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Oversmoothing in hypergraph neural networks is transverse-energy dissipation, and a reaction term can provably stop it.

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Hypergraph diffusion provably collapses node representations, and a reaction term that exactly cancels diffusion dissipation keeps a designed transverse energy level nonzero in Hypergraph Neural Reaction–Diffusion (HNRD).

T0 review reviewed 2026-08-01 challenge →

load-bearing objection The pure-diffusion contraction result is clean and worth citing, but the flagship non-collapse guarantee is a designed exact-cancellation property proven for the continuous ODE, not for the discrete HNRD layer that is actually trained. the 4 major comments →

arxiv 2607.15773 v1 pith:U45TA3YH submitted 2026-07-17 cs.LG

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

Zhiheng Zhou , Mengyao Zhou , Yancheng Chen , Dengyi Zhao , Xingqin Qi , Guiying Yan This is my paper
classification cs.LG
keywords hypergraph neural networksoversmoothingreaction-diffusionDirichlet energycontinuous-depth modelshypergraph diffusionnull-mode subspacetransverse energy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that oversmoothing in hypergraph neural networks is not just a nuisance but an inevitable feature of hypergraph diffusion: the diffusion dynamics exponentially shrink every component of node representations orthogonal to a one-dimensional null mode, driving the network's Dirichlet energy to zero. To counter this, the paper introduces Hypergraph Neural Reaction-Diffusion (HNRD), a continuous-time model with a reaction term that cancels the diffusion's dissipation at each instant and replaces it with bounded feedback toward a learnable positive energy level. The paper proves HNRD's trajectories stay globally well-posed, that its null-mode-free Dirichlet energy has a positive lower bound, and that a forward-Euler layer with step size less than one remains stable. If correct, this gives a principled recipe for building deep hypergraph networks that do not lose discriminative signal with depth.

Core claim

The central claim is that pure hypergraph neural diffusion, written as ∂_t X = -G^T Aθ(X) G X, exponentially attracts every trajectory to a null-mode subspace spanned by the square-root-degree vector. In the orthogonal complement, the distance to that subspace decays as e^{-γ_D t}, so the Dirichlet energy E_φ(X) = ½⟨Qφ X, L_H Qφ X⟩ converges to zero. The paper then shows that adding the reaction Rη(X) = (Rθ(X) + tanh(τη - ∥QφX∥²)) QφX exactly compensates the instantaneous Rayleigh dissipation, making the transverse energy s(t) obey the scalar ODE ṡ = 2 tanh(τη - s)s. Consequently s(t) tends to τη > 0, and the Dirichlet energy is bounded between (λ_H,2/2)τη and (λ_H,max/2)τη in the limit. Thi

What carries the argument

The load-bearing object is the incidence-level hypergraph gradient-divergence pair. The gradient ∇ lifts node features to variations over node-hyperedge incidences; the divergence div aggregates them back. The diffusion operator is -G^T Aθ(X) G, with G the weighted incidence matrix and Aθ a learnable diagonal softmax-normalized modulation. Its null space is exactly the span of D_v^{1/2}1, which becomes the null-mode subspace S_φ. The projection Qφ onto the orthogonal complement extracts the node-discriminative transverse component. The reaction term is built from the instantaneous Rayleigh quotient Rθ(X) = ⟨GQφX, Aθ(GQφX)⟩/∥QφX∥², so that when it is added, the dissipation and compensation ca

Load-bearing premise

The result that the Dirichlet energy stays positive depends on the reaction term exactly compensating the diffusion dissipation at every instant, which only holds if the operator used to compute the compensation is identical to the one in the diffusion, and the projection Qφ is exact.

