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Tackling Over-smoothing on Hypergraphs: A Ricci Flow-guided Neural Diffusion Approach
Mengyao Zhou, Zhiheng Zhou, Xiao Han, Xingqin Qi, Guanghui Wang, Guiying Yan
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Summary
The paper introduces Ricci Flow-guided Hypergraph Neural Diffusion (RFHND), a novel message-passing paradigm for hypergraph neural networks. By applying discrete Ricci flow to hypergraph structures, the method adaptively regulates node feature evolution based on local curvature, effectively mitigating the over-smoothing problem common in deep hypergraph networks while maintaining high-quality node representations.
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RFHND → mitigates → Over-smoothing
confidence 98% · Experimental results show that RFHND significantly outperforms existing methods... while also effectively mitigating over-smoothing.
RFHND → isbasedon → PDE system
confidence 95% · Specifically, RFHND is based on a PDE system that describes the continuous evolution of node features on hypergraphs
Discrete Ricci Flow → regulates → Node feature evolution
confidence 95% · introducing discrete Ricci flow into hypergraph structures can effectively regulate node feature evolution
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Abstract
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.
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- Source: https://arxiv.org/abs/2603.15696v1
- Canonical: https://arxiv.org/abs/2603.15696v1
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Tackling Over-smoothing on Hypergraphs: A Ricci Flow-guided Neural Diffusion Approach Mengyao Zhou, Zhiheng Zhou, Xiao Han, Xingqin Qi, Guanghui Wang, Guiying Yan This work was supported by the National Natural Science Foundation of China (No. 12231018).(Mengyao Zhou and Zhiheng Zhou contributed equally to this work.) (Corresponding authors: Guiying Yan.)Mengyao Zhou and Guiying Yan are with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences and also with the University of Chinese Academy of Sciences, Beijing 100190, China (e-mail: zhoumengyao@amss.ac.cn; yangy@amss.ac.cn).Zhiheng Zhou and Xingqin Qi a with the School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China (e-mail: zhouzhiheng@amss.ac.cn;qixingqin@sdu.edu.cn).Xiao Han is with School of Artificial Intelligence, Beihang University, Beijing 100191,China (e-mail: hx2210@buaa.edu.cn).Guanghui Wang is with the School of Mathematics, Shandong University, Jinan, Shandong 250100,China (e-mail: ghwang@sdu.edu.cn). 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. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. I Introduction In recent years, hypergraphs have attracted widespread attention as an important tool for modeling complex systems [1, 2]. Unlike traditional graphs, which can only represent pairwise relationships between nodes, hypergraphs can connect multiple nodes through hyperedges, naturally capturing higher-order relations [3]. This capability gives hypergraphs unique expressive power in many domains such as social networks [4, 5], recommendation systems [6, 7], and biological networks [8, 9]. To fully exploit the information from hypergraph data, researchers have proposed various hypergraph neural network methods [10], achieving significant progress in tasks such as node classification [11], link prediction [12, 13], and representation learning [2]. Although existing hypergraph neural networks provide powerful tools for hypergraph learning, the problem of over-smoothing remains a key challenge [14, 15]. As the network deepens, node features tend to converge and become increasingly indistinguishable, which degrades the model’s performance [16]. Current mitigation strategies fall into two main paradigms. The first paradigm modifies the network architecture by incorporating mechanisms such as residual connections to preserve initial node features [17, 18]. The second paradigm refines the aggregation step, often using attention mechanism to selectively weigh nodes and hyperedges, thereby preventing feature homogenization [19, 20]. However, while existing methods mitigate over-smoothing to some extent, most of them are essentially operator-level fixes and lack rigorous theoretical guarantees, which limits their efficacy in complex scenarios. Inspired by the work [21], which establishes a direct link between geometric curvature and over-smoothing, we identify the feature convergence in hypergraph neural networks as an unconstrained geometric evolution process, analogous to heat diffusion. This perspective motivates us to seek a mechanism from differential geometry that can intrinsically control such diffusion dynamics. Ricci flow serves precisely this purpose [22], as it describes how the metric tensor evolves according to the local curvature. Recently, its generalization, discrete Ricci flow, has proven effective for tasks on graphs [23]. However, its potential for hypergraphs remains unexplored. In this work, we generalize this evolution to hypergraph and theoretically demonstrate that the evolution constrains the diffusion of node features via local curvature, which in turn effectively mitigates over-smoothing. Motivated by the preceding theoretical insights, we propose Ricci Flow-guided Hypergraph Neural Diffusio(RFHND). Grounded in partial differential equations, RFHND models node feature evolution as a continuous dynamical process and adaptively regulates information diffusion based on local curvature, effectively preventing over-smoothing. The core idea of RFHND is shown in Fig. 1. Specifically, we assign weights to hyperedges proportional to the feature similarity among their nodes, enabling an adaptive flow of updates that mitigates feature homogenization. This design preserves node distinctions while enabling efficient feature fusion, enhancing model expressiveness and stability. Finally, the efficacy of our algorithm is validated through a comprehensive series of experiments. The main contributions of this work are as follows: • Theoretical foundation: We introduce discrete Ricci flow to the hypergraph learning domain, theoretically proving that it mitigates over-smoothing by adaptively controlling feature diffusion via local curvature. • Methodological contribution: We propose RFHND, a discrete Ricci flow–guided hypergraph neural diffusion method that transforms traditional message passing into an adaptive, curvature-guided diffusion process, thereby effectively preventing feature homogenization. • Empirical validation: Experiments on several hypergraph datasets demonstrate that RFHND achieves superior node classification accuracy while exhibiting improved stability, robustness, and a significant reduction in over-smoothing compared to existing methods. Figure 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. I Related Work In this section, we review literature closely related to our work, focusing on two key areas: Hypergraph Neural Networks(HGNNs) and Over-smoothing in HGNNs. I-A Hypergraph Neural Networks Hypergraph neural networks have emerged as a powerful generalization of graph neural networks (GNNs), specifically designed to capture complex high-order relationships among entities beyond pairwise connections. The development of this field was initiated by the pioneering work of the foundational HGNN model [24]. This model utilizes a spectral convolution framework on hypergraphs to aggregate information from nodes connected with the same hyperedge. Following this, a wave of message-passing-based models, such as HNHN [25], HyperGCN [26], HyperSAGE [27], and UniGNN [17] was proposed to enhance the expressive power of HGNNs. These methods extend aggregation and propagation mechanisms to accommodate heterogeneous and weighted hyperedges, leading to significant improvements in tasks such as node classification. Meanwhile, architectures like AllDeepSets [18] and AllSetTransformer [18] abandoned spectral assumptions entirely. They instead employ permutation-invariant set functions over hyperedges to facilitate more flexible, set-level reasoning. Recent HGNN advances include more sophisticated modeling perspectives. For instance, dynamic system models like HDS [28] use ordinary differential equations (ODEs) to improve the stability and control of the learning process. Additionally, models such as KHGNN(employs a nested convolution module named HyperGINE) [29] are designed to capture long-range dependencies by extracting features from nodes, hyperedges, and the intermediate paths between them. I-B Over-smoothing in HGNNs In deep hypergraph networks, node representations tend to become identical, which severely degrades model performance. Current mitigation strategies, designed to combat this issue, can be categorized into two main paradigms. Architectural