Let There be Direction in Hypergraph Neural Networks
Authors: Stefano Fiorini
Stefano Coniglio
Michele Ciavotta
Alessio Del Bue
Venue: Accepted by TMLR ()

Abstract: Hypergraphs are a powerful abstraction for modeling high-order interactions between a set of entities of interest and have been attracting a growing interest in the graph-learning literature. In particular, directed hypegraphs are crucial in their capability of representing real-world phenomena involving group relations where two sets of elements affect one another in an asymmetric way. Despite such a vast potential, an established solution to tackle graph-learning tasks on directed hypergraphs is still lacking. For this reason, in this paper we introduce the Generalized Directed Hypergraph Neural Network (GeDi-HNN), the first spectral-based Hypergraph Neural Network (HNN) capable of seamlessly handling hypergraphs with both directed and undirected hyperedges. GeDi-HNN relies on a graph-convolution operator which is built on top of the Generalized Directed Laplacian} $\vec{L}_N$, a novel complex-valued Hermitian matrix which we introduce in this paper. We prove that $\vec L_N$ generalizes many previously-proposed Laplacian matrices to directed hypergraphs while enjoying several desirable spectral properties. Extensive computational experiments against state-of-the-art methods on real-world and synthetically-generated datasets demonstrate the efficacy of our proposed HNN. Thanks to effectively leveraging the directional information contained in these datasets, GeDi-HNN achieves a relative-percentage-difference improvement of 7% on average (with a maximum improvement of 23.19%) on the real-world datasets and of 65.3% on average on the synthetic ones.

Published in Transactions on Machine Learning Research (10/2024)
Let There be Direction in Hypergraph Neural Networks
Stefano Fiorini
stefano. Sigffmoo.foriniiiii.....
Pottern Analysis 8 Computer Vision (PAVIS)
Italian Institute of Technology (IIT)
Stefano Coniglio
stefano. wtt.:oo..ooigiio@unbb...oo it
Department of Economics University of Bergamo
Michele Ciavotta
michele. clovotta michele.ciavottta
@unimibb it
Deportment of Informatics, Systems and Communication University of Hellno-Bicocee
Alessio Del Buc
alessio. delbue it
Pottern Analysis & Computer Vision (PAVIS)
Italian Institute of Technology (IIT)
Reviewed on OpenReview: https: // openreview. net/ forum? id= h48Ri 6pmvi
Abstract
Hypergraphs are powerful abstraction for modeling high-order interactions between of entities of interest and have been attracting growing interest in the graph-learning literature. In particular, directed hypegraphs are crucial in their capability of representing real-world phenomena involving group relations where two sets of clements affect one another in an asymmetric way. Despite such a vast potential, an established, principled solution to tackle graph-learning tasks on directed hypergraphs is still lacking. For this reason, in this paper we introduce the Generalized Directed Hypergraph Neural Network (GeDi-HNN), the first spectral-based Hypergraph Neural Network (HNN) capable of seamlessly handling hypergraphs featuring both directed and undirected hyperedges. GeDi-HNN relies graph-convolution operator which is built on top of novel complex-valued Hermitian matrix which we introduce in this paper: the Generalized Directed Laplacian ΣN. We prove that IN generalizes many previously-proposee Laplacian matrices to directed hypergraphs while enjoying several desirable spectral properties. Extensive computational experiments against state-of-the-art methods on real-world and synthetically--eneee
datasets demonstrate the efficacy of our proposed HNN. Thanks effectively leveraging the directional information contained in these datasets, GeDi-HNN achieves a serrriveperrentte-ddiffeeee improvement of 7% on average (with maximum improvement of 23.19%) the real-world datasets and of 65.3% on average on the synthetic ones.
