HyperMagNet: A Magnetic Laplacian based Hypergraph Neural Network
Authors: Tatyana Benko
Martin Buck
Ilya Amburg
Stephen J. Young
Sinan Guven Aksoy
Venue: Accepted by TMLR ()

Abstract: In data science, hypergraphs are natural models for data exhibiting multi-way or group relationships in contrast to graphs which only model pairwise relationships. Nonetheless, many proposed hypergraph neural networks effectively reduce hypergraphs to undirected graphs via symmetrized matrix representations, potentially losing important multi-way or group information. We propose an alternative approach to hypergraph neural networks in which the hypergraph is represented as a non-reversible Markov chain. We use this Markov chain to construct a complex Hermitian Laplacian matrix — the magnetic Laplacian — which serves as the input to our proposed hypergraph neural network. We study $\textit{HyperMagNet}$ for the task of node classification, and demonstrate its effectiveness over graph-reduction based hypergraph neural networks.

Published in Transactions on Machine Learning Research (04/2025)
HyperMagNet: A Magnetic Laplacian based Hypergraph
Neural Network
Tatyana Benko
thenko Suoregon. edu Mortin.Buck Buck-tufts. edu
Martin Buck
University of Oregon Tufts Universily
Ilya Amburg Pacific Northwest National Laboratory
slyo. Myy..mmburg@pnnl..goo
Stephen J. Young Pacific Northwest National Laboratory
stephen. wwapheen..ounggpnn.ggo
Sinan G. Aksoy
sinam. www..aksoy@@pml...o
Pucific Northwest National Laboratory
Reviewed on OpenReview: https: // openreview.
/ttp::///peerevieu net/ forum? id=Gdf777sE2
Abstract
In data science, hypergraphs are natural models for data exhibiting multi-way or group rela- tionships in contrast to graphs which only model pairwise relationships. Nonetheless, many proposed hypergraph neural networks effectively reduce hypergraphs to undirected graphs via symmetrized matrix representations, potentially losing important multi-way or group information. We propose an alternative approach to hypergraph neural networks in which the hypergraph is represented as a non-reversible Markov chain. We use this Markov chain to construct a complex Hermitian Laplacian matrix the magnetic Laplacian - which serves as the input to our proposed hypergraph neural network. We study HyperMagNed for the task of node classification, and demonstrate its effectiveness over graph-reduction based hypergraph neural networks.
1 Introduction
A fundamental limitation of graphs in machine learning is the relationships that graph models sarily pairwise: edges connect exactly two vertices in graph. For example, graph edge may indicate two documents within a corpus share common vocabulary, two pixels in an image belong to the same neighbor- hood, or two discases result from mutations in the same gene. However, these relationships are emphatically not only pairwise, but involve interactions between groups documents, pixels, and discases. Across these and other domains, modeling multi-way relationships with graph therefore loses key information in the data. Hypergraphs, which allow more than two vertices be connected by an edge, faithfully capture the multi-way relationships graphs cannot capture (Aksoy al. 2020 Bretto 2013 Fong al. 2019). Data that is sel-valued, tabular, or bipartile is best modeled with hypergraph.
Both authors contributed equally this research. PNNL. Information Release: MINL-SAA194429 Please direct all inquiries
about code availability Sinan Aksoy.

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Despite being a more expressive and general model than graph, analyzing hypergraph data presents chal- lenges. In machine learning tasks, one must. first choose a represenlation of the hypergraph to be processed by a chosen algorithm. For example, convolutional neural networks use a Hermitian matrix representation of the hypergraph in order to learn optimal node embeddings via matrix multiplication with learnable weight matrices. However, classic matrix representations of graphs such as the adjacency matrix or the graph Lapla- cian have no direct analogues for hypergraphs. To overcome this challenge, representations of a hypergraph via a random walk have proved convenient. Zhou et al. (2006) demonstrated this, defining a clustering algo- rithm based on reversible random walk, where there is symmetry in the transition probabilities between any two vertices, and the eigenvectors of an associated hypergraph Laplacian. Building on this representation and inspired by the success of graph convolutional neural networks (GCN), the seminal hypergraph neural network HGNN (Feng et al. 2019) uses Zhou's hypergraph Laplacian in a traditional GCN defined by Kipf & Welling (016)) for classification tasks in visual object recognition and citation network classification.
Unfortunately, it is known (Agarwal et al., 2006) that Zhou's hypergraph Laplacian indoed reduces to a graph Laplacian on the star graph corresponding to a hypergraph, and other Laplacians reduce to those of the clique graph. In this sense, these Laplacians and the aforementioned hypergraph neural network HGNN reduce a hypergraph to a graph. The reason this information loss occurs is because these matricos are based on reversible random walks, which always reduce to a random walk on an undirected graph (Chitra & Raphael 2019). Thus, a more faithful approach is to define Laplacians based on non-reversible random walks. As shown by Chitra & Raphael (2019), hypergraph random walks which utilize edge-dependent verter weights (EDVW), in which transition probabilities are guided by hetex-hyperedge specific weightings, may be non-reversible. These weights, which allow vertices to have varying importance across the hyperedges to which they belong, often naturally present in data or may also be derived from structural properties of unweighted hypergraph data.
In this work, we build and study a hypergraph neural network that doesn't 't rely on a star graph or clique expansion reduction Laplacian. Rather, our proposed HyperMagNet begins with a non-reversible hypergraph random walk that more faithful capturing nuances within hypergraph data, which are reflected as asymmetries in transition probabilities. However, the transition probability matrix representing this random walk is not Hermitian which complicates their use within traditional convolutional neural networks. Instead of overcoming this by only symmetrizing the transition matrix (thereby converting to reversible, graph- matrix called
based random walk), we instead encode it as a complex-valuedi, Hermitiaan-but--symmeed the magnetic Laplacian. This Laplacian has been successfully applied in the graph learning community to directed graphs (Fanuel et al.] 2017 Zhang et al.) 2021 Shubin 1994). Studying its application in a hypergraph setting, we explain how to learn parameters of the magnetic Laplacian from hypergraph data, and build a hypergraph noural network around it. Finally, we investigate the efficacy of this approach, comparison to HGNN and related graph-reduction methods for the task of node classification. On varied data, we find that HyperMagNNet outperforms competing graph-based methods, sometimes modestly and sometimes significantly, and include experiments to test whether this increase in performance is due to the utilization of edge-dependent vertex weights, the magnetic Laplacian, or both. Although slightly more expensive lo run, MaperMagNel is worth using due to its increase in performance over graph-reduction based models, as shown across several data modalities.
