How Particle System Theory Enhances Hypergraph Message Passing
Authors: Yixuan Ma
Kai Yi
Pietro Lio
Shi Jin
Yu Guang Wang
Venue: NeurIPS 2025 poster ()
Keywords: Hypergraph message passing
Hypergraph neural networks
Primary area: deep_learning
TL;DR: Hypergraph Message Passing Inspired by Particle System Theory

Abstract: Hypergraphs effectively model higher-order relationships in natural phenomena, capturing complex interactions beyond pairwise connections. We introduce a novel hypergraph message passing framework inspired by interacting particle systems, where hyperedges act as fields inducing shared node dynamics. By incorporating attraction, repulsion, and Allen-Cahn forcing terms, particles of varying classes and features achieve class-dependent equilibrium, enabling separability through the particle-driven message passing. We investigate both first-order and second-order particle system equations for modeling these dynamics, which mitigate over-smoothing and heterophily thus can capture complete interactions. The more stable second-order system permits deeper message passing. Furthermore, we enhance deterministic message passing with stochastic element to account for interaction uncertainties. We prove theoretically that our approach mitigates over-smoothing by maintaining a positive lower bound on the hypergraph Dirichlet energy during propagation and thus to enable hypergraph message passing to go deep. Empirically, our models demonstrate competitive performance on diverse real-world hypergraph node classification tasks, excelling on both homophilic and heterophilic datasets. Source code is available at \href{https://github.com/Xuan-Elfin/HAMP}{the link}.

How Particle System Theory Enhances Hypergraph
Message Passing
Yixuan Ma Shanghai Jiao Tong University mayx5901@sjtu. edu. en
Kai Yi
University of Cambridge ky3470cam.. ac
Pietro Liò
Shi Jin
Yu Guang Wang*
University of Cambridge p12198cam ac uk
Shanghai Jiao Tong University shijin - edu. cn
Shanghai Jiao Tong University yuguang- wangus edu, cn
Abstract
Hypergraphs effectively model higher-order relationships in natural phenomena, capturing complex interactions beyond pairwise connections. We introduce novel hypergraph message passing framework inspired by interacting particle systems, where hyperedges act as fields inducing shared node dynamics. By incorporating attraction, repulsion, and Allen-Cahn forcing terms, particles of varying classes and features achieve class deeendet equilibrium, enabling separability through the particle-driven message passing. We investigate both first-order and second- order particle system equations for modeling these dynamics, which mitigate over-smoothing and heterophily thus can capture complete interactions. The more stable second-order system permits deeper message passing. Furthermore, we enhance deterministic message passing with stochastic element to account for interaction uncertainties. We prove theoretically that our approach mitigates over- smoothing by maintaining a positive lower bound on the hypergraph Dirichlet energy during propagation and thus enable hy hypergraph message passing go deep. Empirically, our models demonstrate competitive performance on diverse real-world hypergraph node classification tasks, excelling on both homophilic and heterophilic datasets. Source code is available at the link.
Introduction
Hypergraph Neural Networks (HNNs) [20], built upon Graph Neural Networks (GNNs) [29], have demonstrated remarkable success in modeling higher-order relationships involving multiple nodes [28]. In complex systems such social networks [5, 32, 50] and biomolecular interac- tions [41, 46), hypergraphs can effectively capture complex group dynamics compared to pairwise graphs.
The multi-node, higher-order nature of hypergraphs naturally lends itself to an analogy with particle dynamical systems. This is because hypergraph message passing, much like the inherent interactions in particle motion, fundamentally involves multiple interacting components. Driven by this insight, introduce Hypergraph Atomic Message Passing (HAMP), novel framework that reframes hypergraph message passing through the lens of interacting particle systems. Our key innovation in HAMP is the conceptualization of each hyperedge as dynamic field that governs the shared dynamics of nodes. As Figure shows, HAMP updates hypergraph embeddings by superimposing the forces exerted by the hypergraph's 's nodes (particles). This framework offers significant advantages, including
"Corresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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dp: dr
Dp) p.) + f((D.) +
dB. di
Hypergraph Embedding
Superposition Principle
Updated Embedding
damping potential contour
nodes with a hyperedge
attraction
repulsion
noise
uncertainty
Figure 1: An illustration for HAMP framework. The property can account for feature or velocity.
the ability to mitigate over-smoothing and facilitate deep message passing on both homophilic and heterophilic datasets, claim supported by experimental findings in Section 6.3.
Particle theory takes various forms, traditionally categorized into first-order and second-order systems. While first-order systems offer direct approach to defining evolution, the second-order systems are inherently more stable, converging towards an asymptotically stable state. Both first-order and second-order systems exhibit desirable separability properties stemming from the balance between attractive and repulsive forces [27, 31]. Repulsive forces are crucial for separating distinct features, while attractive forces encourage particles of the same category to cluster, allowing for the refinement of essential features. Through this mechanism, we construct the interaction forces within HAMP.
Inspired by particle systems and informed by experimental observations, we identify a critical need for a delicate balance between attractive and repulsive forces. Excessive attraction often leads to feature over-smoothing [34, 35], while unchecked repulsion can cause feature explosion [44]. To address these issues and ensure system stability, we introduce the Allen-Cahe force [1] as balancing mechanism. We theoretically demonstrate that the resulting system exhibits favorable separability and that its solutions converge to an equilibrium state, thereby mitigating both over-smoothing and feature explosion. Our focus second-order systems is further motivated by their inherent theoretical stability, which ensures robustness even with deeper network layers. In practical tasks, ambiguous cases where distinct states hard to differentiate often lead to uncertainty [42]. To address this, we incorporate a stochastic term into HAMP, driven by Brownian motion, which results a stochastic differential equation. Our main contributions are summarized as follows:
We propose HAMP, a novel hypergraph message passing framework based on particle system theory, filling the theoretical gap in understanding hypergraph message passing from the perspective of particle system.
We design two hypergraph message passing algorithms, HAMP-I and HAMP-II, constructed through first-order and second-order particle dynamical systems.
Theoretically, we prove that HAMP maintains a strictly positive lower bound on the hyper- graph Dirichlet energy, effectively resisting over-smoothing.
Empirically, through numerical experiments, should that HAMP achieves competitive results on node classification benchmarks. Notably, on heterophilic hypergraphs, HAMP consistently outperforms the current state-of-the-art baselines by margin of 1-3%.
Preliminaries
Hypergraphs. A hypergraph is generalization of graph in which an edge join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. We can denote a hypergraph {V,E), with |V] nodes and [E] hyperedges, where V {fi...., B is the set of nodes and E {e1..... } is the set of hyperedges. E(i) denotes set containing all the nodes

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sharing at least one hyperedge with node i. The incidence matrix H E R Mix|e is a common notation in hypergraph. Its entries are defined as H. 1 if node i belongs hyperedge e, and 0 otherwise.
Message Passing in Hypergraphs. Neural message passing [23, 3] is the most widely used propagator for node feature updates in GNNs, which propagates node features while taking into account their neighboring nodes. Next, we generalize graph message passing to hypergraph, where multiple nodes interaction reflected in a hyperedge is considered,
11+1) -101 x
D2 (z!)) 21 (1xx))) (e)ee())))))))eeeo
where _ denotes the feature embedding of hyperedge in the 1th layer. $ and @ denote differentiable permutation invariant functions (e.g., sum, mean, or max) for nodes and hyperedges, respectively. (1() denotes a differentiable function such as MLPs.
Neural ODEs. Neural ODEs [18, 10] are essentially ordinary differential equations (ODEs), where the time derivative of the hidden states are parameterized by neural network fe.
dx(() fo (x) x(0) x, dt
where t denotes time, x(t) are the system state, and @ is learnable parameter. It can be understood as the continuous limit form of the residual network, e.g. x = So (x(1), 1) + x(), where denotes the step size of time. Thus, Neural ODEs model the continuous dynamics system that evolve hidden states over a continuous range of "depths", analogous to layers in traditional deep networks.
3 Hypergraph Message Passing based on Particle Dynamics
Viewing HNNs as continuous systems allows for the application of physical models to elucidate and investigate their properties. A compelling conceptual analogy arises when comparing hypergraph message passing in deep learning with interacting particle systems studied in statistical physics. These disparate fields display remarkable structural and dynamical resemblances: the local potential field-goveel governed interactions among particles in many-body systems are analogous to the aggregation operations in HNNs. Concurrently, the emergent collective behaviors observed in self-organizing dynamics demonstrate functional equivalence to information propagation mechanisms in HNNs. As message passing delineates the transfer of information between entities, particle systems can represent this by having particles acting as entities that exchange information through the transition of specific attributes. Moreover, such particle systems can be endowed with hypergraph structures by interpreting hyperedges as distinct fields that exert influence over all nodes encompassed by them.
Fields and Hypergraphs. The hypergraph uses hyperedges which reflect the common relationship among multiple nodes. Such multi-nodal relationships are aptly described using "fields". Each field imposes a common dynamic on the nodes under its influence, which is equivalent to a hyperedge defining a unified information propagation step for its constituent nodes. A node participating in multiple hyperedges thus receives information from diverse sources due to field superposition. Thus, hyperedges reflect both the interactions of particles within a specific region and the concurrent influence of the external environment, which together dictate the propagation dynamics. Based on these observations, we propose hypergraph message passing framework for describing information dynamics. We formalize composite field for each node i by F. = Σ F + Fa to encapsulate the
esiEe
hypergraph structure. This field F; aggregates influences akin to those in a particle system, where F represents the interaction energy for node with the edge e, and Fa is the damping energy.
Interaction Force. Interaction force forms basic association between hypergraph message passing and particle dynamics. We consider particles identified by their specific attributes p; E I in the particle system. Then, we define the interaction energy as following
ΣF:= 3
pp)) Tolp.,p, fotp:
-P;), EEE. E.
(3)
e3ECc
cice jEe
3