What would settle it

Run HNRD on a connected hypergraph with a known initial transverse energy s0, and measure ∥Qφ X(t)∥². The theorem says it converges to τη > 0 for all t; if for any initial condition with QφX0 ≠ 0 the quantity decays to zero, the non-collapse claim is false. A more targeted test: evaluate the scalar ODE ṡ = 2 tanh(τη - s)s and check that its solution reaches τη; a divergence from this trajectory indicates the compensation is not exact in the implemented layer.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Deep hypergraph networks can be built without heuristic residual shortcuts; a single learnable scalar τη controls the target transverse energy.
  • The non-collapse bound provides a measurable guarantee: at infinite depth, the Dirichlet energy of HNRD is at least (λ_H,2/2)τη, so node representations remain distinguishable.
  • The discrete layer with step size h<1 has uniformly bounded transverse energy, giving a practical stability condition for training very deep stacks.
  • Because the null space of any connected hypergraph is one-dimensional, the same reaction construction carries over to any connected hypergraph, regardless of edge sizes.
  • The theory suggests a diagnostic: measuring ∥QφX∥² along propagation directly reveals whether oversmoothing is occurring in any hypergraph architecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exact-cancellation mechanism is likely portable to pairwise graph neural networks: for a standard graph Laplacian, the null mode is the constant vector, and the same Rayleigh-quotient reaction would provably preserve Dirichlet energy.
  • The proof relies on using the same Aθ for both diffusion and compensation; if a network updates Aθ at discrete steps, the lower bound becomes approximate, and the gap between theory and practice is an empirical question.
  • A stress test suggested by the theory: choose a hypergraph with a very small spectral gap λ_2; Theorem 3 predicts exponential contraction for pure diffusion with rate a_min λ_2, so a slow-decaying baseline in that setting would contradict the model.
  • The bounded feedback term tanh(τη - s) makes the equilibrium τη globally attractive; this suggests that HNRD's behavior is insensitive to initial conditions, which could be validated by random restarts.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. This paper studies oversmoothing in hypergraph neural networks from a continuous-dynamics perspective. Using an incidence-level gradient–divergence pair, it models message passing as ∂X/∂t = −G^T A_θ(X) G X. The authors prove (Theorems 3–4) that, under connectedness and uniform positive definiteness of A_θ, the pure-diffusion semiflow exponentially contracts the null-mode-free component and drives the Dirichlet energy to zero. They then introduce HNRD (Eq. 20), adding a reaction term proportional to Q_φ X that combines the instantaneous Rayleigh dissipation rate R_θ with a bounded feedback tanh(τ_η − ∥Q_φ X∥²) toward a learnable positive energy level. They prove global well-posedness (Theorem 6), convergence of the transverse energy to τ_η (Theorem 7), and consequently positive Dirichlet-energy lower bounds (Theorem 8). A forward-Euler discretization is proposed as the practical layer, with a boundedness result under h < 1 (Theorem 9). Experiments on 11 benchmarks plus synthetic heterophilic hypergraphs show HNRD ranks first on 8/11 datasets and maintains nonzero Dirichlet energy with depth.

Significance. If the continuous-time results are taken as the contribution, the paper provides a clean and largely correct dynamical-systems characterization of hypergraph oversmoothing, and the pure-diffusion contraction proof is well executed. The HNRD non-collapse theorem is a designed exact-cancellation property rather than an emergent phenomenon: R_θ is defined as the dissipation rate, so the cancellation in Eq. (77) is true by construction. That still yields a valid control-theoretic guarantee for the ODE. The main gap is that the load-bearing practical claim — that the actual trained architecture provably avoids collapse — is not proved for the discrete layer; Theorem 9 only gives boundedness. The learnable energy level τ_η also makes the lower bound conditional and potentially vacuous. The paper is well positioned methodologically, with detailed appendices and a code release, but the advertised "provable" depth-resistance claim needs either a discrete-level non-collapse proof or explicit qualification.