1 Introduction
In recent years, ground-breaking research in the graph-learning literature has been prompted by seminal works Graph Neural Networks (GNNs) such as (Scarselli et al. 2009 Micheli 2009 Li et al. 2016 Kipf and Welling 2017 Veličković et al. 2018p. Since representing a set of complex relationships solely through undirected or directed graphs can prove too restrictive in many real-world scenarios, generalizations to graphs allowing for higher-order (group) relationships, i.e. hypergraphs, have been considered. Hypergraphs gencralize the notion of a graph to the case where an edge (a hyperedge) can connect an arbitrary number of nodes, thus allowing capture not just pairwise (dyadic) relationships but also group-wise (polyadic) dynamics (Schaub et al.) 2021 This has led to stream of research devoted to the investigation of

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Published in Transactions on Machine Learning Research (10/2024)
Hypergraph Neural Networks (HNNs) Feng et al. 2019 Chien et al. 2021 Huang and Yang 2021
Wang
et al. 2023a
While most of the literature on HNNs has focused on undirected hypergraphs, many real-world phenomena such as chemical reactions are naturally modeled on hypergraphs whose hyperedges have a notion direction. Despite this, the directed case has been addressed only sporadically, and often only in application-specif scenarios such as traffic forecasting (uo et al. 2022) and music recommendation (La Gatta et al. 2022). To the best of our knowledge, general solution based convolution operator which is solidly grounded in spectral graph theory while not being gooblem-dependenn is missing. We aim at bridging such gap.
In this paper, we introduce the Generalized-Direee spectral-based HNN capable of seamlessly handling hypergraphs featuring both directed and undirected hyperedges. GeDi-HNN relies on a Laplacian matrix which we introduce this paper: the Generalized Directed Laplacian LN- IN generalizes various Laplacian matrices: the one proposed in ZZhou al. (2006) and used in Feng et al. (2019) for undirected hypergraphs, the Sign-Magnetic Laplacian proposed for directed graphs in (Fiorini et al. 2023 and the classical Laplacian matrix (Chung and Graham 1997) used for undirected graphs.
Hypergraph Newral Network (GeDi-HNN), the first
operator which is built on top of a novel Hermitian
Main Contributions of the Work
We extend the literature on spectral-based HNNs by introducing GeDi-HNN, the first spectral HNN capable of handling hypergraphs with both directed and undirected hyperedges.
We introduce the Generalized Directed Laplacian matrix Ly; we prove that it enjoys several desirable properties, among which admitting an eigenvalue decomposition, and that it generalizes many existing Laplacian matrices.
Compared to state-of-the-art methods, GeDi-HNN achieves a relative-percentage-ddfieeeee improve- ment of 7% on average (with a maximum improvement of 23. 19%) on the real-world datasets and of 65.3% on average on the synthetic ones. This demonstrates its efficacy in extracting and utilizing the information encoded the hyperedge directions.
The paper is organized follows. Preliminaries and previous works are summarized in Section IN is introduced in Section 3 along with its properties. Section 4 provides an overview of GeDi-HNN's architecture. which is built upon LN- Experimental results are reported in Section 5 Conclusions are drawn in Section 6 The proofs of our theorems and additional details are provided in the Appendix.
2 Preliminaries and Previous Work
Undirected and Directed Hypergraphs
A hypergraph is an ordered pair H = (V, F), with n= [V] and m = [F], where V is the set of vertices (or nodes) and E' C 2 \ O is the (nonempty) set of hyperedges. The hyperedge weights are stored in the diagonal matrix W E Rmxm The vertex and hyperedge degrecs are defined as d, = Ee Eivee weel for u E V and & = le| for e e E and are stored in two diagonal matrices D, E R"X" and De E R=X Hypergraphs E are called k-uniform. Graphs are 2-uniform hypergraphs.
where 5(e) = le for some k N for all Following Gallo et al. (1993), we define directed hypergraph as hypergraph where each edge e E E is partitioned in a head set H(e) and tail set T(e). If T(e) is empty, is undirected edge.
Graph Fourier Transform and Graph Convolutions
Let L be a generic Laplacian matrix of a given 2-uniform hypergraph H. which embeds its topology. We assume that admits an cigenvaluc docomposition of U, and A E R is a diagonal matrix. Let x E C be graph signal, i.c., a function V + C whose domain coincides with the vertices of H. Following Shuman al. 2013), we call f = F(x) U the graph
UAU*, with E Caxn, U the conjugate transpose
2

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Fourier transform of and (#) Uø its inverse transform. The convolution y x between and another graph signal C (taking the role of filler) has natural construction in the frequency space, where it is defined as y @ r = "oaagU"""U"*' Letting Y == UGU with G == diag(U"y)) y), we can write @ in the vertex space as the linear operator Yz.