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If (p) > O, node is attracted by node j. If fs(p:, Py, < 0, node i is repulsed by node j. They correspond to two basic interaction forces: attraction and repulsion. Note that in the previous model similar to Eq. 3, the only interaction is attraction. It relies on an implicit belief that all the hyperedges only connect similar nodes. But this is not always true in the real world. Hence, introducing repulsion into hypergraph neural networks is significant.
Damping Force. The specific formulation of the damping term can vary. The inclusion of damping in neural networks serves two principal purposes. Firstly, it is crucial for ensuring system stability, If inter-particle repulsive forces do attenuate adequately with increasing distance, particles risk unbounded separation or unbounded Dirichlet energy, which could lead to network collapse. A damping term mitigates this by exerting a dominant influence on the dynamics, especially near the system's 's boundaries. Secondly, the damping term has been demonstrated to promote various collective dynamics in numerous systems [30, 31J. For damping, we take double-well potential Fa ((1 p²)² with the coefficient Ç. The corresponding damping force is given by the Allen- Cahn force fa(p) Fa = 6(1 p pP)p [38, 191, where á denotes the strength coefficient. This formulation exhibits favorable separation properties, as will be demonstrated in Section 5.
Hypergraph Dynamics. Interaction term can be formulated using two primary approaches: re- ductive and non-reductive expressions. Reductive methods, such as those based on convolution coefficients [22] or attention coefficients [2], often simplify hyperedges into clique-like structures, which can lead to the loss of crucial high-order information. In contrast, non-reductive expressions, like star expansion, aim to preserve this high-order information by explicitly connecting each node to the hyperedges it belongs to. We take the non-reductive expression and Allen-Cahn force to design interaction force f& and damping force fa. In fact, the partiele attribute p can represent diverse properties, such as features or velocity. This flexibility in defining p naturally allows for the formulation of both first-order and second-order particle systems.
First-order ODEs. Inspired by the opinions dynamics [33], we have P. + X; E Id that interact with each other according to the first-order system. Taking gradient of composite field F, we have dx; dt e:fEe JEE
Es(x, e),)xx - -x:) + fa(x:);
where fa is parameterized function. This form has connection with diffusion process of hypergraph. We defer the detailed analysis to Section 4.
Second-order ODEs. Inspired by the flocking dynamics [33, 15], this is second-order model where the attribute is the velocity, + V E gd, which is coupled to the feature x, E RR In this sense, information can be propagated by the evolution of feature and velocity of nodes, dv; dt EiiEe jEe
- -v)) + fa(v;), where dx; dt V
(5)
From ODEs to SDEs. To capture incompleteness and uncertainty within hypergraph data, stochas- ticity is introduced into the particle system. For instance, such uncertainty arise when particles from different classes exhibit similar features. To model this inherent randomness, these stochastic processes often described using stochastic differential equations (SDEs). We assume these SDEs are driven by Brownian motion B,:
(6) where Ia denotes the x d identity matrix. In essence, SDEs describe the changes in system state under infinitesimal time variations. Here, particle states undergo continuous evolution through both deterministic drift term and stochastic diffusion component. can describe hypergraph evolution of neural message passing in more realistic scenario. According to [15, 24], this model can achieve self-organization in a finite time, which enables the system to adjust its own state in the shortest time.
dp(t -) (F(p(t)) + edB,, where B[dB,] = 0 and Cov(dB,) Las
Hypergraph Message Passing. We introduce unified hypergraph message passing framework, called Hypergraph Atomic Messuge Passing, or HAMP, given Eq. 6. By instantiating this frame- work with either a first-order or a second-order system formulation, we obtain two variants: HAMP-1 and HAMP-II, respectively. Detailed implementations of both approaches are provided systematically in Appendix A to ensure clarity and maintain manuscript focus.

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Scale Translation of Hypergraph: Diffusion and Particle Dynamics
We examine hypergraph message passing through two complementary lenses-microscopic and macroscopic. From the microscopic viewpoint, HAMP is cast as particleddnammis system, as previously discussed. From the macroscopic standpoint, message propagation seen as diffusion process, which reveals that the particle-basee formulation naturally subsumes diffusion-basee models and overcomes their intrinsic limitations.
Hypergraph Diffusioo. Consider node feature space ! = Id and tangent vector field space TΩ Rd. For x, f2 F, 1 E TS2, and Fi,j = Fjii we adopt the following inner products:
(x, X [I
ΣΣ CEE
hig Fi.
Here hig is tuple related node i, and hyperedge and hi, 0 Of HL, O. We set hig to satisfy Σ ΣΣ hi 1. For any u E TΩ, by the adjoint relation [u, Vxj (x, divu), u), where
3EE
Vx= = X, X,, we can derive (divu), = = Σ hi Then, the hypergraph diffusion process
3EE
dx, df divVx = ΣΣ hi (x, -x)).
(8)
AEE
For simplicity, we rewrite Eq. in matrix form
dx di
Lx, where -
h5 is hypergraph 3EE
operator. If C is positive semi-definite, we interpret Eq. as a diffusion-type process hypergraph. Different parameterizations of hi, then yield distinct diffusion equations. For example, applying
forward Euler discretization to Eq. 8 and setting
[Σ h4g) D HWD, D. recovers the 333
simplified HGNN [20] without channel mixing.
Connecting Particle Dynamics. Since Eq. 8 coincides with the self-organized dynamics in particle system [33], we reinterpret Eq. 8 not standard hypergraph diffusion process, but rather as particle dynamics where &e denotes the interaction force between nodes i and j under field e. In fact, Eq. a special case of Eq. 4 where only attractive forces are considered in the message propagation. As we show in Section 5, this simplification is atypical in particle systems and leads to over-smoothing.
5
Theory of Anti-over-smoothing
For diffusion-type hypergraph networks, we define the hypergraph Dirichlet energy of x e Nxd as
E(x) = Σ H., - x1 tr(x [x). Furthermore, we will define wver-sooohhng 8.f=
in hypergraph message passing.
Definition 5.1. Let x denote the hidden features of the 4 layer. We define over-smoothing in HNNs as the exponential convergence zero of the layer-wise Dirichlet energy as a function of I, ie., E(x(") < Cie- -C2, with positive constants C₁ and C2-
Why Do HGNN Cause Over-smoothing? The normalized hypergraph Laplacian matrix is defined by I-I-PP where P is the propagation matrix derived from the incidence matrix H. In an HGNN, the feature at layer without activation a evolves according to x(() = (0) ((L))) However, this purely diffusive propagation inevitably induces over-smoothing: one can show E(x()) < C [9], where ? is the smallest non-zero positive eigenvalue of L. This over- smoothing behavior stems from the intrinsic diffusion dynamics. Because L is symmetric positive semi-definite, repeated application of P causes node representations to decay exponentially towards a limiting state-which is zero, thereby eroding discriminative power.
(1-1).
Why Do HAMP Avoid Over-smoothing? We now derive theoretical guarantees for the collective behavior of models Eq. 4 and Eq. 5. They show that the addition of repulsive force enforces
5

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positive lower bound on the Dirichlet energy. Technically, we suppose there exists {f such that I = {1,... ,N can be divided into two disjoint groups with NI, N₂ particles respectively: ( 2 0, for {i,j} E I or L2 and [a(h) S O, otherwise. We designate { ( = {x,]i } and { = {xyli I2). Finally, we impose the symmetry [a(M( (h(,) which reflects equal-and-opposite interactions under the same field. Under these conditions, one can show that the Dirichlet energy of our system admits strictly positive lower bound, as follows. We leave the detailed proofs in Appendix B.
Proposition 5.2 (I22 separation of HAMP-I). For Eq. 4, suppose the above assumptions are sat- N
N
isfied. Define the mean value x == N ==1
E X4, and the second moments M2(x)= = Σ x² Then Ex,,
i=1
for sufficiently large N1, N2. there exist constants X_, Xt, such that if the initial data satisfies M2((0) ((0) = (0) [O]I S X4, then, there holds that the Lo separation
M2(() X(1) (t()()) - (1)12
S + (2(0) - ---
(9)
with a positive constant μa, where M²(t)= == M2)x((((t) + M2(x(2)(t))
Proposition 5.3 (L2 separation HAMP-II, [19]). For Eq. 5, we set 0 < <SS fa(h?,) {i,}} EII and 0 S (a(h & D otherwise, with k == max (IE())) If the initial (ix x(2)
> 1,
and if there exists positive constant such that
a(S - D)k min (N1, N2) 2 8 +
(10)
Then the system has a bi-cluster flocking.
Proposition 5.4 (Lower bound of the Dirichlet energy). If the hypergraph H. is a connected one, for Eq. 4 with the conditions of Theorem 5.2, or for Eq. 5 with conditions of Theorem 5.1 [19], there exists a positive lower bound of the Dirichlet energy.
6 Experiments
6.1 Experiment Setup
We conduct comprehensive experiments to evaluate the proposed models on node classification task. For more experimental details such as datasets and hyperparameters, please refer to Appendix C.
Datasets. Following ED-HNN[43), the real-world hypergraph benchmarking datasets span diverse domains, scales, and heterophiilic levels. They can be divided into two groups based on homophily. The homophilic hypergraphs include academic citation networks (Cora, Citeseer, and Pubmed) and co-authorship networks (Cora-CA and DBLP-CA). The heterophilic hypergraphs cover legislative voting records (Congress, House, and Senate) and retail relationships (Walmart).
Baselines. The selected baselines cover two types of hypergraph learning frameworks, compris- ing both reductive and non-reductive approaches. The reductive methods include HGNN [20], HCHA [2], HNHN [14], HyperGCN [47], and HyperND [37]. The non-reductive methods include UniGCNII [26], AllDeepSets [11], AllSerTransfonmer [11], ED-HNN [43], and HD [48].
6.2 Node Classifications on Hypergraphs
In this section, we evaluate HAMP-I and HAMP-II on nine real- weal-world hypergraph benchmarks for the node classification task. Tab. reports the accuracy on both homophilic and heterophilic datasets. As HDSode does not provide results on these benchmarks, we reproduce its performance using the official open-source code and perform hyperparameter tuning following the original paper. For other baselines, our results are consistent with those reported by ED-HNN. Overall, our models demonstrate competitive performance across all nine datasets. Notably, the improvement is more pronounced on heterophilic datascts, with the largest accuracy gain of 3% observed on Walmart datasct.

