Tackling Over-smoothing on Hypergraphs: A Ricci Flow-guided Neural Diffusion Approach

Mengyao Zhou; Zhiheng Zhou; Xiao Han; Xingqin Qi; Guanghui Wang; Guiying Yan

cs.LGMar 16, 2026

Tackling Over-smoothing on Hypergraphs: A Ricci Flow-guided Neural Diffusion Approach

Authors: Mengyao Zhou, Zhiheng Zhou, Xiao Han, Xingqin Qi, Guanghui Wang, Guiying Yan

Organizations: the Academy of Mathematics and Systems Science, Chinese Academy of Sciences and also with the University of Chinese Academy of Sciences, Beijing 100190, China · the School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China · School of Artificial Intelligence, Beihang University, Beijing 100191, China · the School of Mathematics, Shandong University, Jinan, Shandong 250100, China

Abstract

Hypergraph neural networks (HGNNs) have demonstrated strong capabilities in modeling complex higher-order relationships. However, existing HGNNs often suffer from over-smoothing as the number of layers increases and lack effective control over message passing among nodes. Inspired by the theory of Ricci flow in differential geometry, we theoretically establish that introducing discrete Ricci flow into hypergraph structures can effectively regulate node feature evolution and thereby alleviate over-smoothing. Building on this insight, we propose Ricci Flow-guided Hypergraph Neural Diffusion(RFHND), a novel message passing paradigm for hypergraphs guided by discrete Ricci flow. Specifically, RFHND is based on a PDE system that describes the continuous evolution of node features on hypergraphs and adaptively regulates the rate of information diffusion at the geometric level, preventing feature homogenization and producing high-quality node representations. Experimental results show that RFHND significantly outperforms existing methods across multiple benchmark datasets and demonstrates strong robustness, while also effectively mitigating over-smoothing.

Figures & tables

Fig. 1 : Ricci flow-guided evolution. (a) Evolution of the manifold geometry under Ricci flow. (b) Evolution of hyperedge weight guided by Ricci flow. (c) Evolution of node feature distributions driven by the updated hyperedge weights.

Metric	Cora	Citeseer	Pubmed	Cora-CA	DBLP-CA	Zoo	NTU2012	ModelNet40	Walmart	Senate	House
$\|V\|$	$2708$	$3312$	$19\,177$	$2708$	$41\,302$	$101$	$2012$	$12\,311$	$88\,860$	$282$	$1290$
$\|E\|$	$1579$	$1079$	$7963$	$1072$	$22\,363$	$43$	$2012$	$12\,311$	$69\,906$	$315$	$340$
#features	$1433$	$3703$	$500$	$1433$	$1425$	$16$	$100$	$100$	$100$	$100$	$100$
#classes	$7$	$6$	$3$	$7$	$6$	$7$	$67$	$40$	$11$	$2$	$2$

TABLE I : Dataset Statistics Summary

Models	Cora	Citeseer	Pubmed	Cora-CA	DBLP-CA	Rank $\downarrow$
HNHN	76.36 $\pm$ 1.92	72.64 $\pm$ 1.57	86.90 $\pm$ 0.30	77.19 $\pm$ 1.49	86.78 $\pm$ 0.29	12
HGNN	79.39 $\pm$ 1.36	72.45 $\pm$ 1.16	86.44 $\pm$ 0.44	82.64 $\pm$ 1.65	91.03 $\pm$ 0.20	8
HCHA	79.14 $\pm$ 1.02	72.42 $\pm$ 1.42	86.41 $\pm$ 0.36	82.55 $\pm$ 0.97	90.92 $\pm$ 0.22	11
HyperGCN	78.45 $\pm$ 1.26	71.28 $\pm$ 0.82	82.84 $\pm$ 8.67	79.48 $\pm$ 2.08	89.38 $\pm$ 0.25	13
UniGCNII	78.81 $\pm$ 1.05	73.05 $\pm$ 2.21	88.25 $\pm$ 0.40	83.60 $\pm$ 1.14	91.69 $\pm$ 0.19	6
AllSetTransformer	78.59 $\pm$ 1.47	73.08 $\pm$ 1.20	88.72 $\pm$ 0.37	83.63 $\pm$ 1.47	91.53 $\pm$ 0.23	5

TABLE II : Performance comparison on academic hypergraph datasets (Mean accuracy (%) ± standard deviation), with the best results in bold, second-best and third-best are marked with an underline.

Models	Zoo	NTU2012	ModelNet40	Walmart	Senate	House	Rank $\downarrow$
HNHN	93.59 $\pm$ 5.88	89.11 $\pm$ 1.44	97.84 $\pm$ 1.25	81.35 $\pm$ 0.61	50.93 $\pm$ 6.33	67.80 $\pm$ 2.59	6
HGNN	92.50 $\pm$ 4.58	87.72 $\pm$ 1.35	95.44 $\pm$ 0.33	80.33 $\pm$ 0.42	48.59 $\pm$ 4.52	61.39 $\pm$ 2.96	11
HCHA	93.65 $\pm$ 6.15	87.48 $\pm$ 1.87	94.48 $\pm$ 0.28	80.33 $\pm$ 0.80	48.62 $\pm$ 4.41	61.36 $\pm$ 2.53	10
HyperGCN	N/A	56.36 $\pm$ 4.86	75.89 $\pm$ 5.26	81.05 $\pm$ 0.59	42.45 $\pm$ 3.67	48.32 $\pm$ 2.93	13
UniGCNII	93.65 $\pm$ 4.37	89.30 $\pm$ 1.33	98.07 $\pm$ 0.23	81.12 $\pm$ 0.67	49.30 $\pm$ 4.25	61.70 $\pm$ 3.37	7
AllSetTransformer	97.50 $\pm$ 3.59	88.69 $\pm$ 1.24	98.20 $\pm$ 0.20	81.38 $\pm$ 0.58	51.83 $\pm$ 5.22	69.33 $\pm$ 2.20	4

TABLE III : Performance comparison on real-world hypergraph datasets (Mean accuracy (%) ± standard deviation), with the best results in bold, second-best and third-best are marked with an underline.