major comments (4)
  1. [V.B / Theorem 9 / Eq. (89)] The non-collapse guarantee advertised in the abstract and introduction — that the null-mode-free Dirichlet energy remains bounded away from zero — is established only for the continuous semiflow of Eq. (20). The practical layer is the forward-Euler discretization in Eq. (25). For this discrete map, Theorem 9 proves only uniform boundedness of s_k (Eq. 27), not a positive lower bound. Indeed, Eq. (89) contains the extra nonnegative term h²∥(B_k − R_θ(X_k)I)Z_k∥², so s_k does not follow the exact scalar ODE (78); the continuous convergence s(t) → τ_η does not transfer. As written, Section V.B overstates the discrete result ("preserves the stabilizing behavior"). Either prove a positive lower bound for {s_k} under explicit additional conditions, or explicitly restrict the non-collapse theorem to the continuous model and describe the discrete layer as an approximation whose non-collapse is e
  2. [Appendix H / Theorem 9 / Eq. (93)] The proof of Theorem 9 uses, without proof, the bound 0 ≤ λ_max(L_H) ≤ 2 (Eq. 93). This bound is not a consequence of the setup: L_H = G^T G = P^T Ω_I P, with G = Ω_I^{1/2} P as in Appendix A. For a single 2-vertex hyperedge with weight ω, L_H = ω P^T P and λ_max = ω, which can be arbitrarily large. Therefore the step-size condition h < 1 does not imply the contraction used in Eqs. (96)–(97) for general weighted hypergraphs. The theorem must either assume normalized hyperedge weights (or another bound on λ_max(L_H)) and state it, or replace Eq. (93) with a condition involving λ_max(L_H).
  3. [Appendix E / Lemma 5 / Eq. (18)] The proof of Theorem 6 asserts that "the map R_θ(X) is locally Lipschitz." This is false at Z = Q_φ X = 0. For example, for a single hyperedge of size 2 with equal node degrees and A_θ = I, R_θ(X) = ∥GZ∥²/∥Z∥² = 1 for all Z ≠ 0, while R_θ(0) = 0 by definition, so R_θ is discontinuous at 0. The product R_θ(X)Q_φ X is locally Lipschitz (it is homogeneous of degree one in Z up to smooth X-dependence), so Theorem 6 may still hold, but the proof as written relies on an incorrect assertion. Replace it with a direct proof for the reaction field R_η(X).
  4. [Theorem 8 / Eq. (24)] The positive lower bound in Eq. (24) is proportional to τ_η = softplus(η), a freely learnable scalar with no constraint in the model or in the training objective. Nothing prevents η from drifting so that τ_η is exponentially small; in that case "bounded away from zero" holds with a bound that is practically vacuous. The statement should be qualified as a bound conditional on the trained value of τ_η, and the paper should either add a regularization or lower-bound constraint on τ_η, or report the learned τ_η values in the depth experiments.
minor comments (4)
  1. [Table III, HND row] NTU2012 and ModelNet40 both report 93.32±0.99 for HND; this is likely a copy-paste error. Please verify and correct.
  2. [Section IV, Eq. (15)] L_H is introduced abstractly, but later appendices identify L_H with G^T G. State this identification explicitly in the main text so that Theorem 9's λ_max bound is traceable.
  3. [Figure 2] The x-axis labels skip from 2 to 4 to 16; if the layers are powers of two, include 8 or note the irregular grid.
  4. [Theorem 2] Theorem 2 is attributed to [27], a preprint by the same authors. For self-containedness, either state the assumptions and result needed or move a proof sketch to the appendix.

Circularity Check

1 steps flagged

HNRD's positive-energy guarantee is the reaction term's own setpoint; the only independent result is the pure-diffusion collapse theorem.

specific steps
  1. self definitional [Section V, Eqs. (18)-(19), Theorem 7, Appendix F Eq. (78)]
    "Rθ(X) := ⟨GZ,Aθ(X)GZ⟩F / ∥Z∥2_F ... Rη(X) = (Rθ(X) + tanh(τη − ∥QϕX∥2_F))QϕX ... d/dt s(t) = 2 tanh(τη − s(t))s(t)."