Modifications. This line of work focuses on altering the network’s architecture to preserve information from earlier layers. Inspired by the effectiveness of residual connections in both CNNs and GNNs, models like UniGCNII [17] and Deep-HGCN [18] integrate skip connections to carry initial node features forward, enabling the construction of much deeper and more effective HGNNs. FrameHGNN [14] introduces a framelet-based hypergraph convolution framework that combines low and high-pass filters with proven deep GNN techniques like residual and identity mapping, thereby maintaining discriminative signals in deep layers. Refining the Aggregation Operator. This paradigm aims to make the feature aggregation process more discerning. Instead of uniform aggregation, attention mechanisms are employed to assign different weights to nodes within a hyperedge or to different hyperedges connected to a node. For example, Hypergraph Attention Networks [19, 20] learn dynamic weights for both nodes and hyperedges, thereby effectively mitigating the issue of indiscriminate feature mixing. ED-HNN [30] suggests that employing equivariant operators to distribute distinct messages across nodes helps preserve feature diversity (or node distinguishability), thereby effectively preventing the over-smoothing problem. Limitations of Existing Methods. While existing methods mitigate over-smoothing to some extent, most of them rely on local framework modifications and lack rigorous theoretical guarantees, which limits their efficacy in complex scenarios. In contrast, we introduce discrete Ricci flow to globally regulate the evolution of node features, thereby effectively alleviating the over-smoothing issue. I Preliminary I-A Notations Let ℋ=(,ℰ)H=(V,E) denote a hypergraph, where V is the vertex set containing n unique vertices and ℰE is the edge set containing m hyperedges. The hypergraph can be represented by an incidence matrix H∈Rn×mH∈ R^n× m where Hij=1H_ij=1 if the vertex vi∈v_i is contained in the hyperedge ej∈ℰe_j , otherwise 0. Let =[1,2,…n]TF=[f_1,f_2,...f_n]^T denote the node feature matrix, and fif_i is associated with the node viv_i. Similarly, let (t)=[1(t),2(t),…n(t)]X(t)=[x_1(t),x_2(t),...x_n(t)] denote the node representation evolved to time t, and (0)=Linear()X(0)=Linear(F). Each hyperedge ej∈ℰe_j is assigned a positive time-varying weight wej(t)w_e_j(t), all the weights formulate a diagonal matrix W(t)∈Rm×mW(t)∈ R^m× m. The vertex and edge degree of hypergraph can be expressed as di=∑j=1mHijd_i= _j=1^mH_ij and dj=∑i=1nHijd_j= _i=1^nH_ij, all the degrees formulate the diagonal matrix DvD_v and DeD_e. I-B Hyperedge Curvature on Weighted Hypergraph Let ℋH denote a weighted hypergraph. Each hyperedge e∈ℰe is assigned a weight wew_e, and a curvature value κe _e . Specifically, κe _e reflects the closeness of the nodes within the hyperedge. Several discrete notions of curvature have been extended from graphs to hypergraphs, including Forman–Ricci curvature κeFR _e^FR [31], Ollivier–Ricci curvature κeOR _e^OR [32] and Resistance curvature κeRC _e^RC [33]. Notably, our method is not restricted to a specific choice of curvature measure. For the precise mathematical formulations of these curvature measures, please refer to the supplementary material. I-C Dirichlet Energy The Dirichlet energy of the hypergraph ℋ=(,ℰ)H=(V,E) is used to characterize the smoothness of features between nodes [34]. The expression is as follows [18]: E((t))=12∑e∈ℰ∑i,j∈e1|e|(i(t)di−j(t)dj)2.E(X(t))= 12 _e _i,j∈ e 1|e| ( x_i(t) d_i- x_j(t) d_j )^2. (1) The equation (1) shows that a higher similarity between neighboring node features leads to an energy value closer to 0, which reflects the over-smoothing problem. I-D Discrete Ricci Flow(DRF) Ricci flow, originally introduced by Hamilton in differential geometry [35], evolves a Riemannian metric gij(t)g_ij(t) according to the partial differential equation: ∂gij(t)∂t=−2Ricij(g(t)), ∂ g_ij(t)∂ t=-2\,Ric_ij(g(t)), (2) where gij(t)g_ij(t) denotes the element of the metric tensor g(t)g(t) and RicijRic_ij denotes the element of the Ricci curvature tensor. This process can be regarded as a geometric heat equation, smoothing out irregularities of the manifold metric over time, and has played a fundamental role in the development of modern geometry and topology. To extend this concept to discrete structures, discrete Ricci flow has been proposed as an analogue of the continuous Ricci flow on graphs and networks [36]. The specific expression is as follows: ∂wij(t)∂t=−κij(t)wij(t). ∂ w_ij(t)∂ t=- _ij(t)w_ij(t). (3) Here, wij(t)w_ij(t) and κij(t) _ij(t) represent the weight and the curvature of the edge (vi,vj)(v_i,v_j) at time t. Discrete Ricci flow has found applications in diverse fields, including community detection [37], network analysis [38], graph embedding [39], and machine learning. Building upon these advances, we further generalize discrete Ricci flow to hypergraphs. The core of this generalization lies in reformulating the notions of edge weight and curvature from pairwise edges to multi-node hyperedges. The specific expression of the hypergraph Ricci flow is as follows: ∂we(t)∂t=−κe(t)we(t). ∂ w_e(t)∂ t=- _e(t)w_e(t). (4) Here, we(t)w_e(t) and κe(t) _e(t) represent the weight and the curvature of the edge e at time t. By formulating discrete Ricci flow on hypergraphs, we aim to capture higher-order structures and provide a new theoretical framework for representation learning in Hypergraph. IV Applying DRF on Hypergraph In this section, we apply discrete Ricci flow to both the hypergraph structure and its node features to construct a dynamic evolutionary system. Similar to the attribute discrete Ricci flow in [39] , we construct the attribute discrete Ricci flow on the hypergraph. Specifically, We define the hyperedge weights as a function of the nodes features it connects, i.e., we(t):=we(e(t))w_e(t) :=w_e(X_e(t)), Where e(t)=i(t)|i∈eX_e(t)= \x_i(t)|i∈ e \ denotes the set of node features on the edge e. Then the attribute discrete Ricci flow of the hypergraph can be written as follow: ∂we(e(t))∂t=−κe(t)we(e(t)). ∂ w_e(X_e(t))∂ t=- _e(t)w_e(X_e(t)). (5) As established in I-B, the curvature of a hyperedge is jointly determined by both structural connectivity and its weight. In turn, this curvature governs the dynamic evolution of node attributes according to Equation (5). Unlike traditional heat diffusion, which models node representation propagation , Ricci flow can be viewed as heat diffusion of the metric. From now on, we define we(t)w_e(t) as follows: we(t)≡1αe[1|e|∑i∈e,j∈ecos(i(t),j(t))didj]+1+ϵ,αe=1|e|∑i∈e,j∈e1didj. split&w_e(t)≡ 1 _e[ 1|e| _i∈ e,j∈ e cos(x_i(t),x_j(t)) d_id_j]+1+ε,\\ & _e= 1|e| _i∈ e,j∈ e 1 d_id_j. split (6) Here, we(t)w_e(t) is a non-negative weight, ϵε is a small positive number. Furthermore, we enforce a unit norm on (t)x(t) (i.e., |(t)|≡1|x(t)|≡ 1, which is crucial for maintaining stability during computation. In this case, we(t)w_e(t) satisfies: we(t)∈[ϵ,2+ϵ]w_e(t)∈[ε,2+ε]. IV-A Dirichlet Energy Bound By applying the attribute discrete Ricci flow on hypergraph, we can prove that the system’s energy value remains bounded when the node feature evolution follows the Ricci flow. This demonstrates that our method can effectively prevent over-smoothing and maintain controlled feature differentiation during propagation. The specific conclusions are as follows: Theorem 1. Consider the attribute discrete Ricci flow with hyperedge weight as equation (6), and |(t)|≡1|x(t)|≡ 1. If H is a non-regular hypergraph, and we(t)w_e(t) are monotonic on [t1,t2][t_1,t_2] for all e in H, then the average Dirichlet energy within [t1,t2][t_1,t_2] has following bound: ρmax(∑e∈ℰ∑i,j∈e12|e|(1di+1dj)+∑e∈ℰαe)≥t∈[t1,t2](E((t)))≥∑e∈ℰ∑i,j∈e12|e|(1di+1dj)−∑e∈ℰαe>0, split& _max ( _e _i,j∈ e 12|e| ( 1d_i+ 1d_j )+ _e _e )\\ & _t∈[t_1,t_2](E(X(t)))\\ &≥ _e _i,j∈ e 12|e| ( 1d_i+ 1d_j )- _e _e 0, split (7) Here, did_i denotes the degree of node i, ρmax=maxeρe _max= _e _e and ρe _e denotes the ratio of the maximum to the minimum value of we(t)w_e(t) . As demonstrated, Theorem 1 provides crucial theoretical support for addressing the over-smoothing problem in hypergraph neural networks. The theorem’s strictly positive lower bound ensures that the differentiation among nodes will not completely vanish throughout the feature evolution process. This, in turn, preserves meaningful feature distinctions and prevents the excessive assimilation of node representations. The full proof is available in the supplementary material. IV-B Convergence Analysis Following the preceding analysis of the method’s effectiveness against over-smoothing, we further investigate its convergence. The specific results of this analysis are as follows: Theorem 2. Consider the attribute discrete Ricci flow with hyperedge weight as equation (6). Assume that there exists a