The contributions of this paper are outlined as follows:
We propose a new hypergraph neural network, HyperMagNel, that doesn't rely on a graph repre- sentation of the hypergraph. Instead, HyperMagNNet uses a non-reversible hypergraph random walk that more faithful in capturing muances within hypergraph data.
We explain how to learn parameters of the magnetic Laplacian, a complex-valued Hermitian-brg but- asymmetric matrix, and build hypergraph neural network around it.
We investigate the efficacy of HyperMogNet in comparison to HGNN and related graph-reduction methods for the task of node classification.
The paper is structured as follows: in Section 2 we provide background on hypergraphs, hypergraph random walks, and the magnetic Laplacian. Tn Section we show how the magnetic Laplacian is an appropriate
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hypergraph Laplacian for the EDVW random walk and include comparisons with traditional hypergraph random walks and Taplacians based on clique graphs. We also introduce the neural network architecture of HyperMagNet in this section. Section | contains related work on hypergraph neural networks with and without EDVW. In Section 5 are experimental results in the tasks node classification on a variety of hypergraph structured data sets, where performance is compared against a variely of machine learning models based on traditional graph-basod representations.
+
HMN
84
Non-reversible Markov Chain
Spectral graph convolution with magnetic Laplacian
11
Standard
P'
HNN
17
approach
1Ug
EDVW Hypergraph
Weighted Graph
Spectral graph convolution with Laplacian
Predicted node labels
Figure 1: Haper (HMN) uses non-reversible Markov chain to build hypergraph Laplacian which avoids Laplacians associated with the star graph or clique expansion.
2 Background
2.1 Hypergraphs and Random Walks.
A hypergraph H EV, is set of vertices = (V1, ( vn) and hyperedges E == (ei,..., where each e V for i i= 1,2..... .., A graph is special case of hypergraph where all hyperedges have size 2. The vertex-hyperedgge relationship is given by the incidence malrir, Y E {0, ()VI)1,)) where y(v,e) is 1 if E and is 0 otherwise.
A simple approach to analyze a hypergraph is to represent as a graph in a way that retains important information from the hypergraph, thereby enabling the application of graph machine learning methods. One popular such graph representation the clique expansion which replaces hyperedges by sets of edges forming cliques. This graph has weighted adjacency matrix YYT, where the weights correspond to the number of shared hyperedges between two vertices. Another popular graph representation of hypergraph is the star exponsion G = (V*, E*) which introduces a new vertex for each hyperedge E E so that V = VU E and E- = ((v,e) E e, E).
Unsurprisingly, graph representations hypergraphs may lose information. A simple example is when many small hyperedges are contained within a larger hyperedge; the clique expansion will erase the relationships between vertices in the sub-hyperedges. Furthermore, non-isomorphic hypergraphs may have identical clique illustrate later in Figure 2b However, working with hypergraph-to-grapp reductions
and line graphs, is convenient because it enables application of graph learning methods. Graph random walks and Laplacians are used throughout machinc learning, such as in recommendation systems, spectral clustering, and graph convolutional neurul networks (GCN). In order to develop similar tools for hypergraphs, a spectral theory is necessary. Here, Zhou et al.) (2006) proposed a spectral clustering algorithm for hypergraphs based on a hyperedge partitioning problem rooted in simple hypergraph random walk. Letting w(e) > 0 denote the hyperedge weight associated with e, d(v) = ElegEleee) w(e) denote the vertex degree of v, and le| 5(e) denote the hyperedge degree, the random walker in this construction at time and vertex is proceeds by:
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1. Choose hyperedge 3 2 with probability proportional to hyperedge weight w(e)
2. Select vertex E e uniformly at random
3. Move to vertex Ut+1 = at time ++1
The transition matrix P for this random walk has the form =) w(e) glu.e ale Letting Dv, DE: and W denote the diagonal vertex degree, hyperedge degree, and hyperedge weight matrices, respectively, P is DW DET Zhou et al. then define hyperyyaph Laplacian as:
d(u)
A -
DV
This process of selecting a vertex uniformly at random an example of an edge-independent verter weighting (EIVW). For a hyperedge weighting function Te : V + R,, if 76(v) = (v) for all pairs e, e' containing v, then the collection feel is an edge-independent vertex weighting. Otherwise if the (e(v) varies across hyperedges, the vertex weighting is referred to as edge-dependent verter weighting (EDVW). EDVW appear across applications: in NLP where the weights are tf-idf values representing the importance of a word to a document (Bellaachia & Al-Dhelaan 2013); in e-commerce, where weights correspond to the number of items in a shopper's basket (Li et al. 2018|; or in biology, where the weights correspond to association scores between gene and a discase (Feng et al. 2021). In the absence of weight data, EDVW may also be generated from hypergraph structure, as explored later in Section 5.2
2.2 The Representative Digraph of Hypergraph.
Chitra Raphael (2019) show any hypergraph random walk using EIVW is equivalent to a random walk the clique graph. Consequently, hypergraph learning methods that use EIVW to construct hypergraph random walk hypergraph Laplacian may lose higher-order relationships in the data. To remedy this, Chitra & Raphacl (2019) show EDVW random walks are not necessarily equivalent to a random walk on any undirected graph such as the clique expansion. Inspired by Hayashi et al. (2020), we use the EDVW to represent the hypergraph via an EDVW random walk on the hypergraph where, if at vertex i2 at time t, the next step is chosen as follows:
Select hyperedge - U, with probability proportional to hyperedge weight w(e)
Select verlex v E with probability proportional to EDVW 7c(v)
3. Move to vertex ut+1 = v at time ++1
A key difference between the EDVW random walk and that defined in Zhou et al. (2006) that the former may be non-reversible (Chitra & Raphael 2019), while the latter is always reversible. Reversibility means that, in stationarity, the probability of transitioning from i to is the same from j to i (more formally, P.j. = P3 where æ is the stationary distribution and P is the transition matrix). Reversible random walks are cquivalent to random walks on graphs, whereas non-reversible random walks are not. Hence, the EDVW random walks camnot be characterized as random walks on the hypergraph's clique graph. This EDVW random walk generalizes the simple random walk defined by Zhou et al. (2006) since we allow for a larger class of probability distributions in the sccond step above, rather than limiting uniform distribution. The EDVW information is stored in weighted incidence matrix, R, where Rie is (e(v) if v e and is 0 otherwise. The transition matrix P for this EDVW random walk is given by p(u, u) = E wie) Ta(u) 2()) d(e) or P DD7'YWDD R² in matrix notation. We represent as a directed graph called the representative digraph of the hypergraph (Hayashi et al. 2020), with vertex set V and weighted edge set E - {(u,v) | Puv >0 0}. As this random walk may be non-teversible, special tools from spectral graph theory are necessary to construct a spectral-based neural network and define graph convolutions (e.g.. Defferrard et al. 2016;; Kipf & Welling (2016)) that rely on the spectral theorem.