In the context of GNN, there are two drawbacks to learning y explicitly as a non-parametrie filter: i) deriving the eigenvalue decomposition of could be computationally too intensive (Kipf and Welling 2017); di) learning y explicitly would require learning number of parameters proportional to the imput size, which could be inefficient for tasks of high dimension DDfferrard et al. 2016).
For these reasons, it is customary in the GNN literature, see Kipf and Welling 2017 and Defferrard et al. (2016), to work with filters whose graph Fourier transform is a degree-K K polynomial function of A with a small K. This leads to a so-called localized filter thanks to which the output (i.e., filtered) signal at a vertex u ee V lincar combination of the input signals within K-hop neighborhood of (Shuman et al. 2013). Using either Chebyshev polynomials as done by Hammond et. al. (2011) and Kipp and Welling (0017) power monomials done by [Singh and Chen (2022), with K (as typical in the literature) one obtains parametric family of linear operators with two (learnable) parameters Og and 01
y : 0o1 + G.L.
(1)
Discrete Laplacians for Undirected Hypergraphs
In a hypergraph H. = E), the relationship between vertices and hyperedges classically represented via an incidence matrix B of size [V] [F]. When H undirected, B is defined as:
Byc {1
E
otherwisc VEVE
(2)
1o
From B, one can derive Q, the Signless Laplacion Malrix Chung and Grahami 1997], as well as its normalized counterpart QN:
Q = BWB
QN = D B. B D.
(3)
When restricting to undirected graphs (i.c., 2-uniform undirected hypergraphs), alternative Laplacian matrix, the so-called Signed Laplacian Malria, can be obtained with a similar construction to equation 3 This involves applying an arbitrary orientation to the edges of the graph (i.e., arbitrarily multiplying by -1 exactly one entry per column of B). Calling such a matrix B', the Signed Laplacian matrix L and its normalized counterpart Ln are defined as follows:
:: B'T
Dr B. D.
(4)
By utilizing the standard definitions of weighted adjacency matrix A Raxx where Aue we if {u, and Age 0 otherwise, for undirected graphs we have:
Q D. A
L D,
QN=I= L
= T-=
(5)
While the definition of in equation does not extend nicely to general (not 2-uniform) hypergraphs, the definition of Ly in equation 5 does generalization of the Signed Laplacian to general undirected hypergraphs which follows LN = QN from equation 5 is proposed by Zhou et al. (2006), and reads y
A A=I-Q QN.
we employ the approximation &= =K ACA from which we deduce Vx UGU*x U(EK 0=A""U"> Σi= (<(UA+U*))r EK 02 L. 2For instance, the choice of which entries of B should be multiplied by 1 would drastically affect L (rendering the orientation not arbitrary anymorc) and L may fcature both positive and negative off-diagonal entrics, thereby violating LN QN (notice that Qx holds by construction). Tn Zhou 2006). QN called e.
FFollowing w.l.o.g. Singh and Chen 2022
3

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Notably, all the Laplacian matrices we introduced satisfy some key properties: they real and symmetric and thus diagonalizable with real-valued eigenvalues; ti) they are positive semi-definite; and si) their normalized versions possess a bounded spectrum.
Discrete Laplacian Matrices for Directed 2-Uniform Hypergraphs
In directed 2-uniform hypergraphs, the presence of edge directions renders the graph asymmetric and none of the previous definitions of the graph Laplacian apply. Indood, those equation B equation [ and equation
O would symmetrize the graph and destroy its direcctins,, while the one in equation an asymmetric matrix which does not admit an eigenvalue decomposition and, thus, would prevent the application of the graph Fourier transform.
S would lead
The Magnelic Laplacian T proposed by Lieb and Loss 1993)) in the context of electromagnetic fields and adopted within a spectral GNN by Zhang et al (2021b a), is complex-valued, Hermitian Laplacian matrix. encodes the directional information the graph while enjoying an cigenvalue docomposition with a nonnegative, real spectrum. This Laplacian matrix generalizes the Laplacian L defined in equation b Letting A, == (A+ ) be the symmetrized version of and letting D, = diag( (A, e), where e the all-one vector, the Magnetic Laplacian and its normalized version are defined follows:
( = D. (4) and ( = 1 - D. (1 DJ with ((9) = A. 3 exp (i 2 (A - A
where is the imaginary unit and q E [0, 1].