constant L such that |κe(t2)−κe(t1)|≥L|we(t2)−we(t1)|| _e(t_2)- _e(t_1)|≥ L|w_e(t_2)-w_e(t_1)| holds for all arbitrary t1,t2t_1,t_2. Let |κe(0)|>0| _e(0)|>0, then for any arbitrarily small positive number δ, it holds that: mint∈[0,+∞)t∣|κe(t)|=δ≤1Lϵ−δln(2L+δ(2+ϵ)). _t∈[0,+∞) \t\; |\;| _e(t)|=δ \≤ 1Lε-δ ( 2L+δ(2+ε) ). (8) The Theorem 2 prove that the discrete Ricci flow possesses the property of exponential convergence. This implies that the hyperedge curvature can rapidly approach zero, regardless of the system’s initial state. The Theorem 2 thus provide a theoretical guarantee for the efficiency and stability of our method. The proof of Theorem 2 can be referred to [39]. V Methodology V-A Ricci Flow-guided Hypergraph Feature Diffusion As established in section IV, applying attribute discrete Ricci flow to hypergraphs yields several key advantages. The process drives the curvature of hyperedges toward uniformity while facilitating the learning of high-quality, non-smooth node representations. Moreover, the method is theoretically proven to have strong convergence properties. Inspired by these strengths, in this section we design a new hypergraph feature diffusion architecture. By expanding Equation (5) using the chain rule, we obtain: ∑i∈e⟨∂we∂i,∂i(t)∂t⟩=−κe(t)we(t). _i∈ e ∂ w_e _i, _i(t)∂ t =- _e(t)w_e(t). (9) The left side of Equation (9) contains |e||e| parts. To weigh these parts, we introduce several scaling functions: ⟨∂we∂j,∂j(t)∂t⟩=λj(1,…|e|)⟨∂we∂i,∂i(t)∂t⟩. ∂ w_e _j, _j(t)∂ t =λ^j(x_1,...x_|e|) ∂ w_e _i, _i(t)∂ t . (10) Then, We focus on one side: ⟨∂we∂i,∂i(t)∂t⟩=−κe(t)we(t)1+∑j∈e,j≠iλj(1,…|e|). ∂ w_e _i, _i(t)∂ t =- _e(t)w_e(t)1+ _j∈ e,j≠ iλ^j(x_1,...x_|e|). (11) Under the the constraint of Equation (11) and |i(t)|≡1|x_i(t)|≡ 1 , we minimize ‖∂i(t)∂t‖\| _i(t)∂ t\|, which means that i(t)x_i(t) always applies only the slightest change to satisfy the attribute discrete Ricci flow, which guarantees that the evolution of i(t)x_i(t) is numerically stable. On this basis, the resulting optimization objective can be formally expressed as follows: min‖∂i(t)∂t‖,s.t.⟨∂we∂i,∂i(t)∂t⟩=−κe(t)we(t)1+∑j∈e,j≠iλj(1,…|e|),|i(t)|=1. split& \| _i(t)∂ t \|, \\ &s.t.\;\; ∂ w_e _i, _i(t)∂ t =- _e(t)w_e(t)1+ _j∈ e,j≠ iλ^j(x_1,...x_|e|),\\ &|x_i(t)|=1. split (12) Proposition 3. The optimization objective presented in Equation (12) has a closed-form solution as follows: ∂i(t)∂t=−κie′(t)[∑j∈ej(t)−cos(i(t),j(t))i(t)didj], _i(t)∂ t=-κ _ie(t) [ _j∈ e x_j(t)- (x_i(t),x_j(t) )x_i(t) d_id_j ], (13) where κie′(t)=μie(−(i(t)Tmie)2)(∑i,j∈e1didj),κ _ie(t)= _ie(1-(x_i(t)^Tm_ie)^2)( _i,j∈ e 1 d_id_j), (14) μie=−κe(t)we(t)1+∑j∈e,j≠iλj(1(t),…|e|(t)), _ie=- _e(t)w_e(t)1+ _j∈ e,j≠ iλ^j(x_1(t),...x_|e|(t)), (15) mie=1∑i,j∈e1didj(∑j∈ej(t)didj).m_ie= 1 _i,j∈ e 1 d_id_j ( _j∈ e x_j(t) d_id_j ). (16) To reduce computational complexity, we select appropriate scaling functions λj(⋅)λ^j(·) such that κie′(t)=κje′(t)κ _ie(t)=κ _je(t) holds for all i,j∈ei,j∈ e. Consequently, Equation (13) can be simplified to: ∂i(t)∂t=−κe′(t)[∑j∈ej(t)−cos(i(t),j(t))i(t)didj]. _i(t)∂ t=-κ _e(t) [ _j∈ e x_j(t)- (x_i(t),x_j(t) )x_i(t) d_id_j ]. (17) The Equation (17) defines the local influence of a single hyperedge e on node i, while a node’s representation is updated based on its entire neighborhood. Accordingly, we generalize this equation from a single hyperedge to the entire hypergraph by iterating over all hyperedges incident to node i and aggregating the corresponding information. We can rewrite Equation (17) as follows: ∂i(t)∂t=∑e:i∈e−κe′(t)[∑j∈ej(t)−cos(i(t),j(t))i(t)didj]. _i(t)∂ t= _e:i∈ e-κ _e(t) [ _j∈ e x_j(t)- (x_i(t),x_j(t) )x_i(t) d_id_j ]. (18) We can rewrite the equation (18) in matrix form as follows: ∂(t)∂t=[diag((S((t))⊙C((t)))N)−S((t))](t), (t)∂ t= [diag ( (S(X(t)) C(X(t)) )1_N )-S(X(t)) ]X(t), (19) S((t))=Dv−12HK′((t))HTDv−12,S(X(t))=D_v^- 12HK (X(t))H^TD_v^- 12, (20) K′((t))=diag(κ1′(t),κ2′(t)…κm′(t)),K (X(t))=diag(κ _1(t),κ _2(t)...κ _m(t)), (21) Cij((t))=cos(i(t),j(t)).C_ij(X(t))=cos(x_i(t),x_j(t)). (22) Here, H represent the incidence matrix of hypergraph, DvD_v represent the vertex degree matrix. The node feature update process in Equation (18) and (19) mitigates over-smoothing via a geometry-aware adaptive mechanism. It aggregates not the raw features of neighboring nodes, but rather the feature dissimilarities between neighboring and central nodes. The sign of the curvature is leveraged to dynamically modulate the direction of this aggregation: in homophilous communities with positive curvature, it reduces feature discrepancies to achieve local smoothing, whereas across heterophilous bridges with negative curvature, it amplifies these differences to preserve node distinctiveness. This bidirectional regulatory capacity, combining both positive and negative effects, transcends the unidirectional averaging process inherent to conventional HGNNs. By preserving the uniqueness of critical nodes, it fundamentally inhibits the convergence of all node features towards uniformity, thereby addressing the over-smoothing problem. Existence and Uniqueness of Solutions: The Equation (19) offers an innovative framework for modeling hypergraph feature evolution. To ensure its validity, we conduct a rigorous theoretical analysis, from which we derive the following conclusion: Theorem 4. Suppose the following conditions hold: 1. For every node i, the degree of node i is strictly bounded: 0<dmin≤di≤dmax<∞0<d_ ≤ d_i≤ d_ <∞, where dmind_ and dmaxd_ denote the minimum and maximum node degrees of the hypergraph. 2. λj(⋅)λ^j(·) is bounded on bounded sets and locally Lipschitz. 3. For every e, κe(t) _e(t) and we(t)w_e(t) are continuously bounded. 4. For every e, 1−(i(t)Tmie)2≥ε∈(0,1]1-(x_i(t)^Tm_ie)^2≥ ∈(0,1]. Then, for any initial condition (0)∈ℝnx(0) ^n and any T>0T>0, there exists a unique solution (t)∈C1([0,T],ℝn)x(t)∈ C^1([0,T],R^n) to Equation (19). Theorem 4 guarantees the reliable local evolution behavior of the dynamical system under study within a strict mathematical framework. Specifically, the local solution (t)x(t) of Equation (19) is guaranteed to exist and be continuously differentiable, which ensures that the system is well-defined in the short term. Simultaneously, the theorem guarantees the solution is unique which ensures that the system trajectory emanating from any initial condition (0)x(0) is one-of-a-kind, thereby lending reliability to analyses and predictions based on this equation. The full proof is available in the supplementary material. V-B Ricci Flow-guided Hypergraph Neural Diffusion As shown in Equation (18), the evolution of node features is tightly coupled with the aggregation weight function κe′(t)κ _e(t). However, in practice, the exact computation of this function for all hyperedges is computationally prohibitive. Therefore, to address this challenge, we propose leveraging the function approximation capabilities of a neural network to fit the hyperedge function. This leads us to propose the Ricci Flow-guided Hypergraph Neural Diffusion(RFHND) model, a framework for node representation evolution inspired by Ricci flow. Based on the universal approximation theorem [40] of neural networks, we can draw the following conclusions. Theorem 5. Suppose for any j∈ej∈ e, λj(⋅)λ^j(·) is a continuous function, When Forman-Ricci curvature κFRκ^FR, Ollivier-Ricci curvature κORκ^OR are used as the definition of hyperedge curvature in RFHND, there exists an HyperNet that can approximate the aggregation weight of RFHND κe′(t)κ _e(t) with arbitrarily high precision. Proof. The goal is to demonstrate that the aggregation weight κe′(t)κ _e(t) is a continuous function of the node features (t)X(t), which in turn allows for its approximation by a neural network. We begin with the premise that for any node j∈ej∈ e, its associated function λj(⋅)λ^j(·) is continuous. As established in the supplementary material, the curvature κe(t) _e(t) is a continuous function of the node features. Similarly, the hyperedge weight we(t)w_e(t), defined in Equation (6), is also a continuous function of the node features. The expression of κe′(t)κ _e(t) is a composition of the functions κe(t) _e(t), we(t)w_e(t), and λj(⋅)λ^j(·) through arithmetic operations that preserve continuity (assuming a non-zero denominator). Since each of its constituent functions is continuous with respect to the node features (t)X(t), it follows that κe′(t)κ _e(t) is also a continuous function of (t)X(t). Therefore, by the Universal Approximation Theorem, there exists a neural network with sufficient capacity that can approximate the aggregation weight κe′(t)κ _e(t) from the node features with arbitrary precision. ∎ This theoretical result provides a rigorous foundation for our architectural design. It guarantees that parameterizing the curvature-guided aggregation weights via neural networks (e.g., MLPs) is a valid approach. In practical implementation, we drive feature updates by numerically solving differential equations. The pseudocode for the forward propagation of RFHND is shown in Algorithm 1. To ensure the robustness of the solution process, we further derive the stability conditions for the explicit Euler method, as detailed below: Theorem 6. The explicit Euler method for the RFHND is stable if the step size τ≤1maxi∑jsij.