We stress EDVW random walks which are key to our magnetic Laplacian approach can provably capture higher-order structure otherwise lost by graph approaches. We provide two examples that illustrate this in
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e
e2
e2
U1
7
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1
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@1
e2
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((e)) 1
U/
1
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((e2) = 5
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0
(a) Edge dependent vs. independent vertex weightings and the resulting reversible and non-reversible Markov chains
(b) Non-isomorphic hypergraphs with identical weighted clique and line graphs
Figure 2: Left: EDVW (top) vs EIVW (bottom) random walks yielding graph vs digraph random walks, Right: hypergraphs (top) with identical, crarsoction-weightee clique and line graphs (bottom) yet are dis- tinguished by their EDVW random walk properties and magnetic Laplacian spectra.
two senses. Figure 2a illustrates how EIVW hypergraph random walks may collapse to graph random walk, whereas factoring in EDVW instead may avoid this. Observing the differences between the representative digraphs also provide intuition about the different weightings: the transition probabilities from V/ i or 12 are equal in the EIVW case, whereas the large EDVW weight Te, (v) 7 yields larger probability of transitioning to e1 than 222 Figure 2b is an example from Aksoy al. (2024) of subtly-different hypergraphs with the same underlying graph information (as given by identical weighted clique and line graph expansions). yet are distinguished by EDVW random walk properties. A witness to their non-isomorphism is that an alternating varte--hyperede walk, the diameter of hypergraph S is 5, whereas for it is 4. Using the structure-derived EDVW weighting (i) deg(n)/E deg(ve), the maximum expected hitting time in an EDVW random walk on S is about 12.24 between nodes A and F, whereas this hitting time is 11.41 in hypergraph R. It can also be verified the magnetic Laplacian spectra differ across the two examples, which we define next.
2.3 The Magnetic Laplacian.
A remarkable tool for analyzing digraphs, such as this representative digraph of the hypergraph, is the magnetic Laplacian. The magnetic Laplacian is built on a complex-valued Hermitian adjacency matrix and has its origins in the quantum physics literature as the Hamiltonian of charged particle confined to a lattice under the influence of magnetic forces. An obvious difference between the magnetic Laplacian and the standard array of graph Laplacians is that direction information appears as a signal in the complex- plane. However, it still enjoys the properties that convolutional neural networks are built on such as being Hermitian and positive semi-definite.
Previous work using a complex-valus valued Hermitian adjacency matrix or Laplacian can be found in Cucuringu et al. 2019), Fanuel et al. 2018), Zhang et al. (2021), Fiorini et al. (2022). In Cucuringu et al. (2019) the eigenvectors a complex- valued Hermitian matrix are used to cluster migration networks in the United States, revealing long-distance migration patterns that traditional clustering methods miss. Similarly, Fanuel et al. (2018) the spectrum of the magnetic Laplacian is shown to be related to solutions of the segguaar-yynchrooiizatooo problem akin to how the spectrum of the graph Laplacian is related to solutions of the graph cut problem. The authors also successfully use the eigenvectors to cluster word-adjacency and political blog networks. [Zhang et al.] (2021) use the magnetic Laplacian to define a new graph convolution and build a neural network for directed graphs that beats state-of-the-art directed graph learning algorithms on a number of NLP data sets. There is also recent work on incorporating the magnetic Laplacian into graph neural networks that process signed and directed graphs, see Fiorini et al. (2022). He et al. (2022). In both Fiorini et al. (2022) and He al. (2022). a novel Laplacian derived from the magnetic Laplacian is introduced, specifically designed to handle signed and directed graphs in a spectral GCN. In
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Fiorini et al. (2022): the authors propose the sign-Magnetic Laplacian, which is parameter-free, unikee the magnetic Laplacian. In contrast, He et al. (2022) present the signed magnetic Laplacian which retains a single charge parameter.
The magnetic Laplacian defined as follows. Let A, be the symmetrized adjacency matriz A, = (A++A) and let D, denote the corresponding diagonal degroe matrix. Now, define (A) as the skew-symmetric phase matrix (A) == (rq(A-A²). For a given parameter q 2 O, the complex-valued Hermitian adjacency matrix (A) is given by an entrywise product between the symmetrized adjacency matrix and an entrywise exponential of the phase matrix:
H (A) = A, o exp(ie
(2)
The unnormalized magnetic Laplacian is D, H(()) while the normalized magnetic Laplacian is
o exp(ie()))
(3)
The parameter q 2 is called a charge parameter and determines how direction information is represented and processed. When q 0, the phase matrix (() and so (A) As, thereby removing direction information in the graph. When q * O, (A) may be complex-valued, whose imaginary components caplure direction information through the A - A² term in the phase matrix. As shown in Fanuel et al. (2018), varying changes how patterns and motifs in the graph are captured via the spectrum. Choosing an optimal value of priori is a difficult task and previous work has treated q as a hyperparameter. Since we consider weighted digraphs, a single value of q may be insufficient to capture direction information. To better accommodate different edge weightings when training HyperMag.Vet, we introduce a novel charge matrix Q (given later in Eq. 6 and ( of learnable parameters that replace the single charge parameter q in the Laplacian. As far the authors know, this is the first work to analyze hypergraph via the EDVW random walk and spectrum of a magnetic Laplacian which opens up applications within hypergraph signal processing by mimicking approaches to clustering, de-noising, and visualization as seen in Cucuringu et al. (209)) and Fanuel et al. (2018), for example.
3 HyperMagNet
3.1 A Hypergraph Laplacian and Convolution.
We define a hypergruph Laplacion using the non-reversible Markov chain captured in the representative digraph P,
=1 -,, exp(i0(0)
(4)
To simplify notation, the hypergraph Laplacian (()) will be written as (()), Since ( is a positive semi-definite matrix, by the spectral theorem, we have cigendecomposition of the hypergraph Laplacian [(A &A where & is the matrix whose columns its cigenvectors {di...., @n); and A is the diagonal matrix containing its corresponding non-negative eigenvalues, {AI,...,AaI) )n). A hypergraph Fourier transform of a function x E Ca on the vertices of the hypergraph is then defined as the representation of the signal in terms the hypergraph Laplacian cigenbasis, = Tx This mimics how a graph Fourier transform was defined in Shhman el al. (2016) and for directed graphs in ZZang el al. (202T). Furthermore, because the matrix + unitary, we can express the original function in terms of its hypergraph Fourier transform: o4 <(0'x) ΣN (k))(x. In classical Fourier analysis, a convolution of two functions f R² + C and g: R + C has the key property that convolution corresponds to multiplication of the Fourier transforms, f * = fg. Following Shuman et al. (2016) and Zhang et al. (2021), we define a hyperyraph convolution as pointwise multiplication of two functions C", y E C on the vertices of the hypergraph the eigenbasis:
x (k) = (k))
This can be expressod in matrix notation as y y F''. Therefore, for fixed function y its corresponding convolution matrix is C. @ Then, convolution with x can be expressed as the matrix-vector product y x Cyz. Following the construction of convolutio in Kipf & Welling (2016)

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and Feng et al. (2019), we approximate convolution in Eq. by truncated Chebyshev polynomial and renormalize the hypergraph Laplacian to have eigenvalues in the range [-1.1]. This circumvents the cost of computing an eigenbasis and performing the Fourier and inverse Fourier transform. The resulting hypergraph convolution has the form Cyæ= 0 (I D, D. P.D.
