hyp.	Sph.	$\mathcal{L}_{\text{align}}$	$\mathcal{L}_{\text{contr}}$	Cora	CiteSeer	PubMed	Photo	WikiCS	Airport
(a)	✓					82.02	70.24	78.45	92.84	74.29	83.19
(b)		✓				79.71	69.41	75.56	93.38	77.62	89.25
(c)			✓			76.68	70.57	76.85	91.61	75.85	85.63
(d)	✓	✓		✓	✓	82.97	71.12	78.73	93.88	78.13	91.97
(e)	✓		✓	✓	✓	82.75	71.37	78.84	93.42	75.76	87.28

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

\theta

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	#Nodes	#Edges	#Classes	#Features
Cora	2,708	5,429	7	1,433
CiteSeer	3,327	4,732	6	3,703
PubMed	19,717	44,338	3	500
Photo	7,650	119,081	8	745
WikiCS	11,701	216,123	10	300
Airport	3,188	18,631	4	4

Appendix

Table 3: Statistics of the experimental datasets.

Figure 5: Impact of negative sample count

K

Ollivier-Ricci Curvature

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Aug 6, 2026J/K move · Enter open · S save

Dynamic Graph Prompting via Topology-Routed Mixed-Curvature Experts

Quanxin Wang, Xuanting Xie, Bingheng Li, Xingtong Yu+3

University of Electronic Science and Technology of China · 2Michigan State University · 3Singapore Management University+2

Link Prediction Few-Shot Learning Geometric Representation Learning Dynamic Graph Learning Prompt Engineering

Open paper AskPDF arXiv

Dynamic graph prompting freezes a pre-trained temporal backbone and adapts it to label-scarce downstream tasks using lightweight prompts. However, existing methods operate within a single, fixed embedding space. In this work, we reveal that temporal shifts in local clustering and degree heterogeneity actively reorganize the edge curvature spectrum---indicating that the optimal representation geometry dynamically evolves with local topology over time. We formalize this unaddressed mismatch as geometry under-adaptation. To overcome this limitation, we propose CurvPrompt, a topology-routed geometry prompting framework for dynamic graphs. Instead of relying on a single space, CurvPrompt maintains a bank of curvature-diverse Riemannian experts, each paired with a learnable prompt. A topology-aware gate dynamically routes each node--time instance to a sparse subset of experts, constructing a personalized mixed-curvature representation. To ensure parameter efficiency and training stability under extreme label scarcity, CurvPrompt employs soft routing during pre-training to build a continuous topology--geometry mapping, and transitions to hard Top-K routing with uniform weights during downstream adaptation. Extensive experiments across four benchmark datasets show that CurvPrompt significantly advances few-shot link prediction while delivering strong, consistent performance on node classification tasks, validating the necessity of geometry-adaptive prompting.

Abstract

Ollivier-Ricci Curvature

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