    Rθ is defined as the per-unit transverse-energy dissipation rate of the diffusion operator. Substituting the reaction definition (19) into the s-dynamics cancels the diffusion contribution identically, leaving the scalar ODE ds/dt = 2 tanh(τη−s)s. Therefore Theorem 7's conclusion lim s=τη is a restatement of the setpoint encoded in Eq. (19), not an emergent property. Theorem 8's lower bound (λH,2/2)τη then inherits the same learnable τη; with τη=softplus(η) unconstrained, the 'bounded away from zero' claim is only as strong as the fitted target. The positive-energy guarantee is equivalent to the definition of the reaction.

full rationale

The pure-diffusion analysis (Theorems 3-4) is self-contained: it uses the incidence gradient/divergence definitions and proves exponential contraction of QφS_D(t)X0 via the spectral gap λ2 and coercivity of Aθ; this part does not reduce to its inputs. The HNRD non-collapse theorem, however, is a different matter: the reaction term is explicitly constructed from Rθ (the instantaneous dissipation rate) plus feedback toward τη, so the proof of Theorem 7 is just the cancellation of the diffusion term by its own definition. The advertised 'provably prevents collapse' conclusion thus tracks the learnable setpoint τη rather than being an independent prediction. I also note two non-circular caveats: (i) self-citations to [27] for well-posedness and the G=Ω^{1/2}P factorization are supporting, not load-bearing; (ii) the discrete HNRD layer (Theorem 9) only proves uniform boundedness of s_k, not a positive lower bound, so the claim that the discrete layer 'preserves' the non-collapse behavior is an extrapolation. These affect rigor but are not themselves circularity.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

No new physical or independently testable entities are introduced. The mathematical objects (Qφ, Rθ, τη) are part of the model definition rather than entities with a falsifiable handle outside the paper. The central free parameters are τη, which directly sets the claimed energy lower bound, and ε, whose positivity is needed for the spectral-gap rate. The hypergraph diffusion operator itself is borrowed from the authors' earlier HND framework.

free parameters (4)
  • τη = softplus(η) = learned per dataset (values not reported)
    Learnable transverse-energy target. Theorem 7's limit and Theorem 8's lower bound are exactly τη, so the non-collapse guarantee is pegged to a fitted scalar.
  • ε (incidence modulation floor) = not reported; small positive constant
    Required in Eq. (9) to keep amin = ε/dmax > 0, giving uniform positive definiteness used in contraction rates and well-posedness.
  • step size h = searched over {0.1, 0.3, 0.5}, or learned via sigmoid
    Controls discrete stability in Theorem 9 and is tuned per dataset.
  • architecture hyperparameters (hidden dim, layers, dropout, weight decay) = selected by validation per dataset
    Standard experimental tuning; not part of the theoretical derivation but needed to reproduce the reported numbers.
axioms (5)
  • domain assumption The hypergraph is connected (Lemma 1).
    ker(G) = span{Dv^{1/2}1} and a positive spectral gap λ2 require connectivity; disconnected hypergraphs have a higher-dimensional null space and Theorem 3 must be applied per component.
  • domain assumption The incidence gradient/divergence operators, the matrix G, and L_H = G^T G from HND [27] have ker G = span{Dv^{1/2}1} and λmax(L_H) ≤ 2.
    All theorems build on this operator definition (Eqs. 1–2) and on the spectral bound used in Theorem 9.
  • ad hoc to paper The modulation Aθ(X) is uniformly positive definite with amin = ε/dmax and locally Lipschitz via the softmax normalization in Eq. (9).
    This is guaranteed by construction, but the contraction rate γD = amin λ2 depends on it; if ε → 0 the rate degenerates.
  • domain assumption Qφ X0 ≠ 0 for non-collapse theorems.
    Theorems 7–9 assume the initial transverse component is nonzero; otherwise s(t) stays at zero and no positive energy is preserved.
  • standard math Picard–Lindelöf, Gronwall's inequality, and spectral properties of symmetric PSD matrices.
    Used in Appendices E–H for global well-posedness and energy estimates.

reviewed 2026-08-01 · how reviews work

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Cite this review

Pith. "Pith review of From Diffusion to Reaction-Diffusion: A Dynamical-Systems View of Oversmoothing in Hypergraph Neural Networks." pith.science (2026). https://pith.science/paper/U45TA3YH

@misc{pith2026260715773,
  author       = {Pith},
  title        = {Pith review of: From Diffusion to Reaction-Diffusion: A Dynamical-Systems View of Oversmoothing in Hypergraph Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U45TA3YH}},
  note         = {Machine review of arXiv:2607.15773}
}
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read the original abstract

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.