1/2 @ exp(i0(0)) where 8g is a learnable parameter, D, and Ps are the diagonal degree matrix and adjaency matrix corresponding to the re-normalized Laplacian [(9)= Amax [(9)- I.
3.2 A Learnable Charge Matrix and HyperMagNet Architecture.
We outline HyperMagNel's nelwork architecture and introduce a modification to the magnetic Laplaciann with a learnable charge matrix Q to accommodate the weighted directed edges present in the hypergraph's reprosentative digraph. Prior work Zhang et al. (2021); Fanuul et al. (2018); Cucuringu et al. (2019) studying the magnetic Laplacian and its applications in clustering and neural nelworks assume the underlying digraph is unweighted, which helps inform the choice of the single charge parameter g, typically in 0 S q S .5. When q ff 0, the phase encodes edge direction and the Hermitian adjacency matrix H (A) takes on four values: Ha (A) = O, exp (2miq), exp ( (-2riq), 1 if, respectively: there are no edges between u and v, only from u to v, only from v to u, and reciprocal edges. When g = 1/4, if there is an edge from u to but not from v - (A). In this case, an edge from to is treated the 'opposite" of an edge from to u (Zhang et al. 2021 Mohar (2020) asserts a natural choice is q == _/ as H (A) = m/// is now a sixth root of unity and thus two oppositely oriented edges have the same effect as a single undirected edge. Nonetheless, it is not clear a priori which choice of 2 0 is optimal.
to u, then H!!0 (A) = 3
For the representative digraph of a hypergraph, the edges are weighted and the justifications for a universal break down. Since @/ (P) = 2Tq(Pur - Paul, there are large phase angles when there is a high probability of moving from v to u but not u to v, ie. when the transition matrix is more asymmetric. All that is accomplished by sctting q 1/4 or q = 1/3 restricting the phase angle between -n/2 and x/2 or -27/3 and 2 /3, respectively. This means the effects of single charge parameter q are not sulficient for the entire graph if we are to mimic the effect of setting q to a fixed value. Therefore q should vary with edge weight in order to accomplish what a single q accomplishes in the unweighted case. As there is no optimal choice of q a priori, we introduce a charge matriz Q E R=*x matrix now has the form
n of learnable charge parameters. The Hermitian adjacency
H (P) = P. O exp(ie(()),
(6)
where (I9) 2TiQ @ (P-P"). except replacing H(9)) with H(9): II II
The weighted hypergraph Laplacian then has the same form as before
exp(ie(9))
To simplify notation, the weighted hypergraph Laplacian used in HyperMogNet (()) is denoted ((Q) and the renormalized version (()), Like with GCNs, each layer of the network will transform the previous layer's vertex feature matrix via matrix multiplication with a matrix of learnable parameters that correspond to the filter weights 0g in Eq. 5 We let L be the number of layers in the network and let X (0) be the the n x fo vertex feature matrix. Here, n corresponds to the number of vertices in the hypergraph and fo is the length of a feature vector associated with each vertex. For 1 l S L, let ft be the number of channels or hidden 1
units in the I-th layer of the network. In matrix notation, if w(ll) self and W(O neigh are the learnable filter weight matrices, and Q is the learnable charge matrix, the output of the I-th layer is then:
x0 = =o(((-1) self + [(() x(1-))("
neigh + BB))
(8)
where B()) is a matrix of real bias weights with the form BB, -) = (b ,..., 19), ) for w E V. Since I(Q) is complex-valued the activation function o is a complex- valued ReLU which zeroes out input in the left half of the complex plane and defined as ((2) = 2 if Arg(z) convolutional layers, the real and imaginary parts of the complex-valus node feature matrix ( e Caxfi.
////2///2), and a2)) = 0 otherwise. After the L
are separated and then concatenated into real-valued feature matrix (L) Rax2/z_ Finally, we apply
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linear layer via multiplication with learnable weight matrix W(L+1) R mapping the learned node features to a vector corresponding to the number of classes ne- This is then transformed into vector of class probabilities via softmax.
4 Related Work
4.1 Hypergraph Neural Networks.
Since the seminal work of Feng et al. (2019 introducing HGNN, several other spoctral-based hypergraph
neural networks (HNN) have been proposed. HyperGCN (Yadati et al. 2019) applies a GCN a weighted
graph representation of an EIVW hypergraph and uses a non-linear Laplacian (Chan & Liang 2020 Louis
2015) to process the graph structure in the convolutional layers. Zhang et al. (2022) introduce unified random walk hypergraph Laplacian which can be used in GCN for both EDVW and EIVW hypergraphs.
Their approach, however, incorporates the EDVW information via reversible random walk, and thus the hypergraph data is still represented by a weighted graph. Other approaches such as Hayhoe et al. (2024) also use a spectral based approach, but in the context of quantifying similarity between hypergraphs via graph expansions such as the clique expansion and line graph.
4.2 Edge-Dependent Vertex Weights in Hypergraph Neural Networks.
Zhang et al. (2022) build the equivalency between EDVW hypergraphs and undirected graphs via a unified random walk. This random walk on the hypergraph incorporates two different EDVW matrices, Qi and Q2. one for each step of the random walk, whose probability transition matrix is given by
P P=Di'QiWo(Dr10)
(9)
where p is a function of the diagonal hyperedge weight matrix Dg- They show both Q1 and Q2 are edge- independent, or Q = kQ2 for some k E R. then there exist weights on the hypergraph's clique expansion such that the random walk given by Eq. g is equivalent. They use this equivalency to build a hypergraph neural network on existing graph convolutional neural networks through this graph representation. The reversllre EDVW random walk |Chitra & Raphael 2019) used in HyperMagNet can be obtained from Eq. g with Q Y (the incidence matrix) and Q2 = R (the EDVW matrix), and thus does not satisfy either of the aforementioned conditions for equivalency to a graph random walk.