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Table 1: Node Classification standard hypergraph benchmarks. The accuracy (%) reported with a standard deviation from 10 repetitive runs. (Key: Best; Second Best; Third Best.)
Homophilic
Cora
Citeseer
Pubmed
Cora-CA
DBLP-CA
HGNN
1.36
72.45-11.16
86.44.10.44
82.64
91.03=0.200
HCHA
1..2
12.42-L1.42
86.4110.36
82.55
90.92-0.22
HNHN
1.92
1.56
86.90-10.00
77.19+1.49
86.78=0.29
HyperGCN
18.45=11.26
71.28=0082
82.84=8.67
79.48=2.08
89.38=0.25
UniGCNII
7..8
73.05=2.21
88.25=0.40
83.60=1.14
91.69=0.19
HyperND
19.20=11114
72.62++1.49
86.68=0.43
80.62=1132
90.35=0.26
AlDDeepSets
76.88++1.80
1..83++163
88.75-+0.3
11.97+1.50
91.27-0.27
AllSet fremorernsorrmmr
1..58+1147
73.08+1.200
88.72+0037
83.63+1147
91.53-0.23
ED-HNN
8..3
73.70+11.8
89.03++053
83.97+
91.90-0.19
oDe
80.65=11.22
74.87=1.12
88.81=0.43
84.95=0,98
91.49=0.25
HAMP-1
81.18 1,30
75.22++1.66
09.02--0.38
85,23 1.15
91.66-0.11
HAMP-II
80.80 1.62
75.33 1.61
1.61
89.05 0.41
84.89 1,53
91.67-0.23
Heterophilic
Congress
Senate
Walmart
House
HGNN
91.26=1.115
48.5944.52
62.00=0.224
61.39+2..6
HCHA
90.43:11.20
48.62.14441
02.25
61.36:12.53
HNHHN
13.35
50.93
07.18
67.80
HyperGCN
55.12=1..96
42.45+3.67
44.74=2.81
48.32
UniGCNII
94.8110.811
49.30.14.25
54.45:10.37
6..25
HyperND
3..63
52.82:1.220
38.10:13.86
51.70
AllDeepSets
91.80=1.53
48.1745.67
64.55=0.33
67.82+22.0
AraSorTransfommer
92.16=11005
51.83+5.22
65.46=0.25
69.33+2.20
ED-HNN
95.00
64.79115.14
66.91-0041
72.45
HDS
90.91
66.90115.52
63.38100448
71.3011.900
HAMP-1
95.09-10.79
69.44.1.099
69.90-10.38
72.72
HAMP-II
9..26=1..4
70.144..08
69.94=0.37
12.60+1.23
6.3 Ablation Studies
In this section, we conduct several ablation experiments on real-world datasets to assess our model design, and provide empirical validation for theoretical findings. For more ablation experiments, see Appendix C.2.
Impact of the Number of Layers on HAMP-1 and HAMP-II. To assess the effectiveness of different methods in deep HNNs, we compare three representative baselines with our proposed HAMP models. Unlike Pubmed, DBLP-CA, Senate, and House, the Cora, Citeseer, and Congress datasets generally perform better shallow networks, but worse in deep networks. Therefore, we focus on these datasets to demonstrate HAMP's advantages in deep architectures. As shown in Fig. 2, HAMP-II consistently outperforms other methods as depth increases, while competitors suffer from accuracy drops. This ability highlights the potential of HAMP to capture complex representations and maintain stability, making it a valuable framework for deep HNNs.
Impact of the Repulsion and Allen-Cahu Forces HAMP-1 and HAMP-II In addition to the thcoretical analysis, we conducted ablation studies to investigate the individual and combined effects of the repulsion force fō and the Allen-Cahn force fa both HAMP-I and HAMP-II. The results given in Tab. 2 show that incorporating the repulsive force significantly improves classification performance. In HAMP-I, enabling repulsion alone yields notable gains over the baseline, while the Allen-Cahi term alone offers moderate improvements. Notably, combining both terms consistently achieves the best accuracy across all datasets. The synergy between the repulsion and Allen-Cahi terms further boosts performance, confirming that these particle system-inspired mechanisms play complementary roles: repulsion term prevents feature over-smoothing by separating node embeddings, whereas Allen-Cahi term balances attraction and repulsion promote class-dependent equilibrium.
7

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Gileserr
Cara
Congress
100
100
100
(%)
Accumic (%)
Accoracy (%a
60
69
HAMP-I
40
49
HDD
BD HNN
BKINN
20
5
15
20
30
40
30
Number of Layers
N Number of Layers
Nomber of Layers
An empirical analysis of the depth-accuracy correlation in deep neural networks. The shaded area represents the standard deviation, helping to show the range accuracy fluctuations.
Figure 2:
Table 2: Node Classification some standard hypergraph benchmarks. The accuracy (%) is reported from 10 repetitive runs. (Key: fg: repulsion; fa: Allen-Cahn; Best.)
J
fa
Cora
Citeseer
Pubmed
Congress
Senate
Walmart
House
x
x
75.67
70.59
87 93
93.47
60.15 14
69.86
69.88
V
x
75.97
70.60
88.23
93.65
61.69
69.73
69.57
HAMP-I
X
80.59
74.67
88.77
94.67
65.63
69.73
71.55
81.18
75.22
89.02
95.09
69.44
69.90
72.72
X
77.18
71.75
88.68
94.35
60.
69.84
70.46
HAMP-II
V
77.40
71.69
88.77
94,63
59.58
69,80
69.63
X
79.50
74.25
88.80
94.12
64.51
69.86
70.96
80.80
75.33
89.05
95.26
70.14
69.94
72.60
Impact Noise on HAMP-I and HAMP-II Fig. illustrates the effect of adding stochastic component into deterministic mes- sage passing on Senate dataset. Notably, incorporating noise consistently improves accuracy for both HAMP-I and HAMP-II. In addition, the stability HAMP-II better than that of HAMP- I, whether not noise is injected, indicating that the second-order particle system more stable. Overall, these results demonstrate that incorporating stochastic component into message passing ef- fectively improves model performance and stability by explicitly capturing data uncertainty.
Scnate
75
70
60
55
50
HAMP-I
HAMP-E
HAME-I
35
HAMP-II noise
30
10
30
35 49
Number of Layers
6.4 Vertex Representation Visualization
Figure 3: Significance plot for noise on Senate dataset.
To more intuitively validate the progressive refinement of vertex representations our HAMP methods, we use t-SNE [39] visualize the vertex evolution process of HAMP-I and HAMP-II on Congress dataset at different epochs. As shown in Fig. 4, we visualize the vertices based on the representations obtained at epoch 1, E, E, 2 E, and E, where E is the total number of epochs. From Fig. 4, we have the following three observations:
When epoch (subgraphs (a) and (f)), the node feature representations exhibit chaotic distribution, and it is difficult to distinguish the number categories. As training progresses, the clustering entropy shows monotonically decreasing trend.
Subgraphs (e) and (j) show the visualization results at convergence for HAMP-I and HAMP- II, respectively. The final category boundaries are clearer HAMP-1I than in HAMP-I, reflecting the geometry refinement enabled by HAMP-II's deeper message passing.
Comparing subgraphs (b) and (g), HAMP-1 is still in the early stages of categorization. while HAMP-II shows a significantly improved clustering effect. Notably, HAMP-II has successfully distinguished six distinct categories, confirming that HAMP-I] achieves faster cluster than HAMP-I due to the second-order mechanism.

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(a) At epoch= 1.
(b) At epoch= E. (c) Atoch = E. (d) At epoch = )
(e) At epoch E.
(i) At epoch = 1. (g) At epoch = E. (h) At epoch E. (i) At epoch = E. (i) At epoch = E.
Figure 4: The t-SNE visualization of vertex representation evolution of HAMP-I (the first row) and HAMP-II (the second row) Congress dataset. The colors represent the class labels.
7 Related Work
Hypergraph Neural Networks. Hypergraph learning was first introduced in [51] as propaga- tion process on hypergraph. Since then, hypergraph learning [21, 28] has developed extensively. As extension of GNNs, Feng et al. [20] proposed HGNN that effectively captures high-order interactions by leveraging the vertex-edge-vene vertex propagation pattern. Further, HGNN* [22] in-
troduced hyperedge groups and adaptive hyperedge group fusion strategy as a general framework for modeling high-order multi-modal/multi-yy
data correlations. Following the spectral theory of hypergraph [47], ShealHyperGNN [16] introduced the sheaf theory to model data relationships in hypergraph more finely. Inspired by Transformers [40], several HÑNs [2, 11, 12] enhanced the feature extraction capability for hypergraph through attention mechanism and centrality for positional encoding. Different from using vertex-vertex propagation pattern, several works [11, 26, 43, 48]
have considered employing multi multi-phase message passing. Among these, UniGNN [26] presented a unified framework that that facilitates the processing of all hypergraph data through GNNs. In contrast, ED-HNN [43] and CoNHD [49] were developed from the perspective of optimization of hyperedge and node potential. Additionally, HDS [48] adopted control-diffusee the hypergraph dynamic system. By contrast, our method is based the particle system theory, employing the first-order and second-order systems to understand and design HNNs.
ODEs to model
Collective Dynamics. Watts and Strogatz [45] first mathematically defined small-works network and explained the reasons behind the collective dynamics. Many researchers are interested in how active/passive media made of many interacting agents form complex patterns in mathematical biology and technology. Battiston et al. [4] showed that higher-order interactions play a crucial role in understanding these complex patterns. These complex patterns
be seen in animal groups, cell clusters, granular media, and celf-organizing particles, as shown in [7, 25, 31] and other references. In many of these models, the agents move into groups based on the attractive-repulsii For example, Fang et al. [19] have studied the Cucker-Smale model [13] with Rayleigh friction and retuulivverreulssive coupling, and [27] showed a similar collective phenomenon with stochastic dynamics. Furthermore, the ACMP [44] was based on Allen-Cahn particle system and incorporated the repulsive force, providing inspiration for our work in hypergraph learning.
forces [8. 17, 6].
Conclusion
In this paper, we introduce a novel hypergraph message passing framework inspired by particle system theory. We derive both first-order and second-order system equations, yielding two distinct models for modeling hypergraph message passing dynamics that capture full hyperedge interactions while mitigating over-smoothing and heterophily. The proposed models further integrate stochastic term to model uncertainty in these interactions and can alleviate over-smoothing in deep layers. By casting

---

HNNS in physically interpretable paradigm, our model balances bigh-order interaction modeling with feature-diversity preservation, offering both theoretical insights and practical advances for complex system analysis. In future work, we plan to extend the proposed methods to protein-structure and sequence design for the discovery of novel antibodies and enzymes.
Acknowledgments and Disclosure of Funding
This work is supported by the National Science Foundation for International Senior Scientists Grant (No. 12350710181), and the Shanghai Municipal Science and Technology Key Project (No. 22JC1402300). We thank the anonymous reviewers for their insightful comments.
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12