Figures

Figures reproduced from arXiv: 2607.15773 by Dengyi Zhao, Guiying Yan, Mengyao Zhou, Xingqin Qi, Yancheng Chen, Zhiheng Zhou.

Figure 1
Figure 1. Figure 1: The upper panel shows that hypergraph diffusion drives node [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Depth-wise oversmoothing analysis on DBLP-CA and ModelNet40. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Parameter analysis of HNRD with respect to hidden dimension and [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of node representation evolution on DBLP-CA and ModelNet40 under different propagation depths. Node embeddings are shown at [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: HNRD requires 39.74 ms per epoch, incurring only a [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Robustness analysis under feature-level and structure-level perturbations on ModelNet40. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Runtime comparison of different HGNNs on DBLP-CA. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

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Reference graph

Works this paper leans on

49 extracted references · 14 linked inside Pith

  1. [1]

    Hypergraph neural net- works,

    Y . Feng, H. You, Z. Zhang, R. Ji, and Y . Gao, “Hypergraph neural net- works,” inProceedings of the AAAI conference on artificial intelligence, vol. 33, no. 01, 2019, pp. 3558–3565

    2019

  2. [2]

    Hypergcn: A new method for training graph convolutional networks on hypergraphs,

    N. Yadati, M. Nimishakavi, P. Yadav, V . Nitin, A. Louis, and P. Talukdar, “Hypergcn: A new method for training graph convolutional networks on hypergraphs,”Advances in neural information processing systems, vol. 32, 2019

    2019

  3. [3]

    Hyper-sagnn: a self-attention based graph neural network for hypergraphs,

    R. Zhang, Y . Zou, and J. Ma, “Hyper-sagnn: a self-attention based graph neural network for hypergraphs,” inInternational Conference on Learning Representations (ICLR), 2020

    2020

  4. [4]

    Unignn: a unified framework for graph and hypergraph neural networks,

    J. Huang and J. Yang, “Unignn: a unified framework for graph and hypergraph neural networks,”arXiv preprint arXiv:2105.00956, 2021

    Pith/arXiv arXiv 2021

  5. [5]

    You are allset: A multiset function framework for hypergraph neural networks,

    E. Chien, C. Pan, J. Peng, and O. Milenkovic, “You are allset: A multiset function framework for hypergraph neural networks,”arXiv preprint arXiv:2106.13264, 2021

    Pith/arXiv arXiv 2021

  6. [6]

    Message passing neural networks for hy- pergraphs,

    S. Heydari and L. Livi, “Message passing neural networks for hy- pergraphs,” inInternational Conference on Artificial Neural Networks. Springer, 2022, pp. 583–592

    2022

  7. [7]

    Deeper insights into graph convolutional networks for semi-supervised learning,

    Q. Li, Z. Han, and X.-M. Wu, “Deeper insights into graph convolutional networks for semi-supervised learning,” inProceedings of the AAAI conference on artificial intelligence, vol. 32, no. 1, 2018

    2018

  8. [8]

    Graph neural networks exponentially lose expressive power for node classification,

    K. Oono and T. Suzuki, “Graph neural networks exponentially lose expressive power for node classification,”arXiv preprint arXiv:1905.10947, 2019

    Pith/arXiv arXiv 1905

  9. [9]

    A note on over-smoothing for graph neural networks,

    C. Cai and Y . Wang, “A note on over-smoothing for graph neural networks,”arXiv preprint arXiv:2006.13318, 2020

    Pith/arXiv arXiv 2006

  10. [10]

    Tackling over- smoothing for general graph convolutional networks,

    W. Huang, Y . Rong, T. Xu, F. Sun, and J. Huang, “Tackling over- smoothing for general graph convolutional networks,”arXiv preprint arXiv:2008.09864, 2020