τ≤ 1max_i _js_ij. Here, sijs_ij denotes the elements of the matrix S((t))S(X(t)). To ensure numerical stability and prevent divergence, the step size of the explicit method must be subject to specific constraints. We formally establish the rigorous form of this criterion in Theorem 6: specifically, the step size τ must be less than the reciprocal of the maximum row sum of the hypergraph edge weight matrix to guarantee the asymptotic stability of the algorithm. For the detailed mathematical proof, please refer to the supplementary material. Computational complexity. To evaluate the computational efficiency of RFHDN, we analyze the time complexity of its forward propagation. The overall time complexity is O(ndindout)+T⋅O(m(r2din+din2))O(nd_ind_out)+T· O(m(r^2d_in+d^2_in)), where n and m denote the number of nodes and hyperedges, respectively. dind_in and doutd_out represent the dimensions of the input and output features, T indicates the number of layers, and r represents the maximum hyperedge degree. The detailed analysis is provided in the the supplementary material. Algorithm 1 Ricci Flow-guided Hypergraph Neural Diffusion Input: Hypergraph ℋ=(,ℰ)H=(V,E) with incidence matrix H, edge index E ,degree matrix DvD_v, features matrix ∈ℝn×dinF ^n× d_in, step size τ, depth T, hidden dimension d. 1:(0)←Linear()X(0) (F) 2:(node_idx,edge_idx)←E(node\_idx,edge\_idx)← E 3:for t=0t=0 to T−1T-1 do 4: Eedge←scatter((t),edge_idx)E_edge (X(t),edge\_idx) 5: Enode←scatter(MLPθ1(Eedge),edge_idx)E_node (MLP_ _1(E_edge),edge\_idx) 6: Eedge←scatter(Eedge,edge_idx)E_edge (E_edge,edge\_idx) 7: K′(t)←diag(MLPθ2(Eedge))K (t) (MLP_ _2(E_edge)) 8: Compute Matrix S((t))S(X(t)) by Eq. (20) 9: F((t))←diag((S((t))⊙C((t)))N)−S((t))F(X(t)) ( (S(X(t)) C(X(t)) )1_N )-S(X(t)) 10: (t+1)←(t)−τF((t))(t)X(t+1) (t)-τ F(X(t))X(t) 11:end for 12:Y←Linear((T))Y (X(T)) 13:Compute loss and back propagation. TABLE I: Dataset Statistics Summary Metric Cora Citeseer Pubmed Cora-CA DBLP-CA Zoo NTU2012 ModelNet40 Walmart Senate House |V||V| 27082708 33123312 19 17719\,177 27082708 41 30241\,302 101101 20122012 12 31112\,311 88 86088\,860 282282 12901290 |E||E| 15791579 10791079 79637963 10721072 22 36322\,363 4343 20122012 12 31112\,311 69 90669\,906 315315 340340 #features 14331433 37033703 500500 14331433 14251425 1616 100100 100100 100100 100100 100100 #classes 77 66 33 77 66 77 6767 4040 1111 22 22 VI Experiment VI-A Results on Benchmark Datasets Datasets. To validate the performance of RFHND, we conducted a comprehensive evaluation on a diverse collection of benchmark datasets representing both academic and real-world scenarios. The academic scenarios evaluation was performed on five well-established hypergraph benchmarks from co-citation and co-authorship networks: Cora, Citeseer, Pubmed, Cora-CA, and DBLP-CA [26]. For these datasets, node features are derived from bag-of-words representations, while labels correspond to the paper’s subject category. To assess the method’s generalizability across different domains, we also utilized several real-world datasets. These include Zoo from the UCI repository [41], the 3D vision datasets ModelNet40 [42] and NTU2012 [43], and the transaction datasets Walmart [44], the social network datasets House [45] and Senate [46]. Following prior works, hypergraph structures were constructed for all datasets, in cases where inherent node features were absent, we initialized them using Gaussian random vectors. For the evaluation, a consistent 50%/25%/25% split was used for the training, validation, and testing sets. To ensure robust and reliable results, the final reported performance is the aggregated outcome of 20 independent trials using different random splits. The statistical summary of all datasets is available in Table I. TABLE I: 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 Cora Citeseer Pubmed Cora-CA DBLP-CA Rank↓ HNHN 76.36± 1.92 72.64± 1.57 86.90± 0.30 77.19± 1.49 86.78± 0.29 12 HGNN 79.39± 1.36 72.45± 1.16 86.44± 0.44 82.64± 1.65 91.03± 0.20 8 HCHA 79.14± 1.02 72.42± 1.42 86.41± 0.36 82.55± 0.97 90.92± 0.22 11 HyperGCN 78.45± 1.26 71.28± 0.82 82.84± 8.67 79.48± 2.08 89.38± 0.25 13 UniGCNII 78.81± 1.05 73.05± 2.21 88.25± 0.40 83.60± 1.14 91.69± 0.19 6 AllSetTransformer 78.59± 1.47 73.08± 1.20 88.72± 0.37 83.63± 1.47 91.53± 0.23 5 AllDeepSets 76.88± 1.80 70.83± 1.63 88.75± 0.33 81.97± 1.50 91.27± 0.27 10 HAN 79.70± 1.77 74.12± 1.52 85.32± 2.25 81.71± 1.73 90.17± 0.65 9 ED-HNN 80.31± 1.35 73.70± 1.38 89.03± 0.53 83.97± 1.55 91.90± 0.19 3 HyperGINE 79.26± 0.41 73.72± 0.52 87.91± 0.28 82.88± 0.48 - 11 KHGNN 80.67± 0.76 74.80± 1.10 88.47± 0.47 84.25± 0.74 - 4 FrameHGNN 81.51± 0.99 74.72± 2.10 88.73± 0.42 85.18± 0.69 - 2 RFHND 81.83± 1.48 76.04± 1.43 88.88± 0.35 85.40± 1.05 91.20± 0.41 1 TABLE I: 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. Models Zoo NTU2012 ModelNet40 Walmart Senate House Rank↓ HNHN 93.59± 5.88 89.11± 1.44 97.84± 1.25 81.35± 0.61 50.93± 6.33 67.80± 2.59 6 HGNN 92.50± 4.58 87.72± 1.35 95.44± 0.33 80.33± 0.42 48.59± 4.52 61.39± 2.96 11 HCHA 93.65± 6.15 87.48± 1.87 94.48± 0.28 80.33± 0.80 48.62± 4.41 61.36± 2.53 10 HyperGCN N/A 56.36± 4.86 75.89± 5.26 81.05± 0.59 42.45± 3.67 48.32± 2.93 13 UniGCNII 93.65± 4.37 89.30± 1.33 98.07± 0.23 81.12± 0.67 49.30± 4.25 61.70± 3.37 7 AllSetTransformer 97.50± 3.59 88.69± 1.24 98.20± 0.20 81.38± 0.58 51.83± 5.22 69.33± 2.20 4 AllDeepSets 95.39± 4.77 88.09± 1.52 96.98± 0.26 81.06± 0.54 48.17± 5.67 67.82± 2.40 8 HAN 75.77± 7.10 83.58± 1.46 94.04± 0.41 79.72± 0.62 - 62.00± 9.09 12 ED-HNN - 88.67± 0.92 97.83± 0.33 - 64.79± 5.14 72.45± 2.28 5 HyperGINE - 88.52± 0.42 97.61± 0.16 - - - 9 KHGNN - 89.60± 1.64 98.33± 0.14 - - - 3 FrameHGNN - 89.98± 2.02 98.41± 0.18 - 67.61± 5.27 72.82± 2.22 2 RFHND 97.70± 2.14 93.36± 1.18 98.56± 0.16 82.40± 1.40 68.12± 4.98 73.52± 2.25 1 Baselines. To evaluate the performance of RFHND, we compared it against a comprehensive set of representative hypergraph neural network baselines. This includes several methods, such as HGNN [24], which applies spectral convolution operations designed for hypergraph data. We also include HCHA [19], a method that incorporates hierarchical attention mechanisms for representation learning, and the widely-used HyperGCN [26], which extends graph convolutions to the hypergraph domain through a clique expansion strategy. Our comparison further incorporates HNHN [25], a model noted for its novel hypergraph-specific normalization techniques, alongside UniGCNII [17], which unifies multiple hypergraph convolution paradigms with residual connections. The well-known HAN [47] model is also included as a key representative of hierarchical attention networks. Additionally, we benchmark against AllSetTransformer and AllDeepSets [18], which adapt deep set theory to ensure permutation invariance in hypergraph learning. The evaluation is extended to more recent and specialized architectures. The baseline set features ED-HNN [30], a method that utilizes equivariant hypergraph diffusion operators. To account for long-range dependencies, we include HyperGINE and KHGNN [29], the latter of which facilitates interactions between distant nodes via K-hop message passing. Finally, the multi-scale approach FrameHGNN [14] is included, which integrates framelet transforms into its architecture. To ensure a fair and standardized comparative environment, all models were implemented within the PyTorch Geometric library [48]. Experiment Setting. For the RFHND model, the Adam optimizer is employed. The main hyperparameters—learning rate, weight decay, dropout rate, hidden dimension, and total training epochs—are individually tuned for each dataset based on validation performance. Specifically, the learning rate is selected from 0.001,0.01 \0.001,0.01 \, weight decay from the range [0.001,0.03][0.001,0.03], input dropout from 0.001,0.01,0.1,0.2,0.3 \0.001,0.01,0.1,0.2,0.3 \, hidden dimension from 16,32,64,128,256,512 \16,32,64,128,256,512 \, and total training epochs from 2,3,4 \2,3,4 \. The optimal configuration for each dataset is determined within these ranges. When applicable, a cosine learning rate scheduler (CosineLR) is adopted. We use the methods implemented by torchdiffeq [49] as the differential equation solver for RFHND. All experiments are conducted using a fixed random seed to ensure reproducibility. Source code is available at https://gitee.com/zmy-ovo/rfhnd.git. Results. As demonstrated in Table I and Table I, the experimental