5 Experiments
To incorporate EDVW into HGNN, we use the EDVW random walk given by Eq. 9 We use p(X) X-1 and Q1 = Q2 = R, the EDVW matrix. We use this reversible EDVW random walk in place of the simple random walk used in Zhou's hypergraph Laplacian, defined Eq. [ i.c.
A A= D7'* D RT DT.
(10)
We use HGNN* to denote HGNN using this Laplacian. While this incorporates EDVW into HGNN, the underlying random walk is still reversible and thus loses higher-order hypergraph structure.
5.1 Term-Document Data.
The 20 Newsgroups data set consists of approximately 18,000 message-board documents categorized accord- ing to topic. To test the performance of HyperMaggNet (HMN) on predicting which topic document belongs to, subsets of four categories were chosen as in Hayashi ct al. (202)). The first set of four categories (G1) includes documents in the OS Microsoft Windows, automobiles, cryplography, and politics-guns topics. The second set (G2) consists of documents on atheism, computer graphics, medicine, and Christianity. The third (G3) contains documents on Windows X, motorcycles, space, and religion. Finally, the last set (G4) contains documents on computer graphics, OS Microsoft Windows, IBM PC hardware, MAC hardware, and Windows X. The categories in G4 are expected to be similar presenting more difficult classification problem.
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For all subsets of categories, the documents undergo standard cleaning by removing headers, footers, quotes, and pruned by removing words that occur in more than 20% of documents using PorterStemmer. Following Hayashi et al. (2020), Chitra & Raphael (2019), the hypergraph is created with documents as vertices, words as hyperedges, tf-idf values as the EDVW, and hyperedge weights as the standard deviation of the EDVW of vertices within each hyperedge. Ti-idf is a common weighting scheme for tokens in a document. Whereas bag-of-words uses a binary representation for a token indicating the presence or absence of a token a document, tf-idf is a scalar representation that incorporates frequency information of the word within a document weighted againts its frequency across documents in the corpus. The formula is term (token) frequency multiplied by an inverse document frequency, seen below. Let t represent a token in document d in the corpus of documents D. Then,
Std (Ereadeea) (1088 d: and lEd)
D
(f-idf(t, d)
The interpretation is that terms that occur frequently within a particular document but are rare in other documents should be given greater weight than those that also occur frequently across all documents. Table (left) shows the resulting hypergraph sizes.
The average classification accuracy over ten random 80% train-test splits for each of the subsets G1 through G4 is recorded in Table []] Both HGNN ands MygNN are two layer neural networks that follow standard hyperparameter settings based that in Kipf & Welling (2016). The dimension of hidden layers set to 128 with ReLU activation functions. For training the Adam optimizer was used to minimize the cross-entropy loss a the learning rate of 0.001 and weight decay at 0.0005. These settings were used for training across data sets. The same settings are used in Kipf and Welling's 's GCN for a baseline comparison across experiments. For another spectral- based graph comparison, we run spectral clustering on the clique expansion and use a majority-vote on the clusters to assign labels.
The following experiments feature two options for Laplacian in HGNN: (1) the standard hypergraph Laplacian using the unweighted hypergraph incidence matrix as defined by Zhou et al. (2006) and seen in Eq. [ (this is the Laplacian that HGNN architecture is built on); and (2) our weighted hypergraph Laplacian defined in Eq. 10 using the weighted hypergraph incidence matrix which incorporates EDVW information. We also include experimental results where the feature vectors ( are the standard bag-of- words representation of a document, and the tf-idf values. Table | shows HyperMagNet outperforms the graph-based models by a large margin across all categories with the exception of when EDVW information is used in the Laplacian for HGNN, where the advantage is smaller.
Table 1: Left: Hypergraph sizes for 20 Newsgroups data in chosen subsets G1 through G4. Right: Average node classification accuracy on 20 Newsgroups. Best result for cach subset G1-G4 is bold, second best is underlined. HyperMagNel (HMN) is highlighted in yellow. HGNN and HyperGCN are other spectral hypergraph methods, HGNN* HGNN with the EDVW incorporated as given in Eq. 10 GCN is a stan- dard graph convolutional network and GCN* uses weighted edges from tf-idf information the Laplacian. Spectral clustering is performed on the clique expansion with majority-votf labeling. Node features are tf-idf values or bag-of-words (BoW) representation.
G1
G2
HMN (tE-idF)
90.33:1..
90.6
0..65
02.66:10.78
78:11.66 1.99
HMN (BoW)
89.2 1.0.9
89.61 1.0.4
92.44.10 0.66
77.87 1.
HGNN (t--idt)
69.31+2.099
69.9741.42
15.69:1.82
40.9::2.11
HGNN (BoW)
79.34.12.01
79.74.12.80
88.44:11.0
39.67:12.31
Nodes
Hyperedges
HGNN (c1-idi)
77.0812.12
76.55 1.65
90.36 1.002
66.71 2.75
G1
2,243
13,031
HGNN (BoW)
89. 89.39:10.95
89.71:00.92
92.85:20.1
76.28:11 11222:::..55 1.58
2,204
9,351
HyperGON (tF-idi)
1..3
75.9912.87
1.25 89.36.11
68.08 0..0881..33
G4
G3
2,110
2,861
12,938
9,766
HyperGCN (BoW)
89.5811.58
89,
00.90-11.12
76.49.12.0
GCN* (ti-idE)
18.11412.00
48.39:42.00
32.008333.00 0.09
51.97:42.0
GCN* (BoW)
49.82.1.2
50.2113.00
52.43.133 3.75
50.88
4.04
GCN (U-idl)
31.89:1.777
00.004
55.75
32.87.12.
GCN (BoW)
32.84:15.89
32.34:144.00
32.38115.22
52.33:1:.69
Spec. Clustering
51.1311.17
59.21 12004
61.13.2.01
47.04.1

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Published in Transactions on Machine Learning Research (04/2025)
5.2 Citation and Author-Papee Paper Networks.
The Cora Citation data set (McCallum et al. 2000) consists of cilations belween machine learning papers classified into seven categories of topics with the BoW representation for each paper. The hypergraph is constructed with papers as vertices and citations as hyperedges. One hyperedge per paper is generated, and this hyperedge contains the paper and all papers it cited. Not all papers cited others in the data set; these degree one hyperedges are not included in the hypergraph. We also run HyperMagNet on the Cora Author (MeCallum et al.) 2000) data set. This hypergraph has papers as vertices and authors as hyperedges; an author's hyperedge contains papers that author appeared on.