The Sign-Magnetis Laplacian L" is matrix proposed by Fiorini et al. 2023) which is well-defined also for graphs with negative edge weights and enjoys some extra desirable properties. If q= 1, L" and L coincide if the latter is first constructed for an unweighted version of the graph and then multipliod component- by Ag. Thus, L is invariant to positive weight scaling which could otherwise alter the sign pattern of L(9) and, thus, the edge direction. Letting D, diag( AA,] e) and sign R - 0, be the component wise signum function, Le and its normalized version are defined as follows:
D. H" and L 1- - D. H 2 with H = = o. (e sign(1A A isign (IA) (IAI-1AT))) I))
To the best of our knowledge, no extensions of the Laplacian matrix are known for the case of directed hypergraphs which are not 2-uniform. Our paper aims to bridge this gap.
3 The Generalized Directed Laplacian
We now introduce our proposed complex-valued Hermitian Laplacian matrix, which is capable of handling hypergraphs featuring both directed and undirected hyperedges. We also establish some of its key properties. We refer to this matrix the Generalized Directed Laplacian. We define it directly in normalized form as:
:=
I- QN
with
D BWD: D D.
where B is the following complex-valued incidence matrix:
E H(e)
B.n {-
if E T(e)
U VEV, DEV,
(8)
0
otherwise
To appreciate how Lx encodes the directions of the hypergraph, we analyze its scalar form for pair of vertices u.v. V:
Σ
We 1
de
du
EEEEE
Juv
(9)
We de
Σ
We
Σ
We
u
+
de
de (EH(O)AVET(e)
Vdl vol.
CEF: (.Tee()Va.PeTTee)
F (e)
RE
4

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The pair u, affects the value of (LN)ue through each hyperedge E where u, U.E.E.e e. Considering the second line of equation g each hyperedge where u, take both the role of head (u, E H(e) or tail (u, E (())) contributes negatively to the real part, R((In)u)) by the opposite of its normalized weight (-we/5e). For undirected hypergraphs, this is the only contribution. Such a behavior is in line with the nature of LN (equation 4 for undirected graphs and of A (equation 6 for undirected hypergraphs. Hyperedges where u, V take opposite roles contribute with their normalized weight negatively u E (e) and v E T(e) and positively if и E T(e) and E Hee). Due to this, the imaginary part, (((xx) uu)), is affected by the net contribution of u and v across all the directed hyperedges that contain them. This is in line with the net flow behavior observed by (Fiorini et al. 2023) for L for the case of directed graphs. An example illustrating the construction of LN for directed hypergraph is provided in Appendix
The relationship between LN and the previously-proposed Laplacian matrices that we introduced in Section as well as its spectral properties are analyzed in more detail in what follows.
On the Relationship between EN and other Laplacian Matrices
The following theorem shows that En and @N generalize the Laplacian matrices proposed by Zhou al. (2006) for undirected hypergraphs which are defined in equation 3 and equation 6
Theorem 1. If H is undirected hypergraph, LN A and ON QN.
Focusing on 2-uniform hypergraphs, we show that LN and @N generalize the Signed and Signless Laplacian matrices LN and QN defined in equation and equation which are classically used for undirected graphs (Chung and Graham 1997]:
Corollary 1. H is on undirected 2-uniform hypergraph, L = LN and @N = 1ºQ-
Focusing on the case of directed graphs, we establish under which conditions ΣN generalizes the Signum- Magnetic Laplacian proposed by Fiorini et al. (2023) and the Magnetic Laplacian introduced by Lieb and Loss
I
Theorem 2. H is a directed 2-uniform hypergraph with antiparallel edges, we have IN 1 with A, = A+ AT.
Corollary If H. is a directed 2-uniform unweighted hypergroph with no antiporallel edges, we have ΣN 2 1 with = and A, = A+A
Key Spectral Properties of LN
We start by showing that EN and @N admit an eigenvalue decomposition with real eigenvalues. The result is structural and follows after showing that both matrices are Hermilian:
Theorem 3. iN and QN are diagonalizable with real eigenvalues.