results fully showcase the superior performance of our proposed RFHND model. By calculating the average ranking across all comparison methods, the RFHND model maintained the first position on both academic and real-world datasets, which strongly attests to its powerful generalization ability. Specifically, the RFHND model achieved State-of-the-Art performance in three out of the five academic benchmark datasets and secured an excellent second-place ranking on one of the others. This performance conclusively confirms the RFHND model’s crucial capability to effectively capture complex relationships within structured data. Furthermore, the RFHND model comprehensively surpassed all comparison methods on the six real-world datasets, which span diverse application domains such as transaction networks and social network analysis. These consistent results across different dataset types collectively validate the outstanding effectiveness of the RFHND model for the hypergraph node classification task. TABLE IV: Model Performance On Synthetic Hypergraphs with Controlled Heterophily α. Model α=1α=1 α=2α=2 α=3α=3 α=4α=4 α=5α=5 α=6α=6 α=7α=7 HGNN 92.58 ± 0.57 87.40 ± 0.89 82.32 ± 0.84 77.18 ± 1.08 70.06 ± 1.35 69.98 ± 0.85 68.41 ± 1.39 HyperGCN 83.40 ± 1.67 78.13 ± 1.57 77.70 ± 1.01 74.90 ± 1.31 50.15 ± 1.94 52.11 ± 0.75 49.40 ± 1.63 UniGCN 90.42 ± 1.04 82.12 ± 0.67 80.04 ± 0.98 76.86 ± 1.10 71.50 ± 1.50 52.30 ± 1.18 49.20 ± 0.74 EDGNN 93.88 ± 0.67 89.89 ± 0.59 83.52 ± 0.85 76.65 ± 1.21 72.18 ± 1.36 52.98 ± 0.78 49.65 ± 1.16 RFHND 95.63 ± 0.62 90.17 ± 0.67 84.21 ± 0.81 77.48 ± 0.98 74.26 ± 1.57 73.51 ± 1.07 73.46 ± 1.35 (a) (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. VI-B Results on Synthetic Heterophilic Hypergraph Dataset Experiment Setting. To assess the capability of RFHND in handling datasets exhibiting varying degrees of structural heterogeneity, we conduct experiments on synthetic datasets with controllable heterophily levels. These datasets are generated following the contextual hypergraph stochastic block model [50, 51, 52]. In particular, we create two node classes, each containing 2,500 nodes, and randomly sample 1,000 hyperedges. Each hyperedge is composed of 15 nodes, among which αi _i nodes are drawn from class i. The heterophily level is quantified as α=minα1,α2α= \ _1, _2\. The node features are sampled from Gaussian distributions conditioned on node labels, with a standard deviation of 1.0. We examine both homophilic configurations (α=1,2,3α=1,2,3 ) and heterophilic configurations (α=4∼7α=4 7 ). The proposed RFHND framework is benchmarked against several baselines, including HGNN, HyperGCN, and ED-HNN. Consistent with prior studies, we employ a 50%/25%/25% partitioning of the data for training, validation, and testing. To ensure statistical reliability, all experiments are repeated over 10 random splits, and the final results are reported as averaged outcomes. Results. The results presented in Table IV indicate that the RFHND model consistently achieves superior performance compared to all baseline methods across the evaluated datasets. Its advantage is particularly evident in heterophilic settings (when α >3), where the model demonstrates markedly improved robustness and generalization ability. These findings substantiate the effectiveness of the proposed architecture. VI-C Ablation Study In this section, we perform ablation experiments on multiple datasets to assess the contribution of each submodule within our model. The detailed findings are summarized as follows. TABLE V: Ablation Study. Variants Zoo NTU2012 ModelNet40 w/o COS 97.16 ± 2.90 90.04 ± 1.09 96.78 ± 0.27 w/o HyperNet 96.89 ± 2.87 91.52 ± 1.12 97.42 ± 0.23 w/o C and H 96.43 ± 2.90 89.79 ± 1.30 96.86 ± 0.27 RFHND (Full Model) 97.70 ± 2.14 93.36 ± 1.18 98.56 ± 0.16 ∙ w/o COS: In this variant of RFHND, the cosine coefficient is removed from the node feature evolution process, so feature updates are performed without cosine-based modulation. ∙ w/o HyperNet: In this variant of RFHND, the HyperNet used to model edge curvature is replaced with a randomly generated numerical value, instead of learning curvature adaptively. ∙ w/o C and H: This RFHND variant excludes both the cosine coefficient in the node feature evolution process and the HyperNet used for edge curvature modeling. As shown in Table V, all variants exhibit degraded performance in label prediction, underscoring the importance of these in enhancing the modeling of interaction types. Specifically, the joint removal of the COS and HyperNet modules (w/o C and H) leads to a substantially more pronounced performance drop compared to the singular ablations, underscoring the critical importance of coordination between these two modules. (a) (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. VI-D Over-smoothing analysis This section examines how model depth affects the performance of hypergraph neural networks. Most existing HGNNs are inherently shallow, which limits their ability to capture information from higher-order neighbors [15]. However, simply increasing the depth often leads to oversmoothing, where node representations become increasingly similar and eventually indistinguishable. To study this issue, we evaluated several models with different depths on Cora, Cora-CA and Citeseer datasets. Using a fixed data split, we tested layer settings of 2, 4, 10, 20, 30, and 40 to observe how performance changes as the network becomes deeper. Meanwhile, we also recorded the Dirichlet Energy across different layer configurations. Specifically, for zero layers, the energy is calculated using the initial node features, while for deeper configurations, it is calculated from the node embeddings output by the trained network. As systematically illustrated in Fig. 7(a), the results yield two key observations: • The proposed RFHND model consistently delivers superior performance across all tested layer depths. Crucially, its performance exhibits remarkable stability as the number of layers is progressively increased. • RFHND shows a stable Dirichlet Energy trajectory on above three datasets, while comparison methods experience a rapid energy decline as depth increases. This is consistent with our theoretical analysis in Section IV-A. (b) (a) shows RFHND accuracy on NTU2012 and ModelNet40 with various hidden dimensions. (b) shows RFHND accuracy on NTU2012 and ModelNet40 with various τ. (c) (a)(b)(c) show the visualization of Pubmed node features at t=0t=0, t=1t=1 and t=2t=2. (d)(e)(f) show the visualization of DBLP node features at t=0t=0, t=1t=1 and t=2t=2. VI-E Robustness analysis To assess the robustness of RFHND under noisy input conditions, extensive experiments are conducted considering both feature-level and structure-level perturbations. The experimental configurations are as follows: • Feature Noise: Node features in the Citeseer dataset are perturbed using three types of noise: (a) Gaussian noise sampled from N(0,σ2)N(0,σ^2) and scaled according to the noise rate; (b) Uniform noise drawn from Unif(−δ,δ)Unif(-δ,δ), where δ is proportional to the noise rate; (c) Mask noise, in which a portion of feature values is randomly set to zero based on the noise rate. • Structure Noise: For the Cora-CA dataset, the hypergraph structure is disturbed by randomly deleting existing hyperedges and introducing hyperedges composed of random node subsets, with the overall perturbation controlled by the noise rate. As illustrated in Fig. 11(a), the proposed approach consistently surpasses baseline methods under all noise settings. When exposed to feature-level perturbations, RFHND demonstrates strong robustness on the Citeseer dataset, maintaining stable performance across different noise types, including Gaussian, uniform, and mask-based disturbances. Notably, all examined models show higher tolerance to mask noise compared with other feature perturbations, possibly because mask noise introduces structured sparsity rather than random distortion, allowing models to better adapt during training. The analysis of structural noise demonstrates comparable benefits on the Cora-CA dataset. Across all examined noise levels (0.1–0.4), the proposed method consistently delivers better performance. While the accuracy of all models decreases as the noise rate grows, RFHND shows only a marginal decline, reflecting its strong robustness against structural disturbances. VI-F Parameter analysis To assess the sensitivity of RFHND to key hyperparameters, an extensive analysis was performed focusing on two crucial factors: the hidden layer dimension and the step size τ. In particular, the model’s performance was evaluated under different configurations of hidden layer dimensions 16,32,64,128,256 \16,32,64,128,256 \ and step sizes 0.2,0.5,1,2,4 \0.2,0.5,1,2,4 \. Specifically, we selected three representative datasets: Zoo, NTU2012 