---

A Algorithms
Complexity Analysis. Here, we analyze the computational complexity of one layer in HAMP-1 and HAMP-II. Analytically, the time complexity is O ([V]lE e² [V]e), where [V], [8] and are the number of nodes, number of hyperedges and number of hidden dimension, respectively. However, the incidence matrix H is sparse matrix, so the time complexity is O ((tr(D.) + tr(D))) + [V(c), where tr (r( is the sum of the degrees of all nodes and tr(De) is the sum of the number of nodes contained in all hyperedges. The detailed process of HAMP-I and HAMP-II are shown in Algorithm 1 and Algorithm 2.
Algorithm 1 The HAMP-1 Algorithm for Hypergraph Node Classification.
1: Input: the incidence matrix H. the node feature X, and the node labels Y.
2: Output: the model prediction accuracy.
3: Initialization: the time T, the step size and all parameters of model.
4: while not converged do
5: Node feature mapping X Linearmap (X);
6: Set the initial time / = 0, the initial node representation X(0) X;
7: while / T do
8: Message passing from V to E: Xy=e(t) = 인 (X H)
9: Message passing from & to V: (t) = (X((), 22 (Xy-s(t), H))
10:
Compute particle dynamics as
X(I + 7) = X(1) TO XE-M(1) - + (L(((()
BBB() +33((0)
Interaction Force
Allen Cahn Force
Noise
11:
Updata t= = T
12:
end while
Input the node representation into the classifier X MLP(X(T)):
Compute the model prediction labels Ý Softmax (X and compute the loss function;
14:
15:
Update all parameter by back propagation using the Adam optimizer;
16:
end while
Algorithm 2 The HAMP-II Algorithm for Hypergraph Node Classification.
1: Input: the incidence matrix H. the node feature X, and the node labels Y.
2: Output: the model prediction accuracy.
3: Initialization: the time T. the step size 7 and all parameters of model.
4: while not converged do
5: Node feature mapping X Linearmap X);
6: Set the initial time t = 0, the initial node representation X(0) X;
7: Set the initial velocity (0) Linear (amnar(X(())) ((()) X(0):
8: while t < T do
9:
Message passing from V to E: (t) = क (V(t), H)
10:
Message passing from E to V: V (t) = v (V(t), t (Vv-s(t), H)); Compute the velocity of particle dynamics system as
V(t V(t)+
VE-v(t) - wV(++ 8fa(V(t)) + +cB(t) +BV(0)
Interaction Force
Allen Cahn Force
Noise
12:
Compute the representation by X(t T) X(t) (V
13:
Updata t end while
14:
15:
Input the node representation into the classifier X MLP(X(T)):
16:
Compute the model prediction labels Ý Coftmax(X""u and compute the loss function;
Update all parameter by back propagation using the Adam optimizer;
18: end while
13

---

Limitations Discussion. Our particle dynamics- hypergraph message passing framework assumes a static hypergraph topology. While this assumption is valid for social and biological hypergraphs with slowly evolving interactions, it may not hold in highly dynamic scenarios like fi- nancial transaction networks, where hyperedge topologies change abruptly. Consequently, effectively modeling temporal hypergraphs with evolving structures remains an open challenge.
B Theoretical Results and Proof
Technically, we suppose there exists {ff} such that I {1, . . can be divided into two disjoint groups with Ni, N₂ particles respectively: fo(hç,)) 2 0, for {i.}} E In or Iz and fo(hg,)) s O, otherwise. We designate
(zf = EI)
(zf)) = {xyli E k2).
(11)
The model is channel wise, hence we use instead of x in the proof.
First, let's define the relevant notation, The mean value:
N
T
Tb
(12)
N
The deviation values:
2, := Zz - B.
(13)
The variance of values within each group:
var( = N1
Σ
(o!
var( = N₂
Σ (of )
(14)
The second moments:
N1
N₂
M2(x(1)) = Σ
(2 12 M2(x12) = Σ (2) 12)/2
(15)
i=1
1=1
And others:
M2 = (M) + M2
var(x + var (2))
(16)
x==== =
Σ
h +
h5
(17)
gEc
337 jEC
h: : max
(18)
DR == }. D = max }.
(19)
Dj = max lla) e
(20)
Technically, we set N₁ = N₂ =
No. This assumption is that N₁ is comparable to N2, i.e. there exists a positive constant x satisfying N₁ S N2 S KN1.
We can rewrite Eq. 4 as
N₂
1
Σ
Σ
(1) 1o
())
Σ
(1
(2)
x. (10) + 5x 2 (1-()()))
N1
N₂
(EE(()
P=1
EEE(1)
j=1
N₂
N
(x
(2)
- 10
Σ
Σ
(at
(1) - ) + 82 ; (1 -
(1- (1)))))))
(2)
N₂
N1
e'==)
REE())
i=1
(21)
14

---

Set the matrix A as
A
(22)
higg
and and designate CA = min F(A)), where F(A") is the Fiedler number A
cEE
Lemma B.1 (L2 estimate for M2).
There exists positive constant M such that
sup 0gtcoo
M2(t) S M S co.
(23)
Proof. Note that 1 = n then
d
M2(x((1)=
2
N₂
(1) (1)
dt
=
N1
Ni
2 N;
2
"i (1 24 x
pett (1)
(1) (1)
Mo (m) (2)
2 )x (1)
N1
N1
1=1 (1) K'E
f= (EE(1) JEE
26
(1
N
2
( 1, 1
2
Σ
h (1 (2)
22 21 (1)
N1
N1
BEC
eEE i,EE
28
N:
(1 (3)
+
x
N
Σ
Σ
1 (1/2) 11)
2 ΣΣ
N (x) -x(")) )o(")( (
N1
EEE Lf'e
3EE IEEO
26 N
(o (1))((11 - (2.
+
N
(24)
Similarly,
N₂
M2(2
(2) (2) j
N₂ j=1
N₂ EEE Ee
N₂
ΣΣ
(a.
x
(2) (2)
)a
(25)
EEE j.Eee
25 N₂
(2)
(1-
N₂
j=1
Define the total second moment M2 M2(x(1)) + (2( Its time derivative is:
d M2 () + d
d
dt (2
(26)
dt dt
By discarding the non-positive squared terms (first two sums), we obtain the inequality:
d
M2 <
2
h (1 (2) -
2
hg (z
x
)ar
dl
N1
N₂
eEE
3EE
JiEe
(27)
28 N₂
28 N₂
+
Σ
(1())
(x
(1(1)(((1)))
(
(2) (1 (2)))))
(2)
N
N₂
i=1
15

---

By the inequality,
N1
( =
Σ
12)
$
NN
Fi
[1]
)
N₂
N2
(22)
Σ
(
(2) }
N2
Σ
[
(2)
(x.") - x; 12 s ((2
1 + (1)
Then we have
N1
2 N₂
25
Σ
2
(1 -1- (x.-) (2( +
12) 2 (1- (x. )2
N
N₂
20 N1
(1)
28 N1
26
N2
(
22)2
2
26
N₂
(x)
))
N1 1=1
N1
N₂
N₂ 1=1
F=1
1=1
20
N
(1) )
28
N1
2
26
No
111/
25
Σ #=1
N2
(zF (
2
(
+
N1
(22)2
N₂
(N2)²
1=1
(28)
M
M221
-28
+
25
+
N1
N2
N1
N2
M (1)
M
M
M22
226
8
N1
N2
N₂
N2
28
6
M2
(M
N7
NO
where N' = min ( N1, N23 and N" = max maxmxxx111 N2).
So, we have
d dt
D M2 S
ΣΣ
(0) ((()))
+ D.
ΣΣ
(100) - 11/22 + (a))))2
Ni
N₂
CEE ijEe
CEE j,iEe
28 N1
1) 2 1 (=f
28
N2
(2)
(1 (1-(2(²)))
+
N
N₂
ΣΣ CEE
3(x; +
Σ Σ
(2(z (1) + 3(a
+ 28
M2
8
M2.
N
2₂
N
N
idE
EEE iJEe
(29)
These relations yield a Riccati-type differential inequality:
d dt
M2 <5D max
1
} 3ΣΣ Σ
(a:)))2 (af 28 1 8(M2)²
N1
N₂
eEE LJEE
5D-k k
M2 +
28
M2
NW (M2)²
(30)
N'
No
5D k + 28
8
M₂
(M2)
N'
N
Let y be solution of the following ODE:
y = ay - by
(31)
Then, by phase line analysis, the solution v() to Eq. 31 satisfies
M2(t) < vt) < max ( a
M2(0)
} = max
(5 k + 28)
M2(00 =: M₂.
(32)
SN
which yields the desired estimate.
16

---

Proposition B.2. For Eq. 4, the distance of the centers of the two clusters is finite.
Proof of Propostion B.2. By Lemma B.1
N1
22
| - -
(2)
N2
==1
==1
N1
N2
(33)
S 2
12
(2)
N₂
j=1
S V2 MF
where the first inequality used the Cauchy-Schwarz Schwarz inequality.
Lemma B.3. Let u, v be the solution to Eq. 21. Then || - -x(2) 1| satisfies
d -
2cm
1 - =122 - A(2g)²
2 20
2120 2
a
No kM2.
(34)
No
Proof. The time evolution of xi(1 is given by
N1 N1 i=1 (EE(()
N1 Σ N1
$00) =
Σ
(1gg - xz (1)
j.s (x) (
i=1 jEE(()
1 ΣΣ
h (z)
(2)
x 1
NI
EEE iJEC
(35)
(2)
(L)
N1
N1 EE7
(2)
(1)
4
x
N
N
3E0
where the first equality uses the relation
(1) 0.
Then we have
1 d [1(1] - 122
(36)
dt
=(x(1) -
(2))
(37)
=(5() - x(2)
Σ
(2)
1
(1)
Σ
1
(1)
N₂
N1
x
N₂
x
+
N₂
x
LEE3
3EE
(38)
=(=(1)- - (20)
N1
(2)
N₂
1
3
(2)
+
N1
11
e. (1)
x
N2
1
Iz
(39)
330
=(F(1) - 1(2))
Σ
(x
+
(2)
_ (2)
+
(2)
+
(z(1)
+
a((1)
N
N₂
N
3EF
(40)
+ N₂
2 (x(1) + +
(41)
=(x(1) - 2(2))
1
((2) +
[
+ 1
ΣΣ
N
N₂.
3EE JEC
iEe
(42)
17

---

1 1
ΣΣ
Vi
1000 +
Ni
1
ΣΣ
(43)
N1
N₂
N₂
jEE
EEE iEe
2
(x(1) x(20)
ΣΣ
x(1)
1 xi(2)
(44)
No
LEEE iEe
EEE jEE
+
2
(I(1) -
ΣΣ
1 (1)
ΣΣ
3 - (
(45)
No
CEE FEE
AEE JEe
We denote
Pes (t) ..(2) = ΣΣ
W
ΣΣ
1 (2)
EEE IEE
EE7 JEE
and
(z) x2))
EΣ
4 2(1)
ΣΣ
6/ (2)
EEE
EEE
Assume there exist constants Cm, Cai, such that
(e)( x(0) 2 cm (0(1) -21))
(46)
Then, by Cauchy's 's inequality, for any 01, we have
1 d
[z(1) - =2(2)12
dd
2
(es((1(1), x00)) - -x(0)) + ((0)((()) - x(2))
2
No
No
2cm
|, - 1 + c (1 11
2
cj
No
No
2Cm
1,
-/2 11
1 2
(Pes(E(1), ,.(221) ²
01
No
a No
2
2cm
(1
122
4
Σ
ws 2
+
Σ
(47)
No
No 3Ef
iie
jEa
2cm
(,((0) =1
4
Σ
||00 112
(1:(1) + ||005=11
112
>x
No
No
33.
BEe
JEE
20m
-a)
(|x(1) - x
4(Dr) )2
Σ
Σ
122 +
Σ
No
No
338
jEc
2Cm
||
122
4(DZ)²
k[M2).
No
No
Lemma B.4. Let u, v be the solution to Eq. 21. Then M2 satisfies
1
M₂ - C₂M₂ + 202l1= - z 122
(48)
2 dl
where
C2 -k (oa
D
D (2-)
(49)
2
402
and O2 is an arbitrary positive constant.
18