Both Cora Citation and Cora Author do not have the raw text available to construct the EDVW from tf-idf values. Therefore, we define the EDVW matrix R based on the degree of the vertices within each hyperedge as
dog(u;)
R, = Evee, deg(v)
(11)
This may be natural EDVW in many contexts: more weight is assigned to verticos with larger degree reflecting greater importance to a particular hyperedge. Interpreted in the context of citation networks, this choice of EDVW measures the "citation-prestige of a paper i relative to that of all papers cited by hyperedge j. For example, j mostly eites niche or obscure research along with one highly-cited impact paper i, the EDVW of paper i with respect to j will be high. In contrast, if is cited in another paper k which cites many highly-cited impact papers, the EDVW for (i, will be lower. We note this choice of EDVW takes one particular view of "relative citation prestige", which (by design) biases the RW towards highly cited articles. Furthermore, it only uses local information to define citation prestige (i.c. the citation count of i and that of its neighbors within j). Using a more nuanced centrality measure beyond degree, like PageRank, to define Raj would yield a more complex notion of citation prestige. This range of expressivity in EDVW is a strength of NeperMagNed approach.
Since HyperMag.Vet can be flexibly run with or without EDVW information, we include results that incor- porate both the above hypergraph degree EDVW and a simple EIVW instead. Table shows HyperMagNet outperforms HGNN on Cora Author by a margin of 4%, whereas HGNN slightly outperforms HyperMagNel on Cora Citation.
Table 2: Left: Hypergraph sizes for Cora Author and Cora Citation. Right Average node classification accuracy (%) on the Cora Author and Cora Citation data sets. Best result is bold, second best is underlined. HyperMaggNet (HMN), with EDVW and EIVW, is highlighted in yellow. HGNN and HyperGCN are other
spectral hypergraph methods, HGNN is HGNN with the EDVW incorporated as given in Eq. 10 GCN is a standard graph convolutional network. Spectral clustering is performed on the clique expansion with majority-vote labeling.
Cora Author
Cora Citation
HMN
88.91
85.62#1. 1. .28
HMN EIVW
87.87 1.066
85.73.11.21
Nodes
Hyperedges
HGNN
83.1110..45
83.82.11.20
Cora Author
2,388
1,072
HGNN
83.01+0..32
85.94+1.55
Cora Citation
1,565
2,222
HyperGCN
83.62+00.88
85.10-11.00
GCN
75.89+1.50
76.81+1.69
Spec. Clustering
84.92+1.92
81.66#2.111
5.3 Computer Vision.
Princcton ModeINet40 (w et al. 2015) and the National Taiwan University (NTU) (Chen et al. 2003 are two popular data sets within the computer vision community to test object classification. The ModelNet-10 data set is composed of 12,311 objects from 40 different categories. The NTU data set consists of 2,012 3D shapes from 67 categories. Within each data set, each object is represented by features that are extracted using two standard shape representation methods called Multi-view Convolutional Neural Network (MV) and Group- View Convolutional Neural Network (GV).
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Following the construction in Feng et al. (2019). the hypergraph is created by grouping vertices together into hyperedge that are within nearest-neighbon distance based on the above features in the data. That is, a probability graph based on the distance between vertices is constructed using the RBF kernel Wyy = exp exp-2Diy/A and the nearest ten neighbors for cach vertex are grouped together in a hyperodge. The NTU hypergraph has 2,012 nodes and hyperedges, while the ModelNet40 hypergraph has 12,311 nodes and hyperedges. The EDVW are the Wi values so that the close vertices are given more weight than distant vertices. Here, HyperMogNeet outperforms the other models in the range of 1 7% depending on the Laplacian and feature vector combination.
Table 3: Average node classification accuracy (%) on NTU and ModelNet40 data sets. Best result is bold, second best is underlined. HyperMagNet (HMN) highlighted in yellow. HGNN and HyperGCN are other spectral hypergraph methods, HGNN* is HGNN with the EDVW incorporated is a standard graph convolutional network. Spectral clustering is performed on the clique expansion with majority-vote labeling. Node features GV and MV are extracted using two standard object representation methods.
given in Eq. I0 GCN
GV for hypergraph construction
MV for hypergraph construction
Node fts
NTU
ModelNet40
NTU
ModelNet440
GV
94.93 + 0.91
91.76 = 0.57
94.4 = 0.88
98.73 = 0.15
HMN
MV
94.87 1 0.90
98.46 1 0.20
91.21 1 1.37
97.22 1 0.31
GV
93.48 1 0.98
89.82 0.40
87.30 1 1.33
91.44 1: 0.54
HGNN
MV
93.24 = 1.13
97.22 + 0.31
86.98 + 1.43
97.14 + 0.26
GV
93:21 + 1.27
90.55 + 0.58
87.84 + 1.85
92.20 + 1.01
HGNN
MV
94.80 + 0.92
95.74 + 0.89
88.40 +: 1.09
96.15 + 0.65
GV
41.8 + 4.95
69.46 = 6.23
32.25 = 5.53
79.05 + 5.72
GCN
MV
55.27 = 6.96
80.97 + 15.6
47.44 + 3.54
95.51 + 1.15
GV
92.53
90.65 1.21
88.86 1 0.97
92.27 1 0.97
HyperGCN
MV
94.81 = 1.23
96.02 = 1.09
88.56 + 1.17
96.13 + 1.32
Spec. clustering
75.55 + 2.34
91.80 + 3.32
63.5 + 3.08
91.95 + 4.17
5.4 Ablation Study.
Finally, we compare the performance of Hyper.MagNet using the weighted hypergraph Laplacian (Eq. 7 and the unweighted hypergraph Laplacian (Eq. 4 The purpose here to demonstrate the utility of introducing the charge matrix Q in the Laplacian as replacement for the charge parameter q. This was motivated in Section X method to account. for the varying edge weights and directionality in the representative digraph that could not be accounted for by previous work with the magnetie Laplacian. Table 4 shows the Cora Citation, Cora Author, and 20 Newsgroups data sets that there is clear performance gain with the weighted hypergraph Laplacian.
Table 4: Performance comparison between HyperMagNNet (HMN) using the ( and ((4) Laplacians defined in Eq. 7 and Eq. 4 average node classification accuracy (%) Cora. Author, Citation, and 20 Newsgroups (G4).
Cora Author
Cora Citation
20 Newsgroups (G4)
HMN [!