and ModelNet40. As illustrated in Fig. 11(b), RFHND remains relatively stable under different hidden layers. However, when the step size becomes excessively large, the model performance deteriorates, indicating that overly large steps may cause unstable feature propagation and weaken the model’s capacity to capture fine-grained relational patterns. This is consistent with the conclusion of Theorem 6. VI-G Feature visualization To gain deeper insights into the dynamic changes of node representations, a feature visualization analysis was conducted on the Citeseer and Pubmed datasets. In this analysis, node features were captured at multiple time points during the ODE integration process, specifically at t=0t=0, t=1t=1, and t=2t=2. The obtained high-dimensional representations were then mapped onto a two-dimensional space using a dimensionality reduction method for visualization purposes. In the resulting visualizations, nodes are color-coded according to their corresponding class labels. As illustrated in Fig. 11(c), it can be seen that the node embeddings gradually form clearer and more compact clusters as time advances. This phenomenon suggests that the model progressively enhances its ability to distinguish nodes from different classes, with the representations evolving into more separable and discriminative structures throughout the integration process. VII Conclusion In this paper, we mitigated the critical challenge of over-smoothing in hypergraph neural networks by generalizing Ricci flow from differential geometry to the hypergraph domain. Specifically, we proposed RFHND (Ricci Flow-guided Hypergraph Neural Diffusion), a novel message passing paradigm that leverages discrete Ricci flow to adaptively regulate information propagation. By formulating node feature evolution as a continuous PDE-based system, our method effectively controls the diffusion rate at the geometric level, thereby preventing feature homogenization. Furthermore, we provided a rigorous theoretical analysis validating the convergence properties and approximation capabilities of our framework. Extensive experiments across multiple benchmark datasets demonstrate that RFHND significantly outperforms state-of-the-art methods and exhibits strong robustness. 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Appendix A HYPEREDGE CURVATURE ON WEIGHTED HYPERGRAPHS Discrete curvature is a key tool for analyzing the geometric properties of networks. In recent years, multiple studies have successfully generalized this concept from graph data to more complex hypergraphs. This section will focus on elaborating two definitions of curvature defined on hyperedges. A-A Forman-Ricci Curvature on Hypergraphs Forman-Ricci (FR) curvature [31], first introduced on cell complexes, has now been extended to hypergraphs. The curvature characterizes the balance between a hyperedge’s internal structure and its external connections, essentially measuring whether it functions as a community core (high curvature) or a bridge (low curvature). A common definition for the Forman-Ricci curvature of a hyperedge e∈ℰe is given by: κeFR=∑i∈ewi−we∑k∈e,el∼kwkwel,κ^FR_e= _i∈ ew_i-w_e _k∈ e,e_l k w_kw_e_l, (23) where wew_e is the weight of the hyperedge e and wiw_i is the weight of the node i. el∼ke_l k represent the represents the hyperedge associated with node k. In this paper, we consider the weights of all nodes as 1. Then, the curvature expression mentioned above can be written in the following form: κeFR=|e|−∑k∈e,el∼kwewel.κ^FR_e=|e|- _k∈ e,e_l k w_e w_e_l. (24) A-B Ollivier-Ricci Curvature on Hypergraphs Ollivier-Ricci(OR) curvature [32], first introduced using optimal transport theory, has now been extended to hypergraphs. The curvature characterizes network robustness by measuring the transport cost between local neighborhood distributions; a low cost implies a robust, community-like structure (high curvature), while a high cost indicates a fragile bridge (low curvature). To define OR curvature on a hypergraph, we first need to define a probability measure μi _i for each node i∈Vi∈ V. This typically represents a two-step random walk from the node. The standard definition is: μi(j)=∑e⊇i,j(we∑i∋fwf⋅1|e|), _i(j)= _e \i,j\ ( w_e _i fw_f· 1|e| ), (25) where j is a neighbor of i , and we′w_e is the weight of the hyperedge e. The OR curvature of the hyperedge e is then defined as: κeOR=1−2|e|(|e|−1)∑i,j∈eW1(μi,μj)d(i,j),κ^OR_e=1- 2|e|(|e|-1) _i,j∈ e W_1( _i, _j)d(i,j), (26) where W1(μi,μj)W_1( _i, _j) is the 1-Wasserstein distance between the two probability distributions, and d(i,j)d(i,j) is the shortest path distance between nodes i and j in the hypergraph. Appendix B PROOF OF THEOREM 1 Lemma 7. Consider the attribute discrete Ricci flow over the interval [t1,t2][t_1,t_2]. If the hyperedge weight we(t)w_e(t) has a finite number of N extrema points within (t1,t2)(t_1,t_2), denoted by T1,…,TNT_1,…,T_N, and we define T0=t1T_0=t_1 and TN+1=t2T_N+1=t_2, then: ρe−1ζet∈[t1,t2](|κe(t)|)>1−ρe−1t2−t1, _e-1 _eE_t∈[t_1,t_2](| _e(t)|)> 1- _e^-1t_2-t_1, (27) where ζe=min0≤i≤N(Ti+1−Ti) _e= _0≤ i≤ N(T_i+1-T_i) represents the length of the minimum monotonic interval within [t1,t2][t_1,t_2], and ρe=wemaxwemin _e= w_e^maxw_e^min denotes the ratio of the maximum to the minimum value of we(t)w_e(t). Specifically, if we(t)w_e(t) is monotonic within (t1,t2)(t_1,t_2), we have: 1minwe(t2),we(t1)|we(t2)−we(t1)|t2−t1>t∈[t1,t2](|κe(t)|), split& 1 \w_e(t_2),w_e(t_1)\ |w_e(t_2)-w_e(t_1)|t_2-t_1\\ &>E_t∈[t_1,t_2](| _e(t)|), split (28) this shows that |we(t2)−we(t1)|→0|w_e(t_2)-w_e(t_1)|→ 0 implies t∈[t1,t2](|κe(t)|)→0E_t∈[t_1,t_2](| _e(t)|)→ 0. Proof. We integrate Equation (13) over an arbitrary time interval [t1,t2][t_1,t_2]: we(t2)−we(t1)=−∫t1t2κe(t)we(t)t.w_e(t_2)-w_e(t_1)=- _t_1^t_2 _e(t)w_e(t)dt. (29) Suppose that within [t1,t2][t_1,t_2], the function we(t)w_e(t) has N extremum points. These N points divide the interval [t1,t2][t_1,t_2] into N+1N+1 adjacent monotonic sub-intervals. Given that we(t)>0w_e(t)>0, the sign of ∂we(t)∂t ∂ w_e(t)∂ t is determined by κe(t) _e(t). Moreover, the κe(t) _e(t) sign remains non-negative or non-positive within each monotonic sub-interval. Let [Ti,Ti+1][T_i,T_i+1] denote any one of these N+1N+1 monotonic sub-intervals. If κe(t)≥0 _e(t)≥ 0 for t∈[Ti,Ti+1]t∈[T_i,T_i+1], then we(t)w_e(t) is monotonically decreasing, which implies we(Ti)>we(Ti+1)>0w_e(T_i)>w_e(T_i+1)>0. Therefore: 0>−we(Ti+1)∫TiTi+1κe(t)t>−∫TiTi+1κe(t)we(t)t>−we(Ti)∫TiTi+1κe(t)t. split&0>-w_e(T_i+1) _T_i^T_i+1 _e(t)dt>- _T_i^T_i+1 _e(t)w_e(t)dt\\ &>-w_e(T_i) _T_i^T_i+1 _e(t)dt. split (30) If κe(t)≤0 _e(t)≤ 0 for t∈[Ti,Ti+1]t∈[T_i,T_i+1], then we(t)w_e(t) is monotonically increasing, which implies we(Ti+1)>we(Ti)>0w_e(T_i+1)>w_e(T_i)>0 and −we(Ti+1)∫TiTi+1κe(t)t>−∫TiTi+1κe(t)we(t)t>−we(Ti)∫TiTi+1κe(t)t>0. split&-w_e(T_i+1) _T_i^T_i+1 _e(t)dt>- _T_i^T_i+1 _e(t)w_e(t)dt\\ &>-w_e(T_i) _T_i^T_i+1 _e(t)dt>0. split (31) Therefore, on any monotonic sub-interval, we have: −we(Ti+1)∫TiTi+1κe(t)t>we(Ti+1)−we(Ti)>−we(Ti)∫TiTi+1κe(t)t. split&-w_e(T_i+1) _T_i^T_i+1 _e(t)dt>w_e(T_i+1)-w_e(T_i)\\ &>-w_e(T_i) _T_i^T_i+1 _e(t)dt. split (32) Take the absolute value of the above inequality: maxwe(Ti),we(Ti+1)∫TiTi+1|κe(t)|t>|we(Ti+1)−we(Ti)|>minwe(Ti),we(Ti+1)∫TiTi+1|κe(t)|t. split& \w_e(T_i),w_e(T_i+1)\ _T_i^T_i+1| _e(t)|dt\\ &>|w_e(T_i+1)-w_e(T_i)|\\ &> \w_e(T_i),w_e(T_i+1)\ _T_i^T_i+1| _e(t)|dt. split (33) Let wemax>maxwe(Ti),we(Ti+1)>minwe(Ti),we(Ti+1)>wemin, split&w_e > \w_e(T_i),w_e(T_i+1)\\\ &> \w_e(T_i),w_e(T_i+1)\>w_e , split (34) we have: |we(Ti+1)−we(Ti)|wemin>∫TiTi+1|κe(t)|t>|we(Ti+1)−we(Ti)|wemax. split& |w_e(T_i+1)-w_e(T_i)|w_e > _T_i^T_i+1| _e(t)|dt\\ &> |w_e(T_i+1)-w_e(T_i)|w_e . split (35) Summing over i from 0 to N and then dividing by t2−t1t_2-t_1: ∑i=0N|we(Ti+1)−we(Ti)|wemin(t2−t1)>1t2−t1∫t1t2|κe(t)|t=t∈[t1,t2](|κe(t)|)>∑i=0N|we(Ti+1)−we(Ti)|wemax(t2−t1). split& _i=0^N|w_e(T_i+1)-w_e(T_i)|w_e (t_2-t_1)> 1t_2-t_1 _t_1^t_2| _e(t)|dt\\ &=E_t∈[t_1,t_2](| _e(t)|)> _i=0^N|w_e(T_i+1)-w_e(T_i)|w_e (t_2-t_1). split (36) Because (N+1)(wemax−wemin)>∑i=0N|we(Ti+1)−we(Ti)|>wemax−wemin(N+1)(w_e -w_e )> _i=0^N|w_e(T_i+1)-w_e(T_i)|>w_e -w_e , and ρe=wemaxwemin _e= w_e w_e : (N+1)(ρe−1)t2−t1>t∈[t1,t2](|κe(t)|)>1−ρe−1t2−t1. (N+1)( _e-1)t_2-t_1>E_t∈[t_1,t_2](| _e(t)|)> 1- _e^-1t_2-t_1. (37) According to the definition of ζe _e, we have (t2−t1)>ζe(N+1)(t_2-t_1)> _e(N+1). Thus, the estimation of the upper bound can be replaced by ζe−1(ρe−1)>(N+1)(ρe−1)t2−t1 _e^-1( _e-1)> (N+1)( _e-1)t_2-t_1. Thus, ρe−1ζet∈[t1,t2](|κe(t)|)>1−ρe−1t2−t1. _e-1 _eE_t∈[t_1,t_2](| _e(t)|)> 1- _e^-1t_2-t_1. (38) ∎ Theorem 8. Consider the attribute discrete Ricci flow with hyperedge weight as equation (6), and |(t)|≡1|x(t)|≡ 1. If for any e in ℋH, we(t)w_e(t) has a finite number of monotonic on [t1,t2][t_1,t_2], then the average Dirichlet energy within [t1,t2][t_1,t_2] has following bound: B1≥t∈[t1,t2](E((t))≥B2,B_1 _t∈[t_1,t_2](E(X(t))≥ B_2, (39) where B1=(t2−t1)∑e∈ℰ(ceρeζe−1)−∑e∈ℰ(αeϵρe),B2=∑e∈ℰ(ce−(2+ϵ)αeζe−1(t2−t1)). splitB_1=(t_2-t_1) _e (c_e _e _e^-1 )- _e ( _eε _e ),\\ B_2= _e (c_e- (2+ε ) _e _e^-1(t_2-t_1) ). split (40) Here, ce=∑i,j∈e(12|e|(1di+1dj)+(1+ϵ)αe)c_e= _i,j∈ e ( 12|e| ( 1d_i+ 1d_j )+ (1+ε ) _e ), did_i denotes the degree of node i, ρe _e denotes the ratio of the maximum to the minimum value of we(t)w_e(t) and ζe _e represents the length of the minimum monotonic interval of we(t)w_e(t) within [t1,t2][t_1,t_2]. Proof. As equation (1) shows E(X(t))=12∑e∈ℰ∑i,j∈e1|e|(i(t)di−j(t)dj)2=12∑e∈ℰ∑i,j∈e1|e|(1di+1dj−2i(t)Tj(t)didj)=∑e∈ℰ(∑i,j∈e12|e|(1di+1dj)+(1+ϵ)αe)−∑e∈ℰαewe(t). splitE(X(t))&= 12 _e _i,j∈ e 1|e| ( x_i(t) d_i- x_j(t) d_j )^2\\ &= 12 _e _i,j∈ e 1|e| ( 1d_i+ 1d_j-2 x_i(t)^Tx_j(t) d_id_j )\\ &= _e ( _i,j∈ e 12|e| ( 1d_i+ 1d_j )+ (1+ε ) _e )\\ &- _e _ew_e(t). split (41) Here,we define ce=∑i,j∈e12|e|(1di+1dj)+(1+ϵ)αec_e= _i,j∈ e 12|e| ( 1d_i+ 1d_j )+ (1+ε ) _e and Ee(t)=ce−αewe(t),E_e(t)=c_e- _ew_e(t), then the Dirichlet energy can be recorded as: E((t))=∑e∈ℰEe=∑e∈ℰ(ce−αewe(t)).E(X(t))= _e E_e= _e (c_e- _ew_e(t) ). (42) Take the partial derivative of t and substitute the equation into Ricci flow: ∂E((t))∂t=∑e∈ℰ(−αe∂we(t)∂t)=∑e∈ℰ(−αe(−κe(t)we(t)))=∑e∈ℰκe(t)(ce−Ee(t)). split ∂ E(X(t))∂ t&= _e (- _e ∂ w_e(t)∂ t )\\ &= _e (- _e (- _e(t)w_e(t) ) )\\ &= _e _e(t) (c_e-E_e(t) ). split (43) By transferring the above results to a hyperedge, we can obtain the following formula: ∂Ee((t))∂t=κe(t)(ce−Ee(t)). ∂ E_e(X(t))∂ t= _e(t)(c_e-E_e(t)). (44) Integrating the above formula within the interval [t1,t2][t_1,t_2] : Ee(t2)−Ee(t1)=ce∫t1t2κe(t)t−∫t1t2κe(t)Ee(t)t.E_e(t_2)-E_e(t_1)=c_e _t_1^t_2 _e(t)dt- _t_1^t_2 _e(t)E_e(t)dt. (45) Then,we consider the N extremum points of we(t)w_e(t) within (t1,t2):T1,…,TN.(t_1,t_2):T_1,...,T_N. For any i, ke(t)k_e(t) does not change its sign within (Ti,Ti+1)(T_i,T_i+1), and Ee≥0E_e≥ 0, so we have |∫TiTi+1ke(t)Ee(t)t|=∫TiTi+1|ke(t)|Ee(t)t| _T_i^T_i+1k_e(t)E_e(t)dt|= _T_i^T_i+1|k_e(t)|E_e(t)dt, so: |Ee(Ti+1)−Ee(Ti)|=|ce∫TiTi+1κe(t)t−∫TiTi+1κe(t)Ee(t)t|=|ce∫TiTi+1κe(t)t|−|∫TiTi+1κe(t)Ee(t)t|=ce∫TiTi+1|κe(t)|t−∫TiTi+1|κe(t)|Ee(t)t≈ce∫TiTi+1|κe(t)|t−t∈[t1,t2]|κe(t)|∫TiTi+1Ee(t)t. split&|E_e(T_i+1)-E_e(T_i)|\\ &=|c_e _T_i^T_i+1 _e(t)dt- _T_i^T_i+1 _e(t)E_e(t)dt|\\ &=|c_e _T_i^T_i+1 _e(t)dt|-| _T_i^T_i+1 _e(t)E_e(t)dt|\\ &=c_e _T_i^T_i+1| _e(t)|dt- _T_i^T_i+1| _e(t)|E_e(t)dt\\ &≈ c_e _T_i^T_i+1| _e(t)|dt\\ &-E_t∈[t_1,t_2]| _e(t)| _T_i^T_i+1E_e(t)dt. split (46) Sum over i from 0 to N and then divide by t2−t1t_2-t_1: 1t2−t1∑i=0N|Ee(Ti+1)−Ee(Ti)|≥t∈[t1,t2]|κe(t)|(ce−t∈[t1,t2](Ee(t))). split& 1t_2-t_1 _i=0^N|E_e(T_i+1)-E_e(T_i)|\\ & _t∈[t_1,t_2]| _e(t)| (c_e-E_t∈[t_1,t_2](E_e(t)) ). split (47) Let ζe=min0≤i≤N(Ti+1−Ti) _e=min_0≤ i≤ N(T_i+1-T_i) represents the length of the minimum monotonic interval within [t1,t2][t_1,t_2], so (N+1)ζe≤t2−t1(N+1) _e≤ t_2-t_1 and we(t)≤2+ϵw_e(t)≤ 2+ε, then we have: 1t2−t1∑i=0N|Ee(Ti+1)−Ee(Ti)|=1t2−t1∑i=0N|αewe(Ti+1)−αewe(Ti)|≤αe(N+1)(maxwe−minwe)t2−t1≤ζe−1αemaxwe(1−ρe−1)≤(2+ϵ)ζe−1αe(1−ρe−1). split& 1t_2-t_1 _i=0^N|E_e(T_i+1)-E_e(T_i)|\\ &= 1t_2-t_1 _i=0^N| _ew_e(T_i+1)- _ew_e(T_i)|\\ &≤ _e(N+1)( w_e- w_e)t_2-t_1\\ &≤ _e^-1 _e w_e(1- _e^-1)\\ &≤(2+ε) _e^-1 _e(1- _e^-1). split (48) Here, ρe _e denotes the ratio of the maximum to the minimum value of we(t)w_e(t). Thus, we have: (2+ϵ)ζe−1αe(1−ρe−1)≥t∈[t1,t2]|κe(t)|(ce−t∈[t1,t2](Ee(t))). split&(2+ε) _e^-1 _e(1- _e^-1)\\ & _t∈[t_1,t_2]| _e(t)| (c_e-E_t∈[t_1,t_2](E_e(t)) ). split (49) Then we can obtain the lower bound for t∈[t1,t2](Ee(t))E_t∈[t_1,t_2](E_e(t)): t∈[t1,t2](Ee(t))≥ce−(2+ϵ)ζe−1(1−ρe−1)αet∈[t1,t2](|ke(t)|)≥ce−(2+ϵ)αeζe−1(t2−t1). splitE_t∈[t_1,t_2](E_e(t))&≥ c_e- (2+ε) _e^-1(1- _e^-1) _eE_t∈[t_1,t_2](|k_e(t)|)\\ &≥ c_e- (2+ε ) _e _e^-1(t_2-t_1). split (50) Take the sum over all edge: t∈[t1,t2](E(t))≥∑e∈ℰ(ce−(2+ϵ)αeζe−1(t2−t1)). splitE_t∈[t_1,t_2](E(t))&≥ _e (c_e- (2+ε ) _e _e^-1(t_2-t_1) ). split (51) Now let’s derive the upper bound of the energy value. From equation (46), we can obtained: ∫TiTi+1|κe(t)|Ee(t)t=ce∫TiTi+1|κe(t)|t−|Ee(Ti+1)−Ee(Ti)|. split& _T_i^T_i+1| _e(t)|E_e(t)dt\\ &=c_e _T_i^T_i+1| _e(t)|dt-|E_e(T_i+1)-E_e(T_i)|. split (52) Sum over i from 0 to N: ∫t1t2|κe(t)|Ee(t)t=ce∫t1t2|κe(t)|dt−∑i=0N|Ee(Ti+1)−Ee(Ti))|=ce∫t1t2|κe(t)|t−∑i=0Nαe|we(Ti+1)−we(Ti)|≤ce∫t1t2|κe(t)|t−αe|maxwe−minwe|≤ce∫t1t2|κe(t)|t−αeϵ(ρe−1). split& _t_1^t_2| _e(t)|E_e(t)dt\\ &=c_e _t_1^t_2| _e(t)|dt- _i=0^N|E_e(T_i+1)-E_e(T_i))|\\ &=c_e _t_1^t_2| _e(t)|dt- _i=0^N _e|w_e(T_i+1)-w_e(T_i)|\\ &≤ c_e _t_1^t_2| _e(t)|dt- _e| w_e- w_e|\\ &≤ c_e _t_1^t_2| _e(t)|dt- _eε( _e-1). split (53) Divide both sides by t2−t1t_2-t_1: t∈[t1,t2](|κe(t)|Ee(t))≤cet∈[t1,t2](|κe(t)|)−αeϵ(ρe−1−1)(t2−t1). split&E_t∈[t_1,t_2] (| _e(t)|E_e(t) )\\ &≤ c_eE_t∈[t_1,t_2](| _e(t)|)- _eε( _e^-1-1)(t_2-t_1). split (54) Applying Lemma 7, we obtain: 1−ρe−1t2−t1t∈[t1,t2](Ee(t))≤ceζe−1(ρe−1)−αeϵ(ρe−1)(t2−t1). split& 1- _e^-1t_2-t_1E_t∈[t_1,t_2](E_e(t))\\ &≤ c_e _e^-1( _e-1)- _eε( _e-1)(t_2-t_1). split (55) Due to E(t)=∑e∈ℰEe(t)E(t)= _e E_e(t), we can derive the upper bound as: t∈[t1,t2](E(t))≤(t2−t1)∑e∈ℰ(ceρeζe−1)−∑e∈ℰ(αeϵρe). splitE_t∈[t_1,t_2](E(t))&≤(t_2-t_1) _e (c_e _e _e^-1 )- _e ( _eε _e ). split (56) Combining the results of Equation (51) and Equation (56), we have: (t2−t1)∑e∈ℰ(ceρeζe−1)−∑e∈ℰ(αeϵρe)≥t∈[t1,t2](E(t))≥∑e∈ℰ(ce−(2+ϵ)αeζe−1(t2−t1)). split&(t_2-t_1) _e (c_e _e _e^-1 )- _e ( _eε _e )\\ & _t∈[t_1,t_2](E(t))\\ &≥ _e (c_e- (2+ε ) _e _e^-1(t_2-t_1) ).\\ split (57) ∎ Based on the above Theorem 8, we can draw the conclusion of Theorem 1, The specific proof is as follows: Proof. When H is a non-regular hypergraph and for all e in ℋH, we(t)w_e(t) are monotonic on [t1,t2][t_1,t_2]. So ζe=t2−t1. _e=t_2-t_1. The upper bound of the Theorem 8 can be written as: (t2−t1)∑e∈ℰ(ceρeζe−1)−∑e∈ℰ(αeϵρe)=(t2−t1)∑e∈ℰρeζe−1(∑i,j∈e12|e|(1di+1dj)+(1+ϵ)αe)−∑e∈ℰ(αeϵρe)=(t2−t1)∑e∈ℰρeζe−1(∑i,j∈e12|e|(1di+1dj))+∑e∈ℰ((t2−t1)ρeζe−1(1+ϵ)−ϵρe)αe=∑e∈ℰρe(∑i,j∈e12|e|(1di+1dj))+∑e∈ℰρeαe≤ρmax(∑e∈ℰ∑i∈e,j∈e12|e|(1di+1dj)+∑e∈ℰαe). split&(t_2-t_1) _e (c_e _e _e^-1 )- _e ( _eε _e )\\ &=(t_2-t_1) _e _e _e^-1 ( _i,j∈ e 12|e| ( 1d_i+ 1d_j )+ (1+ε ) _e )\\ &- _e ( _eε _e )\\ &=(t_2-t_1) _e _e _e^-1 ( _i,j∈ e 12|e| ( 1d_i+ 1d_j ) )\\ &+ _e ((t_2-t_1) _e _e^-1 (1+ε )-ε _e ) _e\\ &= _e _e ( _i,j∈ e 12|e| ( 1d_i+ 1d_j ) )\ + _e _e _e\\ &≤ _max ( _e _i∈ e,j∈ e 12|e| ( 1d_i+ 1d_j )+ _e _e ). split (58) Similarly, the lower bound can be written as: ∑e∈ℰ(ce−(2+ϵ)αeζe−1(t2−t1))=∑e∈ℰ(∑i,j∈e12|e|(1di+1dj)+(1+ϵ)αe)−∑e∈ℰ(2+ϵ)αeζe−1(t2−t1)=∑e∈ℰ(∑i,j∈e12|e|(1di+1dj))−∑e∈ℰαe. split& _e (c_e- (2+ε ) _e _e^-1(t_2-t_1) )\\ &= _e ( _i,j∈ e 12|e| ( 1d_i+ 1d_j )+ (1+ε ) _e )\\ &- _e (2+ε ) _e _e^-1(t_2-t_1)\\ &= _e ( _i,j∈ e 12|e| ( 1d_i+ 1d_j ) )\ - _e _e. split (59) From Equation (6),we have αe=1|e|∑i∈e,j∈e1didj _e= 1|e| _i∈ e,j∈ e 1 d_id_j, So ∑i,j∈e12|e|(1di+1dj)−αe=∑i,j∈e(12|e|(1di+1dj)−1|e|1didj)≥0. split& _i,j∈ e 12|e| ( 1d_i+ 1d_j )\ - _e\\ &= _i,j∈ e ( 12|e| ( 1d_i+ 1d_j )\ - 1|e| 1 d_id_j )≥ 0. split (60) Because ℋH is a non-regular hypergraph, so the lower bound is greater than 0. ∎ Appendix C PROOF OF Proposition 3 Proof. From the condition |i(t)|≡1|x_i(t)|≡ 1, we know that ∂t‖i(t)‖2=0 _t\|x_i(t)\|^2=0, which expands using the chain rule to: 2⟨i(t),∂i(t)∂t⟩=0.2 _i(t), _i(t)∂ t =0. (61) Moreover, due to we(t)≡1αe[1|e|∑i∈e,j∈ecos(i(t),j(t))didj]+1+ϵw_e(t)≡ 1 _e[ 1|e| _i∈ e,j∈ e cos(x_i(t),x_j(t)) d_id_j]+1+ε, we know that, for any e that i∈ei∈ e: ∂we(t)∂i=1∑i,j∈e1didj(∑j∈ej(t)didj). ∂ w_e(t) _i= 1 _i,j∈ e 1 d_id_j ( _j∈ e x_j(t) d_id_j ). (62) We denote ∂i(t)∂t _i(t)∂ t as y. Let μie=−κe(t)we(t)1+∑j∈e,j≠iλj(1(t),…|e|(t)) _ie=- _e(t)w_e(t)1+ _j∈ e,j≠ iλ^j(x_1(t),...x_|e|(t)). Then, the original problem takes the following form: min T ^Ty s.t. 1∑i,j∈e1didj(∑j∈ej(t)Tdidj)=μie, 1 _i,j∈ e 1 d_id_j ( _j∈ e x_j(t)^T d_id_j )y= _ie, (63) i(t)T=0. _i(t)^Ty=0. (64) we denote mie=1∑i,j∈e1didj(∑j∈ej(t)didj).m_ie= 1 _i,j∈ e 1 d_id_j ( _j∈ e x_j(t) d_id_j ). (65) Then consider the method of Lagrange multipliers: ∇yL() _yL(y) =0 =0 →∇y(T+λ(mieT−μie)+τi(t)T) → _y(y^Ty+λ(m_ie^Ty- _ie)+ _i(t)^Ty) =0 =0 →2+λmie+τi(t) → 2y+λ m_ie+ _i(t) =0. =0. (66) Left-multiplying Equation (C) by i(t)Tx_i(t)^T, and combining it with Equation (C) and i(t)Ti(t)=1x_i(t)^Tx_i(t)=1, we get: τ=−λi(t)Tmie.