Next, we show that the spectrum of @N nonnegative. This result is obtained by showing that Øx can be decomposed into the product of the matrix D. Bw D. and its conjugate transpose:
Theorem 4. QN is positive semidefinite.
To show that F, is positive semidefinte, we first derive the equation of ||=|| i.e., the p-Dirichlet energy function with p= induced by the Generalized Directed Laplacian for signal C". In line with the 2-uniform case (Shuman al. 2013), such function provides measure of the global smoothness of across the entire hypergraph.
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Theorem 5. Let a + ib
E C", with a, b R.. The 2- 2-Dirichlet energy function | x LND of x
induced by IN the following quadratic form:
Σ w(e)
Σ
au
av
+
ba
by
1 luveH(evaw,eree)
2
5(E)
dd
vd
ddn
vdn
cE E
uEEEE
Qu
by
2
av
bu
+
vdu
vdn
Vd
vity
Qu
by
2
+
Vdua vin))
bu
)
)uEH(e))ET(e))
(10)
Vdu
vdl/
dd
where is the indicator function.
Corollary 3. Ly is positive semidefonite.
Following the methods used (Kipf and Welling 2017 Zhang et al. 2021b Fiorini et al. 2023), we show that our Laplacian is positive semidefinite and thus can be adopted convolutional operator.
Lastly, we combine Corollary and Theorem 4 derive upper bounds the spectra of LN and QN: Corollary 4. Amax (Lx) s 1 and (Qx) <1.
While these spectral bounds are not required for the construction of the convolution operator defined in equation [ they are necessary to construct localized filters using Chebyshev's polynomials of degree
K > 1 (Kipf and Welling 2017 Defferrard et al. 2016 He et al. 2022a), and could be of independent interest.
The proofs the theorems and corollaries of this section can be found in Appendix
4 Generalized Directed Hypergraph Neural Network (GeDi-HNN)
We embed the Generalized Directed Laplacian IN in GeDi-HNN, the first HNN capable of handling hypergraphs with both undirected and directed hyperedges via spectral-basor convolution operator. For this purpose, we rely on the localized filter of Section 2 which led to equation Letting L = Ln, our convolution operator is Yz = 0,I + Lv.
The adoption of a localized filter with two parameters 6u, plays an important role towards the generality and the flexibility of GeDi-HNN, which we highlight next:
Proposition 1. The convolution operator oblained from equation i by letting L = LN with parameters 0o, Os coincides with the one obtained by letting C L= QN with parameters 0 0, + 02, 84 -0,.
The proposition implies that GeDi-HNN generalizes previously-proposed GNNs and HNNs irrespective of whether they are designed around Signed or Signless Laplacian matrix (provided that cither LN QN generalize the matrix such networks employ). This is because, as the proposition shows, GeDi-HNN can implement the convolution of equation built on top of either Laplacians by learning suitable values for 0o, O,.
In our implementation, GeDi-HNN features the following extension of the convolution operator of equation ! Let X Ca be eg-dimensional graph signal (a graph signal with co input channels), which compactly represent matrix. We combine Oo and 0, with the miring operator that is commonly applied to X to linearly combine the channels of the graph signal. In doing so, we introduce two linear operators Oo 9 E Co which can either upscale (if e > cg) or downscale (if c < co) the number of channels of X. A similar choice is made in other GNN/HNNs such as MagNet (Zhang et al. 2021b
Letting & be an activation function applied component wise to its input matrix, the output Caxe of any of GeDi-HNNN's convolutional layers is:
Z(X) = & (IX0 + +Ixx,.).
(11)
6

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As activation function, adopt complex extension of the ReLU function, in which, for given E C, (b() = z if R=) 2 0 and ((2) = O otherwise. A similar choice is followed in other GNNs/HNNs works such as (Fiorini et al. 2023 2024]. We project the complex-valued output of the last convolutional layer into the reals via an unwind operation by which ( E Caxe is transformed into (((2()))(3(((2)) E R=*2e, where || is the concatenation operator.