---

Proof. Subtracting Eq. 35 from Eq. 21 gives if!)
Then we have
N₂
1 d
2 dt
N1
N
1
(1))
N 1=1
N1
1
Σ
Σ
Net 2
(x 11 -2(-)
Σ
hig (u] - )
N
1=1
(c
K'Ee
jEe
N1
(1) (1 - (3)
+
N
2m
N1
(50)
(1)
Ri (+ - =1(
Σ
Mis (1) (2) - 2.(i)
N1
1=1
(EE(()
L'Ee
JEE
8
(1) (1)
(1- (c) (2)
+
N1
1 N
N₁
Σ
(ag 2(11)
(1)
Σ (EE(() jEe
Σ
M (0
N1 1=1
10
N1
(E) FeEe
Y=1
N1
Σ
Σ
Σ
(Z(
1o (x(2) - m(1)
8 N
(1- (3)
()
+
x
(EE( jEC
1=1
=: 1 1 + 13 14
41 can be defined by
N
11
Σ Σ (EE Mee
NI 1=1
Σ Σ
1 (4 (1)
- yo!
(1)
(51)
EEE FNEE
(
N1 3EE
where (f(1) = (6
x
(1) T and A given by Eq. 22 for each e. Thus 7 is bounded by (
No
1
Σ (((
1 ( S
1 E
G
Σ CAPPP
(52)
N
N EEE
N
3E0
where c, is constant larger than 1 related to the repetition of (3!")) } in all hyperedges. In can be controlled by
N
Ip = Ni
(1)
Σ
Σ
R.j (5f -20)) (
(53)
(EC()) JEc
Σ
(2)
(1) (1)
(54)
N1
EEE LEEE
Σ
(1)
ΣΣ 11?"
(55)
N
Ni
GEE LeEe
GEE i.jee
19

---

Σ
Mi 2 (1000) 1| 13) +
1 Σ
De Σ
11
(56)
N1
N1
BEE IJE
3E0
iEe
1
(3 112 +
3D
12
(57)
2N
2N
cEE JEE
EEE iEe
D
|
10
3D
ΣΣ
11a))
1|2
(58)
2N
2N1
EEE jEC
EEE iEe
N₂
N1 Σ
D
Σ
1
12
3D
1
(59)
2N
2N1
1g has the below estimate for any constant C2 0:
1 N1
1/ - Ni
Σ
Σ
ho (1 (1())=))))) 200)
(EE(1) JEE
|c2||=(1) - x2) 12 +
1
1 1 11
402N1
EEE ijEc
(60)
+ (D-)²
(021((1)=x(0)
Σ
l 10
41
CEE LJEE
(D )
N1
So21| 12
| ( 12
4c2N1
3EE
Define
N1
(1) (1) (1 - 119)
4
N1
N
8
(3(") +
- (100)(1-1011)
N1
(61)
8
N,
No Σ
(1)) ())
+
8 N₂
Σ
2 (1) (1)
8 N₁
Σ
2 (z. (x
(1) 2
N1
N1
N1 1=1
N1
i=1
i=1
8
N1
(a(1))
5
Σ
N₁
(1) (1)
12 (1)
N1
N1
Note that
Ni
N1
11
11) = Σ
|| (11
1112 - -0 (1)
1=1
|a
(1)
(2((( 1111 1,000 (13)
N
(62)
|1a5"21 (
1
Σ
|
1=1
N
11
(1)
11=
1
(11=
(1)
133
2
2.N
1=1
20.
Hence,
14 S (M).
(63)
20

---

Then
Ni
d
1
Σ
|
1
dt
2N
C
3D- -
(D-)²
Σ
Nr
| +
D
Σ
N₂
||
1 + -)||=(1)- - =22 (64)
Nj
2.N1
Nc
2N2 j=1
i=1
k
-- +
3D
+
(u) + k
D
M2(u) + +c21=>!)) - -x(3)12
402
Similarly,
N2
d
|2
1012 1|?
dl
2N2
j=l
(65)
(-cM
3D
)D-)²)
D
<
+ 2
+
+ M2 +
M2(u) + +22l(=(0) x12
402
Summing them together gives
d (M²) S - (ca
D
(D
(M2 202 2021lx111 -z!2) 212)
(66)
di
2
402
K.
This gives an exponential growth estimate or M2 up to an error term of || 122)
Proposition B.5 (L2 separation of HAMP-I). For Eg. 4, suppose the above assumptions are satisfied.
Define the mean value == Σ x; and the second moments M2(x) == Σ x². Thenfor sufficiently
large N1, N2, there exist constants x_ A_ such that if the initial data satisfies
M2
A(O) = (x(1 (03))
S X++=
(67)
then, there holds that the L2 separation
Mz(t)
X(1) ((0 S + (X(0) d_
(68)
with positive constant μ., where Mz(1) (2 + (1))
Proof of Proposition 5.2. Lemma B.3 gives an exponential growth estimate of ||x(1) - x(2) up to error term of M2. If No is large enough, the coefficient of the error term is small.
Lemma B.4 gives an exponential growth estimate M2 up to error term of [|x(1) (2) If
large enough, the coefficient of the error term is small.
Set
2Cm
4(D,)
C4 [Ca_ D- (D-)²)
A11
c1,
A12
A21 202, A22
(69)
No
No
4o2
If Ng is large enough, Eq. 69 is satisfied.
(An + A12)² - 4.221 A12 > 0.
(70)
Apply Lemma 4.1 in [27] then we obtain the T separation.
Remark B.6. We alternatively prove Theorem 5.2 by the method of [[44) Proposition 2 and 3). Proposition B.7 (Lower bound of the Dirichlet energy). If the hypergraph H is connected one, for Eg. with the conditions of Theorem 5.2, or for Eq. 5 with conditions of Theorem 5.1 ( [19], there exists a positive lower bound of the Dirichlet energy.
21

---

Proof of Proposition 5.4. The relative size between | - 1 and M2 is an indicator of group separation in the sense of L2: if || 112 is much larger than M2. then the two groups well-separated in average sense. Since the hypergraph is connected, there is a positive bound between different clusters, hence the Dirichlet energy does decay zero.
Remark B.8. The proof for the second-order system Eq. can be proved in similar way. Remark B.9, The separability of Eq. 6 which is the Eq. 4 or Eq. 5 with stochastic term also holds.
C Experiment Details
C.1 Dataset Details
We selected the most common hypergraph benchmark datasets, and the statistics of nine datasets across different domains are summarized in Tab. 3. The key statistics include the number of nodes, hyperedges, features, classes, average node degree de, average hyperedge size [E], and CE ho- mophily [36]. These variations highlight the diversity dataset scales and structural patterns, which may influence model performance hypergraph-related tasks.
Table 3: The summary of data statistics.
Dataset
# nodes
# hyperedges
# features
f classes
avg. d,
avg. |8|
CE homophily
Cora
2708
1579
1433
1.767
3.03
0.897
Citeseer
3312
1079
3703
1.044
3.200
0.893
Pubmed
19717
7963
500
1.756
4.349
0.952
Cora-CA
2708
1072
1433
1.693
4.277
0.803
DBLP-CA
41302
22363
1425
2,411
4.452
0.869
Congress
1718
83105
100
427.237
8.656
0.555
House
1290
340
100
9.181
34,730
0,509
Senate
282
315
100
19.177
17.168
0.498
Walmart
88860
69906
100
11
5.184
6.589
0.530
C.2 Additional Ablation Studies
Impact of Hidden Dimension HAMP-I and HAMP-II We explore the tolerance of our model to different hidden dimensions, as shown in Tab. For simplicity, we only vary the size hidden dimension, while other parameters remain fixed. Overall, these results demonstrate the robustness of our methods with varying hidden dimensions. is worth noting that high hidden dimension is key to achieving the best performance of HAMP.
Table 4: Impact of hidden dimension evaluated on the hypergraph datasets.
HAMP-1
HAMP-II
128
256
1024
128
256
512
1024
Cora
80.19
80.55
81.18
79.85
79.59
79.54
80.80
79.60
Citeseer
73.39
72.89
75.22
72.40
73.95
74.36
75.33
75.19
Pubmed
88.49
88.85
89.02
88.82
88.48
88.48
89.05
88.78
Senate
63.24
65.35
69.44
66.62
63.52
63.10
70.14
62.96
House
71.21
70.62
72.66
72.72
68.30
69.47
72.60
71.21
Impact of Repulsion, Allen-Cahn Force, and Noise on HAMP-J and HAMP-II. We summary ablation studies to investigate the individual and combined effects of repulsion f5. Allen-Cahn
force fa, and noise B, on both HAMP-I and HAMP-II Tab. 5 reports the average node classification accuracy with a standard deviation across seven standard hypergraph benchmarks over 10 runs. Models with the repulsion term enabled outperform their counterparts in some dataset, indicating
22