Model	$\alpha=1$	$\alpha=2$	$\alpha=3$	$\alpha=4$	$\alpha=5$	$\alpha=6$	$\alpha=7$
HGNN	92.58 $\pm$ 0.57	87.40 $\pm$ 0.89	82.32 $\pm$ 0.84	77.18 $\pm$ 1.08	70.06 $\pm$ 1.35	69.98 $\pm$ 0.85	68.41 $\pm$ 1.39
HyperGCN	83.40 $\pm$ 1.67	78.13 $\pm$ 1.57	77.70 $\pm$ 1.01	74.90 $\pm$ 1.31	50.15 $\pm$ 1.94	52.11 $\pm$ 0.75	49.40 $\pm$ 1.63
UniGCN	90.42 $\pm$ 1.04	82.12 $\pm$ 0.67	80.04 $\pm$ 0.98	76.86 $\pm$ 1.10	71.50 $\pm$ 1.50	52.30 $\pm$ 1.18	49.20 $\pm$ 0.74
EDGNN	93.88 $\pm$ 0.67	89.89 $\pm$ 0.59	83.52 $\pm$ 0.85	76.65 $\pm$ 1.21	72.18 $\pm$ 1.36	52.98 $\pm$ 0.78	49.65 $\pm$ 1.16
RFHND	95.63 $\pm$ 0.62	90.17 $\pm$ 0.67	84.21 $\pm$ 0.81	77.48 $\pm$ 0.98	74.26 $\pm$ 1.57	73.51 $\pm$ 1.07	73.46 $\pm$ 1.35

TABLE IV : Model Performance On Synthetic Hypergraphs with Controlled Heterophily

\alpha

Panel of Figure 6 — Fig. 2 : (a) shows the accuracy(%) with various depths on Cora. (b) shows the accuracy(%) with various depths on Cora-CA. (c) shows the accuracy(%) with various depths on Citeseer. (d) shows the dirichlet energy with various depths on Cora. (e) shows the dirichlet energy with various depths on Cora-CA. (f) shows the dirichlet energy with various depths on Citeseer.

Variants	Zoo	NTU2012	ModelNet40
w/o COS	97.16 $\pm$ 2.90	90.04 $\pm$ 1.09	96.78 $\pm$ 0.27
w/o HyperNet	96.89 $\pm$ 2.87	91.52 $\pm$ 1.12	97.42 $\pm$ 0.23
w/o C and H	96.43 $\pm$ 2.90	89.79 $\pm$ 1.30	96.86 $\pm$ 0.27
RFHND (Full Model)	97.70 $\pm$ 2.14	93.36 $\pm$ 1.18	98.56 $\pm$ 0.16

TABLE V : Ablation Study.

Panel of Figure 8 — Fig. 3 : (a) shows the performance on Citeseer under feature gaussian noise. (b) shows the performance on Citeseer under feature uniform noise. (c) shows the performance on Citeseer under feature mask noise. (d) shows the performance on Cora-CA under structure noise.

Panel of Figure 9 — Fig. 4 : (a) shows RFHND accuracy on NTU2012 and ModelNet40 with various hidden dimensions. (b) shows RFHND accuracy on NTU2012 and ModelNet40 with various $\tau$ .

Panel of Figure 10 — Fig. 5 : (a)(b)(c) show the visualization of Pubmed node features at $t=0$ , $t=1$ and $t=2$ . (d)(e)(f) show the visualization of DBLP node features at $t=0$ , $t=1$ and $t=2$ .

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From Diffusion to Reaction-Diffusion: A Dynamical-Systems View of Oversmoothing in Hypergraph Neural Networks

Zhiheng Zhou, Mengyao Zhou, Yancheng Chen, Dengyi Zhao+2

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Hypergraph Neural Networks Hypergraph Learning Graph Diffusion

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Higher-order couplings enhance the expressive power of hypergraph neural networks (HGNNs), but they also intensify representation collapse in deep propagation due to strong multi-way feature mixing. This work investigates hypergraph oversmoothing from a dynamical-systems perspective and develops a reaction--diffusion framework for depth-resistant hypergraph learning. By defining hypergraph gradient and divergence operators, we interpret message passing as an incidence-level diffusion process. The analysis of pure diffusion shows that its continuous semiflow exponentially contracts the null-mode-free component of node representations and drives the Dirichlet energy to zero, revealing hypergraph oversmoothing as an intrinsic transverse-energy dissipation phenomenon. Motivated by this analysis, we propose Hypergraph Neural Reaction--Diffusion (HNRD), which introduces a reaction mechanism acting on the transverse component to compensate diffusion-induced dissipation and stabilize discriminative variations. We establish global well-posedness of the proposed dynamics and prove that the null-mode-free Dirichlet energy remains bounded away from zero. A forward-Euler discretization provides a practical HNRD layer with a stability condition for deep propagation. Experiments on benchmark and synthetic heterophilic hypergraphs demonstrate that HNRD consistently improves over representative hypergraph baselines. Depth, robustness, and efficiency analyses further show that HNRD preserves stable performance and nonzero Dirichlet energy under deep propagation and perturbations. These results provide a principled dynamical framework for designing deep hypergraph architectures that maintain higher-order expressiveness without representation collapse.

Abstract

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