τ=- _i(t)^Tm_ie. (67) Similarly, left-multiplying Equation (64) by mieTm_ie^T: 2mieT+λ+τmieTi(t)=0 2m_ie^Ty+λ+τ m_ie^Tx_i(t)=0 →2μie+λ−λ(i(t)Tmie)=0 → 2 _ie+λ-λ(x_i(t)^Tm_ie)=0 →λ=−2μie(1−(i(t)Tmie)2)−1. → λ=-2 _ie (1-(x_i(t)^Tm_ie)^2 )^-1. (68) Thus we can obtain the solution to this optimization problem: ∗=−λmie+τi(t)2=−λmie−λi(t)Tmiei(t)2=−λ(−i(t)i(t)T)mie2=μie(−i(t)i(t)T)mie1−(i(t)Tmie)2=μie[mie−cos(i(t),mie)i(t)]1−(i(t)Tmie)2. splity^*&=- λ m_ie+ _i(t)2\\ &=- λ m_ie- _i(t)^Tm_iex_i(t)2\\ &=- λ(I-x_i(t)x_i(t)^T)m_ie2\\ &= _ie(I-x_i(t)x_i(t)^T)m_ie1-(x_i(t)^Tm_ie)^2\\ &= _ie [m_ie- (x_i(t),m_ie)x_i(t) ]1-(x_i(t)^Tm_ie)^2. split (69) So, ∗=μie−(i(t)Tmie)2[mie−cos(i(t),mie)i(t)]=μie−(i(t)Tmie)2[1∑i,j∈e1didj(∑i,j∈ej(t)didj)−cos(i(t),mie)i(t)]=μie(−(i(t)Tmie)2)(∑i,j∈e1didj)[(∑j∈ej(t)didj)−∑j∈ecos(i(t),mie)i(t)didj]=−κie′(t)[∑j∈ej(t)−cos(i(t),mie)i(t)didj]. splity^*&= _ie1-(x_i(t)^Tm_ie)^2 [m_ie- (x_i(t),m_ie)x_i(t) ]\\ &= _ie1-(x_i(t)^Tm_ie)^2\\ & [ 1 _i,j∈ e 1 d_id_j ( _i,j∈ e x_j(t) d_id_j )- (x_i(t),m_ie)x_i(t) ]\\ &= _ie(1-(x_i(t)^Tm_ie)^2)( _i,j∈ e 1 d_id_j)\\ & [ ( _j∈ e x_j(t) d_id_j )- _j∈ e (x_i(t),m_ie)x_i(t) d_id_j ]\\ &=-κ _ie(t) [ _j∈ e x_j(t)- (x_i(t),m_ie )x_i(t) d_id_j ]. split (70) Here, κie′(t)=μie(−(i(t)Tmie)2)(∑i,j∈e1didj)κ _ie(t)= _ie(1-(x_i(t)^Tm_ie)^2)( _i,j∈ e 1 d_id_j). The ‖≥0\|y\|≥ 0 always holds. Therefore, ∗y^* corresponds to the point of minimum y. ∎ Appendix D proof of Theorem 4 Lemma 9. C((t))C(X(t)) is locally Lipschitz continuous. Proof. Given that ‖i‖=1\|x_i\|=1, the gradient norm of cos(i,j) (x_i,x_j) with respect to (i,j)(x_i,x_j) is bounded. Thus, there exists a constant LCL_C and bounded sets such that ‖C((t))−C((t))‖F≤LC‖(t)−(t)‖F.\|C(X(t))-C(Y(t))\|_F≤ L_C\|X(t)-Y(t)\|_F. (71) So, C((t))C(X(t)) is locally Lipschitz continuous. ∎ Lemma 10. Under the assumption of Theorem 4, the functions miem_ie, μie _ie, and κie′(t) _ie (t) are bounded and Lipschitz continuous. Proof. We prove the boundedness and Lipschitz continuity for each function based on the assumptions of Theorem 4. 1. miem_ie is bounded and Lipschitz continuous: By the definition of miem_ie, it can be written as a convex combination of xjx_j: mie=∑jαjj,where αj≥0 and ∑jαj=1.m_ie= _j _jx_j, _j≥ 0 and _j _j=1. (72) From this convex combination, we immediately obtain the Lipschitz property: ‖mie((t))−mie((t))‖F≤‖(t)−(t)‖F.\|m_ie(X(t))-m_ie(Y(t))\|_F≤\|X(t)-Y(t)\|_F. (73) Furthermore, since the vector norm is bounded by unity, miem_ie is also bounded. 2. μie _ie is bounded and Lipschitz continuous: The function μie _ie is defined as: μie=−κe(t)we(t)1+∑j∈e,j≠iλj(1(t),…|e|(t)). _ie=- _e(t)w_e(t)1+ _j∈ e,j≠ iλ^j(x_1(t),...x_|e|(t)). (74) Since κe(t) _e(t) and we(t)w_e(t) are bounded, and λj(⋅)λ^j(·) is bounded and locally Lipschitz, μie _ie is clearly bounded on bounded sets and locally Lipschitz. 3. κie′(t) _ie (t) is bounded and Lipschitz continuous: The function κie′(t) _ie (t) is given by: κie′(t) κ _ie(t) =μie(1−(i(t)Tmie)2)Se, = _ie(1-(x_i(t)^Tm_ie)^2)S_e, (75) where Se S_e =∑p,q∈e1dpdq∈[|e|2dmax,|e|2dmin]. = _p,q∈ e 1 d_pd_q∈ [ |e|^2d_ , |e|^2d_ ]. (76) From the result in part 2, μie _ie is bounded and locally Lipschitz. Assumption guarantees that the denominator, (1−(i(t)Tmie)2)Se(1-(x_i(t)^Tm_ie)^2)S_e is bounded below by the strictly positive quantity εSe S_e, where ε>0 >0. Therefore, κie′(t)κ _ie(t) is the ratio of two bounded, locally Lipschitz functions whose denominator is bounded away from zero, implying κie′(t)κ _ie(t) is also bounded on bounded sets and locally Lipschitz. ∎ Lemma 11. The function S((t))S(X(t)) is locally Lipschitz continuous and bounded. Proof. The matrix S((t))=De−12HK′((t))HTDe−12S(X(t))=D_e^- 12HK (X(t))H^TD_e^- 12 is linear with respect to K′((t))K (X(t)). By Lemma 10 , each diagonal element of K′((t))K (X(t)) is locally Lipschitz. Therefore, S((t))S(X(t)) is locally Lipschitz. Furthermore, S((t))S(X(t)) is bounded: ∥S((t))∥≤cHmaxe|κe′((t))|:=M(t).\|S(X(t))\|≤ c_H _e|κ _e(X(t))| :=M(t). (77) where cHc_H is a constant independent of time t. ∎ Based on the above Lemma, we can draw the conclusion of Theorem 4, The specific proof is as follows: Proof. Let F((t))=diag((S((t))⊙C((t)))N)−S((t)),F(X(t))=diag ( (S(X(t)) C(X(t)) )1_N )-S(X(t)), (78) G((t))=[diag((S((t))⊙C((t)))N)−S((t))](t).G(X(t))= [diag ( (S(X(t)) C(X(t)) )1_N )-S(X(t)) ]X(t). (79) Combining Lemmas 9 and 11 with the bound |C((t))|≤1|C(X(t))|≤ 1, it follows that there exists a constant LH(t)L_H(t) such that the following inequality holds on the bounded set: ‖F((t))−F((t))‖≤LH(t)‖(t)−(t)‖,\|F(X(t))-F(Y(t))\|≤ L_H(t)\|X(t)-Y(t)\|, (80) ‖F((t))‖≤2‖S((t))‖≤2M(t).\|F(X(t))\|≤ 2\|S(X(t))\|≤ 2M(t). (81) Then, the Lipschitz continuity of G((t))G(X(t)) is established as follows: ∥G((t))−G((t))∥=‖F((t))(t)−F((t))(t)‖≤‖F((t))(t)−F((t))(t)‖+‖F((t))(t)−F((t))(t)‖≤‖F((t))−F((t))‖‖(t)‖+‖F((t))‖‖(t)−(t)‖≤(2M(t)+LH(t)‖(t)‖)‖(t)−(t)‖. split\|G&(X(t))-G(Y(t))\|\\ =&\|F(X(t))X(t)-F(Y(t))Y(t)\|\\ ≤&\|F(X(t))X(t)-F(Y(t))X(t)\|\\ &+\|F(Y(t))X(t)-F(Y(t))Y(t)\|\\ ≤&\|F(X(t))-F(Y(t))\|\|X(t)\|+\|F(Y(t))\|\|X(t)-Y(t)\|\\ ≤&(2M(t)+L_H(t)\|Y(t)\|)\|X(t)-Y(t)\|. split (82) So, G((t))G(X(t)) is locally Lipschitz continuous. Furthermore, the linear growth bound is given by: ‖G((t))‖≤‖F((t))‖‖(t)‖≤2M(t)‖(t)‖.\|G(X(t))\|≤\|F(X(t))\|\|X(t)\|≤ 2M(t)\|X(t)\|. (83) By the Picard-Lindelöf theorem, for any initial value (0)X(0), there exists a unique solution on a maximal time interval [0,τ)[0,τ) (where τ>0τ>0). From the growth bound (83) and Grönwall’s inequality, we obtain: ‖(t)‖≤‖(0)‖exp(2∫0tM(s)s).\|X(t)\|≤\|X(0)\| (2 _0^tM(s)\,ds ). (84) Since M(⋅)M(·) is integrable on [0,T][0,T], the solution remains bounded on the interval. This prevents finite-time blow-up, thereby ensuring that the solution can be uniquely extended to the full interval [0,T][0,T]. ∎ Appendix E proof of Theorem 6 Proof. Let F((t))=diag((S((t))⊙C((t)))N)−S((t)),F(X(t))=diag ( (S(X(t)) C(X(t)) )1_N )-S(X(t)), (85) and the explicit Euler method updates the state (t+1)X(t+1) based on the current state (t)X(t). The discrete form of the update is given by: (t+1)=(t)−τ⋅F((t))(t),X(t+1)=X(t)-τ· F(X(t))X(t), (86) where τ is the time step. Next, we expand the squared norm of the updated state: ‖(t+1)‖2=((t)−τF((t))(t))⊤((t)−τF((t))(t)). split&\|X(t+1)\|^2\\ &= (X(t)-τ F(X(t))X(t) ) (X(t)-τ F(X(t))X(t) ). split (87) This can be simplified to: ‖(t+1)‖2=‖(t)‖2−2τ(t)TF((t))(t)+τ2‖F((t))(t)‖2. split&\|X(t+1)\|^2\\ &=\|X(t)\|^2-2τX(t)^TF(X(t))X(t)+τ^2\|F(X(t))X(t)\|^2. split (88) For stability, we require that the norm of the updated state is not greater than the norm of the previous state, i.e., ‖(t+1)‖2≤‖(t)‖2.\|X(t+1)\|^2≤\|X(t)\|^2. (89) This implies: −2τ(t)TF((t))(t)+τ2‖F((t))(t)‖2≤0.-2τX(t)^TF(X(t))X(t)+τ^2\|F(X(t))X(t)\|^2≤ 0. (90) We can rewrite this expression as: τ≤2(t)TF((t))(t)‖F((t))(t)‖2=2(t)⊤F((t))(t)(t)⊤F((t))2(t).τ≤ 2X(t)^TF(X(t))X(t)\|F(X(t))X(t)\|^2= 2X(t) F(X(t))X(t)X(t) F(X(t))^2X(t). (91) The right side of the inequality is the Rayleigh quotient for the matrix F((t))F(X(t)). Now, We denote the largest eigenvalue of F((t))F(X(t)) by λmax(F((t))) _ (F(X(t))). Since F((t))F(X(t)) is symmetric, its spectral radius satisfies λmax(F((t)))=‖F((t))‖2 _ (F(X(t)))=\|F(X(t))\|_2. By the triangle inequality, we can obtain: ‖F((t))‖2 \|F(X(t))\|_2 (92) ≤‖diag(S((t))⊙C((t)))N‖2+‖S((t))‖2 ≤\|diag(S(X(t)) C(X(t)))1_N\|_2+\|S(X(t))\|_2 =‖(S((t))⊙C((t)))N‖∞+‖S((t))‖2. =\|(S(X(t)) C(X(t)))1_N\|_∞+\|S(X(t))\|_2. Moreover, since |cij|≤1|c_ij|≤ 1, ‖(S((t))⊙C((t)))N‖∞ \|(S(X(t)) C(X(t)))1_N\|_∞ ≤maxi∑j|sijcij| ≤ _i _j|s_ijc_ij| (93) ≤maxi∑jsij. ≤ _i _js_ij. Similarly, since S((t))S(X(t)) is symmetric, we have ‖S((t))‖2≤‖S((t))‖∞=maxi∑jsij.\|S(X(t))\|_2≤\|S(X(t))\|_∞= _i _js_ij. (94) Combining the inequalities above yields the following bound: λmax(F((t)))=‖F((t))‖2≤2maxi∑jsij. _max(F(X(t)))=\|F(X(t))\|_2≤ 2 _i _js_ij. (95) According to the Equation (91), the stability condition requires τ≤2λmax(F((t)))τ≤ 2 _ (F(X(t))). Substituting the upper bound derived above, we conclude that a sufficient condition for stability is: τ≤1maxi∑jsij.τ≤ 1 _i _js_ij. (96) Thus, the explicit Euler method for the RFHND is stable provided that this condition is met. ∎ Appendix F Analysis of computational complexity In this section, we conduct a detailed analysis of the complexity of RFHND. Let n, m, and N=∑e∈ℰ|e|N= _e |e| denote the number of nodes, hyperedges, and incidence pairs, respectively. dind_in and doutd_out represent the dimensions of the input and output features, T indicates the number of layers. The per-step computational cost consists of three main components: • Sparse scatter operations and matrix multiplication S((t))(t)S(X(t))X(t), scaling with O(Ndin)O(Nd_in). • Edge-wise MLP transformations, scaling with O(mdin2)O(md^2_in). • Cosine similarity computations. We restrict similarity calculations to node pairs induced by hyperedges, resulting in O(M2din)O(M_2d_in) where M2=∑e∈ℰ|e|2M_2= _e |e|^2. Consequently, the total time complexity for T steps is O(T(Ndin+M2din+mdin2))O(T(Nd_in+M_2d_in+md^2_in)). When the hyperedge size is bouded (|e|≤r|e|≤ r), then N=O(mr)N=O(mr) and M2=O(mr2)M_2=O(mr^2). This ensures the model scales linearly with the number of hyperedges. Adding the complexity of the linear transformations in the input and output layers, the total complexity is O(ndindout)+T⋅O(mr2din+mdin2).O(nd_ind_out)+T· O(mr^2d_in+md^2_in). (97)