---

enhanced ability to distinguish node representations in complex hypergraph structures. The synergy between the repulsion and Allen-Cahn terms further boosts performance, confirming that these particle system-inspired mechanisms play complementary roles. Overall, these improvement confirm the validity of the HAMP construction and further highlight the significant advantages of incorporating particle system theory into the hypergraph message passing learning process.
Table 5: Ablation studies some standard hypergraph benchmarks.
The accuracy (%) is reported with standard deviation from 10 repetitive runs. (Key: fø: repulsion; fa: Allen Cann; B,: noise.)
Homophilic
fa
B.
Cora
Citeseer
Pubmed
Cora-CA
x
x
1..0
70.53+ 1,56
87.98+0.38
83.13+1.22
V
x
76.40=1.56 x
70.85
88.25=0.50
83.15= 1,36
X
V
x
8..11=1.41
74.83+1.70
88.90=0.45
84.77=1.16
HAMP-1
X
x
75.67
1.45
87.93.10.52
1.0
1..4
74.96
88.8711.40
85.21
X
V
1..5
74.67
88.77.10.4
8.05
V
11.18=1.30
7..22+
89.02::0.49
8.12
X
X
x
1..42:
71.50 1.49
88.68
83.60
X
x
1.70
72.20
88.59:10.00
+2.95+11.35
1..13+11.26
1..3
88.86+
84.37
HAMP-II
X
77.18=1..11
1.675=
88.6810.59
12.91
x
19.70=11.36
1..99
88.8210.48
84.30=1.32
X
19.50:11225
74.25
88.80110.40
1..31=1.44
80.80+1162
75.33+1.61
89.05=0.41
84.89
Table 6: Node Classification on standard hypergraph benchmarks. The accuracy (%) is reported with standard deviation from 10 repetitive runs. (Key: fō repulsion; fa: Allen-Cahn; B₁: noise.)
Heterophilic
f3
fa
Congress
Senate
Walmart
House
x
x
x
93.51
60.70-18.38
0..5
70.50 1.45
x
93.70
60.00+18.66
69.80=10.45
70.56:11.96
X
V
x
94.79-:1.14
66.76-t5.44
69.77--0.88
71.55++1.55
HAMP-1
X
x
v
93.47=1.113
60.1417.54
69.86=0033
69.8812.48
x
94.58=:1.25
67.75=18822
69.76=1.37
71.58:11.77
X
94.67=11002
65.6312.98
69.7310.49
71.55
95.09-10.79
69.44+66009
69.90-0.38
72.72+11.77
X
X
94.79-10.73
58.73+7.03
69.84+-.225
69.85+1166
X
x
94.02.11.10
61.9515598
69.91
69.35.1..77
X
X
94.1911.07
62.82.16.44
69.8910.31
70.96.12.06
HAMP-II
X
94.35.11.14
60.14
69.8410.37
70.46
x
94.58=00.6
61.97+8.42
69.92=0.33
1.63
X
94.12+0..3
64.51#4.19
69.86-0.34
70.96
95.26=11.34
70.1426.08
69.94-0.37
72.60
C.3
Time-Memory Tradeoff Analysis
We intuitively reveal the differences of different methods with a single-layer network in the time- memory trade-off Walmart dataset. As shown in Fig. 5, HAMP-I and HAMP-II methods demon- strate a notable trade-off between time efficiency and memory consumption. The experimental results reveal:
Memory usage: HAMP-I and HAMP-II maintain memory consumption within 11000-12000 MiB, achieving 20-26% reduction compared to ED-HNN, while being comparable to HDDSode
Time efficiency: Although the time consumption for HAMP-I and HAMP-II runtime slightly exceeds ED-HNN (0. it outperforms the baseline HDS
23

---

16000
Memory Usage (MiB)
15000
ED ED-HNN
14000
13000
HAMP-II
12000
HAMP-I
HDSode
11000
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Time Consumption (s) SD
Figure 5: Time- Memoryy tradeoff analysis of different methods on Walmart dataset. SD denotes the standard deviation of time consumption.
C.4 Hyperparameters
To ensure fairness, we follow the same training recipe as ED-HNN. Specifically, we train the model for 500 epochs using the Adam optimizer with the learning rate of 0.001 and no weight decay during the training phases. And we apply early stopping with a patience 50. For the stability, we run 10 trials with different seed and report the results of mean and the standard deviation. All experiments are implemented on an NVIDIA RTX 4090 GPU with Pytorch.
We explore the parameter space by grid search, where the search ranges for each critical hyperparam- eter are delineated below:
Dropout rate in (0.1, 0.2. 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9);
Layer number of classifier in 2,
The hidden dimension of classifier (128, 256, 512);
Hidden dimension of model in (128, 256, 512, 1024);
step size of solver in (0.09, 0.1, 0.15, 0.2, 0.25);
Y of repulsive force in (0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.11, 0.12, 0.13, 0.14, 0.15);
Initial values learnable parameters of damping term in (0, 2, 4, 5, 6, 8, 10, 11,
12, 13, 14, 15);
Imitial values of learnable parameters e of noise term in (0, 0.1, 0.3);
Tab. and Tab. 8 summarize the best hyperparameters on standard hypergraph benchmarks using HAMP-I and HAMP-II, respectively. For fairness, linear layer is added to perform feature mapping when conducting odd experiments. The optimal hyperparameters for node classification on standard hypergraph benchmarks is achieved by the HD$ algorithm, as demonstrated in Tab. 9.
24

---

Table 7: The best hyperparameters of Node Classification on standard hypergraph benchmarks using the HAMP-I algorithm.
Dataset
model. hd
cls. hd and # layers
time
step size
?
dropout
Cora
512
128, 1
0.1
12
0,05
0.4
0
Citeseer
512
512, 1
0.6
0.1
6
0.05
0.2
0
Pubmed
512
256,
0.2
0.1
0.1
0.5
0
Cora-CA
512
512, 2
0.2
0.05
0.9
DBLP-CA
256
128, 2
1.1
0.1
11
0.12
0.2
0
Congress
128
128, 2
1.4
0.1
0.08
0.3
0
House
1024
512.
L.55
0.15
0.05
0.8
Senate
512
256, 2
0.6
0.1
0.05
0.7
Walmart
256
128, 2
1.75
0.25
0.02
0.3
Table 8: The best hyperparameters of Node Classification on standard hypergraph benchmarks using the HAMP-II algorithm.
Dataset
model.
cls. hd and layers
time
step size
Y
dropout
Cora
512
512,
0.12
0.3
0.1
Citeseer
512
512,
1.8
0.15
8
0.13
0.6
0
Pubmed
512
256, 1
0.6
0.09
5
0.09
0.3
0
Cora CA
512
128, 2
0.75
0.25
3
0.01
0.7
0
DBLP-CA
256
128,
0.15
0.09
0.3
Congress
128
128,
0.25
0.01
0.3
0
House
512
256, 2
1.6
0.1
8
0.14
0.8
0
Senate
512
256, 2
0.2
13
0.05
0.3
0.3
Walmart
256
128, 2
2.5
0.25
0
0.02
0.3
0
Table 9: The best hyperparameters of Node Classification standard hypergraph benchmarks using the HDSode algorithm.
Dataset
model. hd
cls. hd and layers
# layer model.
alpha..
alpha..
step
Cora
512
256, 2
10
0.05
0.9
20
Citeseer
512
512,
0.05
0.9
20
Pubmed
512
256,
12
0.05
0,9
20
Cora-CA
512
512, 2
0.05
0.9
20
DBLP-CA
256
256, 2
15
0.05
0.9
20
Congress
256
128, 2
0.25
0.9
5
House
256.
0.05
0.9
20
Senate
512
256, 2
0.05
0.9
20
Walmart
256
128, 2
6
0.25
0.9
5
25

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
The abstract and introduction accurately describe the proposed framework (HyperMP), theoretical contributions (hypergraph particle dynamics system, anti-over- smoothing analysis), and experimental scope (hypergraph node classification), which are consistent with what is presented in the paper (Sections 3, 4, 5, 6, Appendices A, B, C).
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is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
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The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
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3. Theory assumptions and proofs
26

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Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
The paper states proposition (Proposition 5.2-5.4) and provides detailed proofs for the main theoretical results in Appendix B.
Justification:
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
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The proofs can either appear the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
The paper describes the experimental setup, datasets (standard hypergraph benchmarks), and baselines in Section The appendix C.2 provides dataset details, more ablation studies, and hyperparameter details. The code has been provided anonymously in the abstract.
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(c) the contribution a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
27