85.62 1.66
38.0110..99
78
HMN L(9), q=
83.71+1.71
81.44+1.52
75.70
HMN T,00) == .15
83.98+1.14
81.47+1.55
75.98-+2.24
HMN ((d) == 25
83.62+1133 81.60=1.70
75.65:22.05
5.5 Runtime Comparison and Scalability Discussion.
The additional computational cost of HyperMogNet can be attributed three factors: (1) complex-valued arithmetic (2) the introduction of the learnable charge matrix and (3) dense blocks within the probability transition matrix P. A complex-valued neural wetwork like Hyperraggee incurs the cost of doing complex-
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valued arithmetic. Matrix multiplication is still naively cubic-order operation, but each multiplication of two complex numbers is four times the number of operations as the multiplication of two real numbers. The charge matrix Q adds an additional O (where n is the number non-zero entries in the transition matrix P) operations to each forward pass in the network. This is due to the Hadamard (entry-wise) matrix product that is introduced via the new phase mairix (PQ) = 2ri @ (P - př) that absent in the original phase matrix Ol+) 2xiq [ P This effect is ameliorated when P is sparse, which unfortunately is not always the case in practice. For example, in the 20 Newsgroups G4 dataset there an order of magnitude more hyperedges than nodes in the hypergraph and large hyperedge sizes k. This contributes to a dense transition matrix P and the order of magnitude, jump in training time of HaperMaggett compared to other models. A possible barrier to scaling MogNet and more generally, all of the hypergraph methods surveyed here is that large hyperedges yield dense matrices. Indeed, any hypergraph random walk in which there is always a nonzero probability of transitioning between vertices within the same hyperedge yields O(k²) nonzeros in P and subsequently ( Several techniques can be applicd to amcliorate this issuc. One could take a hyperedge filtering approach, applying one of the filtering operations defined and studied in [andry et al.] (2024). In contexts where filtering out large hyperedges is undesirable, alternative approach is to retain them but construct sparser transition matrix via hypergraph sparsification algorithms Liu et al. (2022); Benson et al. (2020). For example, Algorithm 2 in Liu et al. (2022) is an efficient and parallelizable approach for computing the s-line graph or s-clique expansion of hypergraph. Interpreted in our context, this may be adapted to restrict transition space to pairs of vertices which share or more hyperedges, yielding sparser transition matrices as 8 is increased.
Table 5: Model runtime comparison reported in minutes.
Cora Author
Cora Citation
20 Newsgroups (G4)
HMN (44)
0.71
0.55
43.7
HMN L( 9=
0.49
0.37
40.6
HMN ( q= .15
0.57
0.44
41.4
HMN ())) .25
0.48
0.37
40.9
HGNN
0.16
0.43
3.62
HyperGCN GCN
0.21
0.54
3.10
0.14
0.34
2.92
Spec. Clustering
0.11
0.44
3.64
5.6
Future Work.
In addition improving runtime and boosting HyperMagNell's capacity process large hypergraphs by incorporating sparsification methods, we have identified two other potential areas for future work: link prediction and directed hypergraphs. While performing link prediction on hypergraphs is feasible, and hypergraph neural networks have been proposed for this task Yadati et al. (2020), several challenges are associated with it. First, predicting the existence of hyperedges is more complex for hypergraphs than for standard graphs, as hyperedges can involve any number of vertices. Second, while HyperrMagNett could be used to learn node and hyperedge embeddings from the input hypergraph, a scoring function would still be necessary to evaluate the probability of existence for both existing and potential hyperedges. This would require defining and integrating such function within the HMN framework.
Incorporating directed hypergraphs into HyperMaggett is task that can be more easily achieved. Given random walk on directed hypergraph with EDVW, it can be represented using the corresponding repre- sentative digraph and processed with the magnetic Laplacian in Hyper.MaNeet. Random walks on directed hypergraphs have been explored, see Ducournau & Bretto (014).. However, there are important considera- tions when defining the random walk for use in Hyper Depending on the dataset and the random walk definition (such as transitioning from a vertex in the head of hyperedge to vertex the tail), the walk may not be ergodic. To address this, we can incorporate a restart mechanism into the random walk ensure its ergodicity.
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6 Conclusion
We proposed HoperMagNNet, hypergraph neural network which represents the hypergraph as a non-reversible Markov chain, and uses the magnetic Laplacian to process the higher-order information for node classilica- tion tasks. In building this Laplacian, we integrated a learnable charge matrix which allows HyperMagNNet to better process the weights associated with the non-reversible Markov chain. We demonstrated the perfor- mance of HyperMagNel against other spectral-based HNNs and GCNs on several real-world data sets. Our results suggest the more nuanced and faithful approach taken by HyperMagget may lead to performance gains, ranging from modest to significant, for the task of node classification. Further investigation of Hly- perMogNett is warranted, including adapting it to perform other tasks such link prediction, incorporating directed hypergraphs (Gallo et. al. 1993) as possible inputs to the model, and using hypergraph sparsification
methods Liu et al. 2022 to improve training time.