    Pith/arXiv arXiv 2008

  11. [11]

    Comprehensive analysis of over-smoothing in graph neural networks from markov chains perspec- tive,

    W. Zhao, C. Wang, C. Han, and T. Guo, “Comprehensive analysis of over-smoothing in graph neural networks from markov chains perspec- tive,”arXiv preprint arXiv:2211.06605, 2022. JOURNAL OF LATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 13

    Pith/arXiv arXiv 2022

  12. [12]

    Dropedge: Towards deep graph convolutional networks on node classification,

    Y . Rong, W. Huang, T. Xu, and J. Huang, “Dropedge: Towards deep graph convolutional networks on node classification,”arXiv preprint arXiv:1907.10903, 2019

    Pith/arXiv arXiv 1907

  13. [13]

    Pairnorm: Tackling oversmoothing in gnns,

    L. Zhao and L. Akoglu, “Pairnorm: Tackling oversmoothing in gnns,” arXiv preprint arXiv:1909.12223, 2019

    Pith/arXiv arXiv 1909

  14. [14]

    Simple and deep graph convolutional networks,

    M. Chen, Z. Wei, Z. Huang, B. Ding, and Y . Li, “Simple and deep graph convolutional networks,” inInternational conference on machine learning. PMLR, 2020, pp. 1725–1735

    2020

  15. [15]

    Towards deeper graph neural networks,

    M. Liu, H. Gao, and S. Ji, “Towards deeper graph neural networks,” inProceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining, 2020, pp. 338–348

    2020

  16. [16]

    Representation learning on graphs with jumping knowledge networks,

    K. Xu, C. Li, Y . Tian, T. Sonobe, K.-i. Kawarabayashi, and S. Jegelka, “Representation learning on graphs with jumping knowledge networks,” inInternational conference on machine learning. pmlr, 2018, pp. 5453– 5462

    2018

  17. [17]

    Gread: Graph neural reaction- diffusion networks,

    J. Choi, S. Hong, N. Park, and S.-B. Cho, “Gread: Graph neural reaction- diffusion networks,” inInternational conference on machine learning. PMLR, 2023, pp. 5722–5747

    2023

  18. [18]

    Graph neural reaction diffusion models,

    M. Eliasof, E. Haber, and E. Treister, “Graph neural reaction diffusion models,”SIAM Journal on Scientific Computing, vol. 46, no. 4, pp. C399–C420, 2024

    2024

  19. [19]

    Preventing over-smoothing for hypergraph neural networks,

    G. Chen, J. Zhang, X. Xiao, and Y . Li, “Preventing over-smoothing for hypergraph neural networks,”arXiv preprint arXiv:2203.17159, 2022

    Pith/arXiv arXiv 2022

  20. [20]

    Deep hypergraph neural networks with tight framelets,

    M. Li, Y . Fang, Y . Wang, H. Feng, Y . Gu, L. Bai, and P. Lio, “Deep hypergraph neural networks with tight framelets,” inProceedings of the AAAI Conference on Artificial Intelligence, vol. 39, no. 17, 2025, pp. 18 385–18 392

    2025

  21. [21]

    Sheaf hypergraph net- works,

    I. Duta, G. Cassar `a, F. Silvestri, and P. Li `o, “Sheaf hypergraph net- works,”Advances in Neural Information Processing Systems, vol. 36, pp. 12 087–12 099, 2023

    2023

  22. [22]

    Implicit hypergraph neural networks: A stable framework for higher-order relational learning with provable guarantees,

    X. Li, G. Tang, and J. Jiang, “Implicit hypergraph neural networks: A stable framework for higher-order relational learning with provable guarantees,”arXiv preprint arXiv:2508.09427, 2025

    Pith/arXiv arXiv 2025

  23. [23]

    Hnhn: Hypergraph networks with hyperedge neurons,

    Y . Dong, W. Sawin, and Y . Bengio, “Hnhn: Hypergraph networks with hyperedge neurons,”arXiv preprint arXiv:2006.12278, 2020