---

(d)
We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
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Reviews:
[{"review_id": "P5LewbJiM1", "reviewer": "Reviewer_9rG1", "summary": "This paper introduces a hypergraph message passing framework for node classification, inspired by interacting particle systems. Specifically, the proposed framework models each node as a particle, each hyperedge as a field that exerts a learnable attraction or repulsion forces, while an Allen–Cahn double-well damping force keeps feature norms bounded throughout the evolving time steps of the neural ODE system (which is analogous to the number of layers in traditional deep networks). The authors derive two architectures: HAMP-I, a first-order ODE that generalizes diffusion, and HAMP-II, a second-order (velocity-aware) ODE/SDE that naturally incorporates a Brownian noise to model uncertainty. The authors further show that this physics-inspired formulation gives a provable lower bound on the hypergraph Dirichlet energy, preventing the over-smoothing issues that occur on many GNN models. Experiments show that the proposed HAMP variants achieve up to 6.5% performance gain on heterophilic hypergraphs, and slight performance gain on homophilic graphs. Ablation studies further confirmed the effectiveness of the attraction–repulsion mechanism, the damping term, and the Brownian noise for modeling uncertainty.", "questions": "See weaknesses.", "limitations": "The proposed framework is only evaluated on the node classification problem. One suggestion is to add discussions about whether the proposed approach can be extended to other graph learning tasks such as link prediction.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## Strengths\n- The proposed approach is elegantly grounded on clear and explainable physical concepts. The authors effectively maps the information in the hypergraph into mathematical terms for modeling particle system framework (e.g., node feature as position or velocity), and the experiment results demonstrate the effectiveness of the modeling design. \n- Leveraging the physical theories backing the model, the framework clearly explains the limitations of existing hypergraph GNNs (e.g., due to the lack of repulsion or damping forces) and provides guarantee of the proposed model in preventing oversmoothing.  \n\n## Weaknesses\n- It is unclear to me how the authors model $f_\\beta(\\mathbf{x}_i, \\mathbf{x}_j, e)$ in their implementation and experiments. It would help with the reproducibility if the exact math formulation can be added in the main paper. \n- On most of the datasets, the proposed method's improvement is not statistically significant upon the best performing baseline based on the reported standard deviation. \n- The definition of homophilic and heterophilic hypergraphs are not given in the paper. This should be added to make the paper self-contained to readers.", "was_revised": true, "final_justification": "The authors sufficiently addressed my concerns for Q3 and Q4 in their rebuttals. \n\nThough I still have remaining concerns for Q1 and Q2, I personally do feel this is a strong paper and it meets the NeurIPS acceptance criteria. I found this paper inspiring and enjoyed reading it.", "rebuttal": "Thanks for your meticulous observation and insightful comments, we are deeply encouraged. All concerns are addressed below point by point. \n\n**Q1: How to model $f_\\beta(\\mathbf{x}_i, \\mathbf{x}_j, e)$.**\n\nThe interaction force in our particle system, $f_\\beta(\\mathbf{x}_i, \\mathbf{x}_j, e)$, integrates both attraction and repulsion. The attractive component is modeled similarly to ED-HNN [1], using a two-phase hypergraph diffusion process that propagates features first from nodes to hyperedges, and then from hyperedges back to nodes.\n\nCrucially, during the second phase (hyperedge-to-node), we introduce a repulsive force as a prior. This repulsion prevents homogenization across all nodes in a hyperedge, i.e., the uniform averaging of features. By leveraging prior information from the dataset to disrupt this averaging, the model can capture far more complex interactions.\n\nFurther details on the formulation and implementation of $f_\\beta$ are provided in Appendix A.\n\n**Q2: Performance improvements.**\n\nOur model, HAMP, demonstrates superior performance through consistent gains and stability across key benchmarks. As detailed in Table 1, HAMP achieves strong results on all nine datasets, including a notable 3% improvement on both the Senate and Walmart datasets.\n\nMore importantly, HAMP-II shows exceptional robustness as network depth increases. Figure 2 illustrates that while competing methods suffer from significant performance degradation at greater depths, HAMP-II  consistently outperforms them. This highlights HAMP's ability to capture complex representations while maintaining stability, establishing it as a valuable and reliable framework for deep Hypergraph Neural Networks (HNNs).\n\n**Q3: Definition of homophilic and heterophilic hypergraphs.**\n\nTo formalize the concepts of homophily and heterophily in hypergraphs, we adopt the Class-Edge (CE) homophily metric $H_{CE}$, as proposed in [2] and used by ED-GNN [1]. This metric quantifies the consistency of class labels among nodes within the same hyperedge: $$H_{CE} = \\frac{1}{|\\mathcal{E}|} \\sum_{e \\in \\mathcal{E}} \\frac{ \\max_{c \\in \\mathcal{C}} |\\{ v \\in e \\mid y_v = c \\}| }{ |e| }.$$ Here, $\\mathcal{E}$ is the set of hyperedges, $\\mathcal{C}$ is the set of node classes, and $y_v$ is the class label for node $v$. A higher $H_{CE}$ score indicates that hyperedges tend to connect nodes of the same class.\n\nFollowing established criteria [1], we classify a hypergraph as homophilic if $H_{CE} > 0.7$ and heterophilic if $H_{CE} \\leq 0.7$.\n\n\n**Q4: Other learning tasks.**\n\nOur current experiments focus on hypergraph node classification to ensure a fair comparison with state-of-the-art methods like ED-HNN and AllDeepSets. This approach also serves to validate our core message-passing mechanism on established hypergraph benchmarks. While this work centers on hypergraph node classification, we recognize the potential of HAMP for other learning tasks. Future work will explore extending its versatility to hyperedge prediction, implemented by scoring pairwise node representations, and to hypergraph classification, using a global pooling mechanism. These extensions will further demonstrate the robustness and adaptability of the HAMP framework.\n\n**References:**\n\n[1] Wang, et al. Equivariant hypergraph diffusion neural operators. In *ICLR*, 2023. \n\n[2] Pei, et al. Geom-GCN: Geometric graph convolutional networks. In *ICLR*, 2020."}, {"review_id": "jMAKe3LTkL", "reviewer": "Reviewer_nNPk", "summary": "This paper presents a framework for hypergraph message passing based on the perspective that the nodes are “particles” with features living in some embedding space, and these features are updated by pairwise “forces” between them mediated by the hyperedges.  \n\nThey highlight the importance of having both attractive and repulsive terms to prevent over-smoothing, and include as well a global damping term for stability of the method and separation of the classes.  \n\nTheir methods, HAMP-I and HAMP-II, update the node/particle features according to a first- and second-order differential equation, and they perform numerical studies demonstrating the effect of the damping terms on their performance.", "questions": "How is the repulsive force $f_\\beta^-$ implemented?  \n\nThe text referencing Table 2 seems to be off.  I see a larger increase from including Allen-Cahn $f_d$ than from the repulsive force $f_\\beta^-$.  \n\nIs the double-well potential needed?  Could one get away with a potential that just grows sufficiently quickly so as to maintain stability?", "limitations": "No other limitations aside from the concerns/questions listed above.", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Overall, the paper is written reasonably well.  And while I appreciate the physics-based perspective of particles interacting via forces, I am uncertain as to how the proposed framework is fundamentally different than message passing.  Equations 4 and 5 are special cases of message passing.  \n\nPropositions 5.2 and 5.3 already assume that within group $I_1$ or $I_2$ forces are attractive and between groups $I_1$ and $I_2$ forces are repulsive.  Though this seems unlikely in practice, if this were the case, then the partitioning problem is already solved.  \n\nIt is unclear to me how the repulsive force for $f_\\beta^-$ is implemented.  \n\nIt would be good to have error bars for the “significance plot” in Figure 3.", "was_revised": true, "final_justification": "## Outstanding concerns: \n\n### **The significance of the results is questionable.**\n\nMany of the numbers are in bold due to an accuracy difference of a fraction of a percent, and nearly all differences are less than the reported standard deviation.  As the only notable improvements are on the heterophilic datasets, I would need to know **how** the authors decided which nodes repel each other in order to assess the fairness of the comparison.  I asked this twice, and their responses did not leave me feeling I had a satisfactory answer.  The Class-Edge homophily metric requires knowing the class labels, and I still don't know how this translates to the magnitude of repulsion.  Is there just a single parameter controling this?  Is it tuned by hand to get higher accuracy?  \n\n*(First rebuttal to me, Paragraph \"Q2: How to implement the repulsive force.\")* \n> \"[...], we introduce a repulsive force. This force acts as a predefined prior to prevent homogenization---the uniform averaging of features across a hyperedge---by leveraging dataset-specific information. While effective, we note that a fully data-driven approach to modeling this repulsive force could further enhance the model's expressiveness, and we leave this for future work.\" \n\n*(Second rebuttal to me, Section \"2. Heterophily via repulsion\")* \n> \"Ideally, repulsion serves to maintain the separation of nodes. However, in practical applications, precisely identifying repelling node pairs remains a challenge. Consequently, in instances where an attractive force exists between nodes, we implement a reduction in the magnitude of this attraction to implicitly account for repulsion. While this approach is a rudimentary implementation, the predefined reduction, leveraging dataset-specific information, effectively counteracts homogenization. We now elucidate the nature of this specific information.  \n> \n> The Class-Edge homophily metric quantifies the consistency of class labels among nodes within the same hyperedge: $H_{CE} = \\frac{1}{|E|} \\sum_{e \\in E} \\frac{ \\max_{c \\in C} | { v \\in e \\mid y_v = c } | }{ |e| }.$ Here, ${E}$ is the set of hyperedges, ${C}$ is the set of node classes, and $y_v$ is the class label for node $v$. A higher $H_{CE}$ score indicates that hyperedges tend to connect nodes of the same class.\n> \n> For heterophilic datasets, such as Senate, where hyperedges predominantly connect dissimilar nodes, strong repulsion is indispensable for enhancing model performance. Conversely, homophilic datasets necessitate minimal repulsive interactions, thereby diminishing the utility of explicit repulsion mechanisms. Future research will explore data-driven methodologies to further refine the implementation of repulsion, aiming to achieve superior model performance.\" \n\n\n### **The lower bound on the Dirichlet energy is not particularly useful.**  \nTheir proof for the lower bound on the Dirichlet energy assumes *very* fine-tuned conditions.  In particular, my issue is with the assumption that the nodes can be partitioned into two groups, with no attraction between the two groups, and no repulsion within a group.  This is not just a technical condition (e.g., mild regularity conditions, or excluding a set of measure zero) --- it seems very unlikely to hold in practice, and seems to require that perfect node classification has already been achieved with zero outliers.  I am not convinced that this result constitutes a \"theoretical guarantee\" about their model's dynamics.  \n\n*(Second rebuttal to me, Section \"1. Theoretical Interpretation\")* \n> \"The incorporation of repulsive forces, provided certain conditions are met, endows the Dirichlet energy with a lower bound, thereby mitigating over-smoothing.\" \n\n\n### **Text does not match figures.**  \nFigure 2: Allen-Cahn seems to help more than pairwise repulsion.  Text says the other way around.  \nFigure 3: Including noise doesn't seem to change much (is the legend correct?).  Text suggests it does.  \n\nOn this note, they provided a table of accuracies with and without noise in (Second rebuttal to me, Section \"3. Impact of noise on HAMP\").  However, like the other numerical results about accuracies, when framed as a z-score, the results are rather lukewarm.  \n\n\n### **Final Justification** \nWhile I appreciate the physics-inspired model for hypergraph message-passing, I still have some issues with the way they have presented their particular implementation.  Given these outstanding concerns, I would like to keep my current score.  \nHowever, if there is significant desire from the other reviewers, I would not want to stand in the way of publication.", "rebuttal": "We appreciate your recognition of the writing and motivation of our work. Please find our detailed responses to your comments below.\n\n**Q1: Fundamental differentiation from classical message passing.**\n\nWhile Equations 4 and 5 can be expressed in a message passing form, the key novelty of HAMP lies in introducing a particle system perspective to message passing: we view message passing as the result of a physically inspired interacting particle system. This perspective enables several conceptual and practical advances beyond standard message passing schemes. \n\nFirst, traditional message passing typically aggregates information based on local neighborhood similarity and implicitly assumes homophily. In contrast, the proposed particle system formulation explicitly models both attractive and repulsive interactions between nodes, allowing it to handle heterophilic structures where neighbors may belong to different classes. The repulsive force mechanism is especially effective in promoting class separation, as demonstrated in our experiments on heterophilic datasets.  \n\nSecond, a known side effect of repulsion is feature explosion in deep networks. To address this, we introduce the Allen-Cahn regularization term, inspired by phase transition dynamics, which stabilizes the system by imposing a double-well potential. This term promotes polarization of node features while keeping them bounded, allowing for deeper and more stable message passing.  \n\nThird, our formulation provides a physically grounded justification for several behaviors observed in hypergraph networks, including over-smoothing. We prove that the particle-driven dynamics maintain a positive lower bound on the Dirichlet energy, analytically mitigating over-smoothing—a known limitation of diffusion-based message passing.\n\n\n\n**Q2: How to implement the repulsive force.**\n\nThe interaction force in our particle system, $f_\\beta(\\mathbf{x}_i, \\mathbf{x}_j, e)$, integrates both attraction and repulsion to model complex interactions. The attractive component operates via a two-phase hypergraph diffusion process, similar to ED-HNN, that propagates features from nodes to hyperedges and back. Crucially, during the second (hyperedge-to-node) phase, we introduce a repulsive force. This force acts as a predefined prior to prevent homogenization---the uniform averaging of features across a hyperedge---by leveraging dataset-specific information. While effective, we note that a fully data-driven approach to modeling this repulsive force could further enhance the model's expressiveness, and we leave this for future work.\n\n\n\n**Q3: Figure 3 without the error bars.**\n\nThank you for this valuable suggestion regarding statistical rigor. We agree completely on its importance. For Figure 3, we found that adding error bars directly to the main plot created significant visual clutter, especially in the deeper layers ($>$25) where the critical performance divergence occurs. This overlap obscured the figure's primary insight. However, the main takeaway is visually unambiguous: our models with stochastic components maintain $>$60% accuracy at layer 35, while the noise-free versions collapse to 50%. This substantial performance gap demonstrates the enhanced stability our method provides. To fully address your point while maintaining the figure's clarity, we have now included a version of Figure 3 with complete error bars in the supplementary material. We believe this approach provides the requested statistical validation without compromising the interpretability of the main plot.\n\n\n**Q4: Functional distinction and necessity of the double-well potential**.\n\nAs shown in Table 2, our framework features a crucial functional distinction between its two core components. The repulsion term, $f_\\beta$, governs pairwise node interactions within hyperedges, primarily enforcing feature separation. In contrast, the Allen-Cahn potential, $f_d$, operates on individual node features, creating a double-well energy potential that preserves their intrinsic stability throughout the learning dynamics.\n\nThis second mechanism ($f_d$) is particularly impactful in heterophilic environments like the Senate and House datasets, as it directly counters the homogenization of node features during aggregation. While simpler monotonic potentials could provide basic stability, they lack the double-well's capacity to induce meta-stable phase separationl, which actively promotes class-aware feature clustering.\n\nThis dual-component design explains the performance variations observed: some datasets benefit more from the robust feature preservation offered by the Allen-Cahn potential, while more interaction-centric datasets rely heavily on the pairwise repulsion from $f_\\beta$. Our framework intentionally unifies these mechanisms to adeptly handle such diverse hypergraph topologies."}, {"review_id": "7BsHbiHrC6", "reviewer": "Reviewer_5ffs", "summary": "This paper introduces Hypergraph Atomic Message Passing (HAMP), a framework inspired by particle systems, to address higher-order relationships in hypergraphs. Utilizing concepts like attraction-repulsion forces and Allen-Cahn potential dynamics, HAMP improves traditional hypergraph neural networks by modeling hyperedges as fields influencing particle-level node interactions. Theoretically, HAMP maintains a positive lower bound on hypergraph Dirichlet energy, ensuring deeper message passing without sacrificing efficiency.", "questions": "1. How does HAMP perform computationally on hypergraphs with millions of nodes and edges? Are there optimizations to alleviate complexity at scale?\n2. HDS^ode also models the hypergraph dynamic system, what are the main advantages of HAMP being more effective compared to it?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- Strengths:\n1. HAMP leverages particle system theory to model interactions, bridging physics concepts with hypergraph learning for deeper insights.\n2. The integration of repulsion and Allen-Cahn dynamics effectively reduces over-smoothing, enabling deeper architectures.\n3. Experimental results show HAMP achieves superior accuracy, particularly on challenging heterophilic hypergraph benchmarks.\n4. The code is available.\n\n- Weaknesses:\n1. The use of first-order and second-order systems adds computational overhead, potentially limiting scalability to larger hypergraphs.\n2. While effective for classification, broader application scenarios (e.g., graph classification) remain unexplored.", "was_revised": true, "final_justification": "Comprehensively considering the original paper's innovation and contribution, and the authors' rebuttal, I keep my score (Borderline accept).", "rebuttal": "We really appreciate you for the valuable insights. All concerns are addressed below point by point.\n\n**Q1: Scalability of HAMP.**\n\nAs mentioned in the appendix A and C.3, we analyze the computational complexity of one layer in HAMP-I/II. This aligns with efficient ED-HNN. We show Time-Memory tradeoff analysis of different methods on Walmart dataset ($|\\mathcal{V}| \\approx 88k$, $|\\mathcal{E}| \\approx  69k$). Results (Figure 5) show: HAMP-I/II use 20–26% less memory than ED-HNN. Runtime is near-identical to ED-HNN ($\\Delta t \\approx 0.1$s) and faster than HDS$^{ode}$.\n\n**Q2: Broader application scenarios.**\n\nOur current experiments focus on hypergraph node classification to ensure a fair comparison with state-of-the-art methods like ED-HNN and AllDeepSets. This approach also serves to validate our core message-passing mechanism on established hypergraph benchmarks. While this work centers on hypergraph node classification, we recognize the potential of HAMP for other learning tasks. Future work will explore extending its versatility to hyperedge prediction, implemented by scoring pairwise node representations, and to hypergraph classification, using a global pooling mechanism. These extensions will further demonstrate the robustness and adaptability of the HAMP framework.\n\n**Q3: The main advantages of HAMP compared to HDS$^{ode}$.**\n\nAs discussed in our related work, HDS$^{ode}$ adopts control-diffusion ODEs to model hypergraph dynamic systems. In contrast, we design hypergraph message passing from the perspective of particle system theory, which inherently alleviates the over-smoothing problem. Previous methods predominantly neglect a key physical insight: **message passing is inherently a particle-level phenomenon, involving local, interactive dynamics that cannot be fully captured by pure diffusion models**. Our core innovation lies in modeling hyperedges as dynamic interaction fields between node-level particles, where class-dependent equilibrium distributions in the feature space are achieved through particle dynamics equations incorporating attractive-repulsive forces and Allen-Cahn potential regularization."}, {"review_id": "J5slluA0GD", "reviewer": "Reviewer_L9c7", "summary": "This work proposes a novel hypergraph message passing framework inspired by particle system theory. The work suggests two models for message passing for HNNs based on first-order and second-order system equations. The suggested models are proved to mitigate over-something and empirically shown to produce good results on the tested benchmarks.", "questions": "What is bi-flocking? (line 200)", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\nThe suggested method has a theoretical background which ensures that it avoids over-smoothing\n\nAn ablation study was conducted, showing that both repulsion and Allen-Cahn improve Node Classification results.\n\nThe suggested method generally performs better than the compared baselines\n\nWeaknesses\n\nAs far as I know, Cora, Citeseer and Pubmed are not Hypergraph datasets but graph datasets.\n\nThe improvement of HAMP is insignificant with respect to the second performing baseline.\n\nFrom the results, HAMP-|| indeed outperforms the other baselines when using deep layers, but its performance converges very fast, even when the network is shallow, so there is no extra advantage in using deep HAMP-||?\n\nIt would be interesting to see the performance of the model on particle system datasets.", "was_revised": true, "final_justification": "Since the rebuttal did not include any new experiments involving additional datasets, and the reviewers did not clearly specify how the new clarifications would be incorporated into the main text, I have decided to maintain my original score.", "rebuttal": "**Q1: Dataset Type (Cora, Citeseer, Pubmed)**.\n\nWe appreciate the reviewer's vigilance regarding data types. The reviewer is correct that Cora, Citeseer, and Pubmed were originally introduced as graph datasets. However, in hypergraph representation learning research, it is standard practice to convert these citation networks into hypergraphs to capture higher-order relationships. The key transformation lies in defining the hyperedges. Specifically, following the methodology of ED-HNN [1] and common practice in the field, for co-citation networks (Cora, Citeseer, Pubmed), all documents cited by a given document are connected by a hyperedge. As previously stated, ED-HNN, which our work builds upon, explicitly follows this established methodology. This conversion allows for the evaluation of hypergraph models on established benchmarks.\n\n**Q2: Performance improvements.**\n\nOur model, HAMP, demonstrates superior performance through consistent gains and stability across key benchmarks. As detailed in Table 1, HAMP achieves strong results on all nine datasets, including a notable 3% improvement on both the Senate and Walmart datasets.\n\nMore importantly, HAMP-II shows exceptional robustness as network depth increases. Figure 2 illustrates that while competing methods suffer from significant performance degradation at greater depths, HAMP-II consistently outperforms them. This highlights HAMP's ability to capture complex representations while maintaining stability, establishing it as a valuable and reliable framework for deep Hypergraph Neural Networks (HNNs).\n\n**Q3: Extra advantage of using deep HAMP-II.**\n\nWe will clarify HAMP-II's depth behavior from both theoretical and empirical perspectives. As our theoretical analysis in the article shows, based on the particle system perspective, HAMP enables stable information propagation through deeper layers. From an empirical standpoint, HAMP-II achieves rapid convergence on locally structured datasets (e.g., Cora, CiteSeer) with shallow architectures (2-3 layers), demonstrating parameter efficiency. However, for datasets exhibiting long-range dependencies (e.g., Congress, Walmart), deep architectures are essential to capture multi-hop interactions. As depicted in Figure 2, on the Congress dataset, a 10-layer HAMP-II model improves accuracy over its 2-layer counterpart, whereas ED-HNN degrades due to over-smoothing. This demonstrates HAMP-II's unique capacity to leverage depth for complex topologies without performance collapse.\n\n**Q4: The performance of the model on particle system datasets.**\n\nExploring HAMP within the context of particle systems is indeed an interesting direction, as the continuous dynamics of such systems naturally align with our HAMP framework. Our current evaluation, however, focuses on discrete hypergraph benchmarks to facilitate direct comparison with the existing hypergraph neural network literature.\n\n**Q5: Definition of bi-flocking.**\nThank you for this insightful question, it has revealed an important conceptual inaccuracy in our original terminology. This prompted us to replace all instances of ''bi-flocking'' with the more precise term ''bi-cluster flocking'' adopted from [2].\nNext, we will elaborate on the concept of bi-cluster flocking, which is characterized by the following two conditions:\n- *Intra-cluster cohesion*: bounded position differences and velocity alignment within each sub-ensemble.\n- *Inter-cluster separation*: diverging positions between sub-ensembles.\n\n\n**References:**\n\n[1] Wang, et al. Equivariant hypergraph diffusion neural operators. In *ICLR*, 2023.\n\n[2] Fang, et al. Emergent behaviors of the Cucker-Smale ensemble under attractive-repulsive couplings and rayleigh frictions. *Mathematical Models and Methods in Applied Sciences*, 29(07):1349–1385, 2019."}]

Author rebuttal:

Meta-review: The paper introduces a hypergraph message passing framework for node classification, inspired by interacting particle systems.  The authors propose two models within the framework which provably mitigate over-smoothing, and evaluate them empirically.  The reviewers appreciate the physically inspired perspective, the provable power to avoid over-smoothing, and the empirical performance.  After extensive discussion, there are still some issues remaining.  Based on the discussion, the AC has formed their own opinion: the paper would bring positive value to the community, and the outstanding concerns -- while they are valid and reasonable -- do not necessarily lead to immediate "reject".  Overall the paper is close to the borderline.  Regardless of the final decision, we encourage the authors to further improve their paper based on the constructive comments (especially those by Reviewer nNPk).
Decision: Accept (poster)