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15


Reviews:
[{"review_id": "XDsnkCV616", "reviewer": "Reviewer_J8KU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\nNovelty and Significance: the authros introduce a learnable charge matrix in magnetic Laplacian for HGNNs, which is novel and well-justified, addressing limitations of fixed charges. To my knowledge, its tje first work to combine EDVW random walks and magnetic Laplacians for hypergraph learning.\n\nStrong Empirical Validation: The authors use diverse datasets to evaluate their method showing competitive and often superior performance compared to HGNN, HyperGCN, GCN, and spectral clustering. HGNN* comparison highlights magnetic Laplacian benefits.\n\nClear Motivation: Effectively motivates non-reversible random walks and magnetic Laplacian. Clearly explains limitations of Zhou et al.'s Laplacian. Strong theoretical grounding with EDVW, representative digraphs and quantum mechanics analogy. Figure 2 effectively illustrates key concepts.\n\nWell-Written and Organized: Clear background, method, and experiment sections. Consistent mathematical notation and effective placement within existing literature.\n\n**Weaknesses:**\n\nReproducibility : Although the authors give a fair description of the architecture, the experimental setup, and the data preprocessing steps, the lack of code, or an anonymous repo hinders me from understanding whether the experiments are reproducible.\n\nComputational Complexity: As far as I understand, Complex-valued networks and learnable charge matrix introduce a computational overhead. Needs discussion of computational complexity of HyperMagNet vs. real-valued HGNNs/GCNs, including runtime scaling analysis.\n\nLimited Theoretical Charge Matrix Analysis: Limited theoretical exploration of the learnable charge matrix's properties (convergence, interpretability, regularization).\nAblation Scope (Scalar Charge): In order to realize matrix form benefits, a version with a learnable scalar charge could be compared with the HMN model.\n\nEDVW Derivation Heuristics: Justification and limitations of degree-based EDVW derivation for citation networks could be expanded.\nHyperparameter Sensitivity: Due to the lack of the code, I cannot identify any sensitivity of hyperparameters. The authors should discuss any sensitivity observations they can yield.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The manscript introduces HyperMagNet (HMN), a hypergraph neural network architecture focused on node classification. HMN employs the magnetic Laplacian, constructed from a non-reversible Markov chain based on edge-dependent vertex weights (EDVW), to represent hyper-graph structure. \n\nThe core contribution is the introduction of a learnable charge matrix, replacing the traditional fixed charge parameter within the magnetic Laplacian. This seems to address the limitation of single charge parameters in capturing complex weighted directed edges in hypergraph representative digraphs. HMN, also, utilizes a complex-valued neural network with a complex ReLU activation function. The author(s) show an empirical evaluation across diverse datasets (term-document, citation networks, computer vision) that demonstrates that HMN outperforms existing hypergraph and graph neural networks, showcasing its ability to better capture multi-way relationships in hypergraphs compared to graph-based reduction methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The most important changes I'd recommend to be integrated are the following:\n\nComputational Complexity Discussion: Include a paragraph discussing computational complexity, comparing HMN to real-valued counterparts and including runtime scaling analysis.\n\nAblation Study Extension (Scalar Charge): Add an ablation experiment comparing HMN to a version with a learnable scalar charge parameter to isolate the benefit of the charge matrix.\n\nReproducibility: It'd be very useful if the authors provide some form of resource (e.g. anonymous repo) to check for reproducibility.\n\nMoreover, it'd be positive if the authors can discuss the following :\n\nTheoretical Properties of Charge Matrix: Expand discussion on theoretical properties of the learned charge matrix Q, exploring convergence, interpretability, and regularization.\n\nEDVW for Citation Networks: Elaborate on the justification and potential limitations of degree-based EDVW for citation networks.\n\nHyperparameter Sensitivity: Include a short paragraph discussing hyperparameter sensitivity and provide practical tuning guidance, especially for the charge matrix.", "broader_impact_concerns": "I do not believe there exist of any concerns for ethical implications  that would require  adding a Broader Impact Statement.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "X942m7IOLH", "reviewer": "Reviewer_pa67", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**  \n\nThe paper is well-written and effectively motivates the proposed approach, making the ideas easy to follow and understand.  \nThe integration of edge-dependent vertex weights (EDVW) adds significant flexibility, enabling the model to handle diverse hypergraph structures and making it applicable to various real-world scenarios.  \n\n**Weaknesses**  \n\nThe proposed method is computationally intensive compared to traditional graph-reduction approaches. This increased cost may limit its practicality for processing very large hypergraphs.  \n\nSome aspects of the experimental setup are not clearly explained, which could hinder reproducibility and understanding (details provided below).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors begin by explaining how existing approaches for analyzing hypergraphs often involve converting the hypergraph into a standard graph, followed by the application of spectral techniques on the resulting graph. They then highlight the limitations of this process and introduce an alternative method: directly applying spectral techniques to hypergraphs using the magnetic Laplacian. Their proposed method, HyperMagNet, demonstrates promising results in node classification tasks across various hypergraph domains.\n\nAs someone who is not an expert in hypergraphs, I recognize that some of my comments may seem basic. That said, I recommend clearly defining certain concepts (even if they might seem straightforward to experts) to make the paper more accessible and engaging for readers without a deep background in hypergraphs. Considering my non-expertise in hypergraphs, the following comments are an educated guess.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Better introduce the notion of non-reversible random walk\n2. Better explain the difference concerning Fiorini et al 2022 and He et  al. 2022\n3. Not clear why table 1 contains 4 GCN items\n4. Details on the difference between (tf-idf) and (BoW)\n5. Citation order:  (Aksoy et al., 2020; Bretto, 2013; Feng et al., 2019) -->  (Aksoy et al., 2020; Feng et al., 2019; Bretto, 2013).", "broader_impact_concerns": "no ethical concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Mi4IPjhO5S", "reviewer": "Reviewer_6krg", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The approach sounds very reasonable, really using the full hypergraph structure\n- The methods are explained well and it is well written\n- Nice ablation study comparing weighted and unweighted Laplacian\n\nWeaknesses:\n- The presentation of the results could use some more clarity in my opinion. This includes the tables e.g. I think that all tables should be readable without having to check in the text for abbreviations. Maybe add explanations to the caption. Also it would be good to highlight clearly which method is yours and which are related works.\n- What does the asterisk mean in Table 1?\n- It is not clear to me why the authors opted for using 10 random splits on the dataset instead of evaluating different randomly initialised models. Could you please elaborate on this?\n- As far as I've seen there is no code base linked. This would help assessing the work and make it useable for others\n- Were hyperparameters tuned individually for all datasets and methods? Because the same parameters could favour specific methods.\n- The authors only evaluate accuracy in their experiments. It would be interesting to see if the method has an impact on other metrics and scores, such as F1 or similar.\n\nMinor writing/formatting issues:\n- “Observe the differences between the representative digraphs also provide intuition about the different weightings:” —> “Observing the differences…” (page 4)\n- The legends of Figure 2 b and 2a (most right) are too small for reading. This should be improved\n- “However, it still enjoys the properties that a convolutional neural networks are built on such as being Hermitian and positive semi-definite.” —> “However, it still enjoys the properties that convolutional neural networks…” (page 4)\n- Repetition: there are no edges between u and v, only from u to v, only from v to u, and reciprocal edges.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose a novel hypergraph neural network that is based on non-reversible Markov chains, allowing for a more detailed distinction between different hypergraphs. The authors show better performance to other methods and evaluate their method on several datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Adapting the presentation of the results (see above) to make them more clear and highlight the specific contribution as well as important findings\n- Potentially, re-doing the experiments with the same train/test splits.\n- The authors note that the runtime increases with their method. It would be good to add those to the experiments to get an idea of the magnitude.\n- The conclusion could be extended to highlight main findings more clearly and show the limitations of this work. Here, also a discussion on the runtime could be added, when the numbers are evaluated in the results section.\n- How easy would it be to extend this method to other methods such as link prediction? The authors mention briefly that this is part of future investigation. Here it would be good to discuss a bit how this would be possible.\n- It would be interesting to discuss in which settings your method out-performs others? Do you know in which settings HMN might be especially suitable to use? why is the performance difference larger for some datasets and smaller in others? Is this linked to the nature of the hyperedges or graph size or sth like that?", "broader_impact_concerns": "none", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: As requested by two reviewers, I would ask the authors to try their best to make the code available. The authors argued that this might not be possible due to one author's employment contract. 

For future submissions, I would recommend highlighting the changes after the review in a different color to simplify the review process for the reviewers and AC.
Decision: Accept (poster)