    Pith/arXiv arXiv 2006

  24. [24]

    From hypergraph energy functions to hypergraph neural networks,

    Y . Wang, Q. Gan, X. Qiu, X. Huang, and D. Wipf, “From hypergraph energy functions to hypergraph neural networks,” inInternational Con- ference on Machine Learning. PMLR, 2023, pp. 35 605–35 623

    2023

  25. [25]

    Hypergraph dynamic system,

    J. Yan, Y . Feng, S. Ying, and Y . Gao, “Hypergraph dynamic system,” in The twelfth international conference on learning representations, 2024

    2024

  26. [26]

    Hypergraph neural diffusion networks,

    F. Lu, M. Ng, and A. Yip, “Hypergraph neural diffusion networks,” Neural Networks, p. 108271, 2025

    2025

  27. [27]

    Hypergraph neural diffusion: A pde-inspired framework for hypergraph message passing,

    Z. Zhou, M. Zhou, X. Lin, X. Qi, and G. Yan, “Hypergraph neural diffusion: A pde-inspired framework for hypergraph message passing,” arXiv preprint arXiv:2604.10955, 2026

    Pith/arXiv arXiv 2026

  28. [28]

    Hypergraph neural sheaf diffusion: A symmetric simplicial set framework for higher-order learning,

    S. Choi, G. Kim, and Y .-G. Oh, “Hypergraph neural sheaf diffusion: A symmetric simplicial set framework for higher-order learning,”IEEE Access, 2025

    2025

  29. [29]

    Understanding oversmoothing in diffusion-based gnns from the perspective of operator semigroup theory,

    W. Zhao, C. Wang, X. Wang, C. Han, T. Guo, and T. Yu, “Understanding oversmoothing in diffusion-based gnns from the perspective of operator semigroup theory,” inProceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V . 1, 2025, pp. 2043–2054

    2025

  30. [30]

    Rethinking over- smoothing in graph neural networks: A rank-based perspective,

    P. Deidda, K. Zhang, D. Higham, and F. Tudisco, “Rethinking over- smoothing in graph neural networks: A rank-based perspective,”arXiv e-prints, pp. arXiv–2502, 2025

    2025

  31. [31]

    Grand++: Graph neural diffusion with a source term,

    M. Thorpe, T. M. Nguyen, H. Xia, T. Strohmer, A. Bertozzi, S. Osher, and B. Wang, “Grand++: Graph neural diffusion with a source term,” inInternational Conference on Learning Representations, 2022

    2022

  32. [32]

    Tackling over-smoothing on hypergraphs: A ricci flow-guided neural diffusion approach,

    M. Zhou, Z. Zhou, X. Han, X. Qi, G. Wang, and G. Yan, “Tackling over-smoothing on hypergraphs: A ricci flow-guided neural diffusion approach,”arXiv preprint arXiv:2603.15696, 2026

    arXiv 2026

  33. [33]

    J. K. Hale,Asymptotic behavior of dissipative systems. American Mathematical Soc., 2010, no. 25

    2010

  34. [34]

    Temam,Infinite-dimensional dynamical systems in mechanics and physics

    R. Temam,Infinite-dimensional dynamical systems in mechanics and physics. Springer Science & Business Media, 2012

    2012

  35. [35]

    Uci machine learning repository, 2017,

    D. Dua, C. Graffet al., “Uci machine learning repository, 2017,”URL http://archive. ics. uci. edu/ml, vol. 7, no. 1, p. 62, 2017

    2017

  36. [36]

    3d shapenets: A deep representation for volumetric shapes,

    Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang, and J. Xiao, “3d shapenets: A deep representation for volumetric shapes,” inProceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 1912–1920

    2015

  37. [37]

    On visual similarity based 3d model retrieval,

    D.-Y . Chen, X.-P. Tian, Y .-T. Shen, and M. Ouhyoung, “On visual similarity based 3d model retrieval,” inComputer graphics forum. Wiley Online Library, 2003, pp. 223–232

    2003

  38. [38]

    Clustering in graphs and hy- pergraphs with categorical edge labels,

    I. Amburg, N. Veldt, and A. Benson, “Clustering in graphs and hy- pergraphs with categorical edge labels,” inProceedings of the web conference 2020, 2020, pp. 706–717

    2020

  39. [39]

    Generative hypergraph clustering: From blockmodels to modularity,

    P. S. Chodrow, N. Veldt, and A. R. Benson, “Generative hypergraph clustering: From blockmodels to modularity,”Science Advances, vol. 7, no. 28, p. eabh1303, 2021

    2021

  40. [40]

    Connecting the congress: A study of cosponsorship networks,

    J. H. Fowler, “Connecting the congress: A study of cosponsorship networks,”Political analysis, vol. 14, no. 4, pp. 456–487, 2006

    2006

  41. [41]

    Grand: Graph neural diffusion,

    B. Chamberlain, J. Rowbottom, M. I. Gorinova, M. Bronstein, S. Webb, and E. Rossi, “Grand: Graph neural diffusion,” inInternational confer- ence on machine learning. PMLR, 2021, pp. 1407–1418

    2021

  42. [42]

    Hypergraph convolution and hyper- graph attention,

    S. Bai, F. Zhang, and P. H. Torr, “Hypergraph convolution and hyper- graph attention,”Pattern Recognition, vol. 110, p. 107637, 2021

    2021

  43. [43]

    Equivariant hypergraph diffusion neural operators,

    P. Wang, S. Yang, Y . Liu, Z. Wang, and P. Li, “Equivariant hypergraph diffusion neural operators,”arXiv preprint arXiv:2207.06680, 2022

    Pith/arXiv arXiv 2022

  44. [44]

    K-hop hypergraph neural network: A comprehensive aggregation approach,

    L. Xie, S. Gao, J. Liu, M. Yin, and T. Jin, “K-hop hypergraph neural network: A comprehensive aggregation approach,” inProceedings of the AAAI Conference on Artificial Intelligence, vol. 39, no. 20, 2025, pp. 21 679–21 687

    2025

  45. [45]

    Fast graph representation learning with pytorch geometric,

    M. Fey and J. E. Lenssen, “Fast graph representation learning with pytorch geometric,”arXiv preprint arXiv:1903.02428, 2019

    Pith/arXiv arXiv 1903

  46. [46]

    Consistency of spectral partitioning of uniform hypergraphs under planted partition model,

    D. Ghoshdastidar and A. Dukkipati, “Consistency of spectral partitioning of uniform hypergraphs under planted partition model,”Advances in Neural Information Processing Systems, vol. 27, 2014

    2014

  47. [47]

    Contextual stochastic block models,

    Y . Deshpande, S. Sen, A. Montanari, and E. Mossel, “Contextual stochastic block models,”Advances in Neural Information Processing Systems, vol. 31, 2018

    2018

  48. [48]

    Community detection in hyper- graphs: Optimal statistical limit and efficient algorithms,

    I. Chien, C.-Y . Lin, and I.-H. Wang, “Community detection in hyper- graphs: Optimal statistical limit and efficient algorithms,” inInterna- tional conference on artificial intelligence and statistics. PMLR, 2018, pp. 871–879

    2018

  49. [49]

    Sur l’application des m ´ethodes d’approximations suc- cessives `a l’ ´etude des int ´egrales r ´eelles des ´equations diff ´erentielles ordinaires,

    E. Lindel ¨of, “Sur l’application des m ´ethodes d’approximations suc- cessives `a l’ ´etude des int ´egrales r ´eelles des ´equations diff ´erentielles ordinaires,”Journal de math ´ematiques pures et appliqu ´ees, vol. 10, pp. 117–128, 1894. JOURNAL OF LATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 14 APPENDIXA PROOF OFLEMMA1 Proof of Lemma 1.First, b...

    2020

This paper was first reviewed by deepseek-v4-flash on August 1, 2026.