To obtain the final result, we add l linear layers GeDi-HNNN's architecture and a residual connection for every convolutional layer except for the first one. These connections have been proven to aid in training deeper models by allowing them to retain information from the input of the previous layers (He al. 2016 Kipf and Welling 2017). GeDi-HNN's architecture is depicted in Figure]
Hypengraph Convolution
Hypengroph Convolution complex function
Uniiidd operation
Fully connected layer
Input directed hypeegraph
irput Featurels)
complex Function
Real
Imag
D
Figure 1: GeDi-HNN's architecture: after two complex convolutional layers and a residual connection, we unwind the real and imaginary parts of the feature matrix and apply a fully connected layer.
Complexity of GeDi-HNN. Let assume w.l.ogg (as it does not affect the asymptotic analysis of complexity) that the input and output data of each convolutional layer except. for the last one have channels and that the output of the last convolutional layer and the number of channels of each linear layer are equal to c'. With l convolutional layers and S linear layers, GeDi-HNN's complexity is O(e(n"c+ ne²) ne (S ()(c?) + né'd + nd), where d is the number hidden node classes (the last channel the last linear layer). Letting ē = max fc, d, be the largest number of channels throughout the network, we have a complexity of Oll(n"e) + + (nn))) Such quantity is quadratic w.r.t. the number of nodes and the largest number of channels c, which is in line with previous GNNs and HNNs architectures. For more details see Appendix
5 Numerical Experiments
We now illustrate the results of an extensive set experiments carried out to evaluate the performance of GeDi-HNN on directed hypergraphs. We compare our proposal against state-of-the-art methods from
the lypergraph--earning lilerature: HGNN (Feng al. 2019), HCHAT
(Bai et. al. 2021). HCHA with
the attention mechanism (Bai et al.
2021). HNHN (Dong et al.
2020
HyperGCN (Yadati al. 2019
UniGCNII (Huang and Yang 2021
HyperDN (Tudisco et al.
2021
AllDeopSets (Chien et al. 2021
T Transformer (Chien et al. 2021 LEGCN Yang et al. (0022). ED-HNN (Wang et al. 2023a and
NN (Wang et al.J 2023b).
The hyperparameters of these baselines and of our proposed model are
selected via grid search (see Appendix
The experiments carried out on the node classification task of predicting the class associated with each node, which is the same task that was consistently used throughout the papers where the 11 baselines were proposed. The comparison is carried out on real-world datasets (Subsection 5.1 and on synthetic hypergraphs (Subsection 5.2
Throughout the following tables, the best results are reported boldface and the second-best are underlined. The datasets and code we used are publicly available on GitHuu (see Appendix A
Among the many versions of HCHA in Dong 2020 we use the implemented
HNN which coincides with HGNN Gao 2022

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5.1 Node Classification Task on Real-World Datasets
We test. GeDi-HNN on real-world datasets from the literature: Cora, Citeseer, and PubMed (Zhang et al. 2022); email-Enron and email-Bu (Benson al. 2018); Texas, Wisconsin, and Cornell (Pei et al.) 2020
WikiCS (Mernyei and Cangea 2020); and Telegram (Bovet and Grindrod, 2020). We lest the 11 baselines (which, we recall, are not designed to handle directed hypergraphs) on undirected version of the 10 datasets which is compiled following the procedure of Feng et al. (2019). Differently, we test GeDi-HNN, which is the only method designed to handle directed hypergraphs, on a directed version of these instances, which we compile following a slight modification to the previous procedure. For every node p sharing a relationship with nodes a, b, c, d, we create the hyperedge e with H(e) = {p} and T(e) = {a, b, c, d]. Considering, e.g., a citation relationship in CiteSeer where paper p cites papers a, b, c, d, in the undirected case we follow Feng et al. (2019) and create the hyperedge {a, c, d] to semantically represent the paper p whereas, in the directed case, set {p} head and {a, 6, c, d] as tail. We adopt the split proposed by Zhang et al.] (2021b for Telegram, Texas, Wisconsin, and Cornell and the split of Chien et al. (2021) for the other ones. All the experiments are conducted using 10-fold cross-validation. More details on the datasets can be found in Appendix D
Table 1: Mean accuracy and standard deviation obtained on the node classificat ion task on the real-world
datasets. The results of Cora, Citeseer, and PubMed are taken from Table 3 of (Wang et al.
2023b
Method
Cora
Citeseer
Pubmed
email- Fu
email Earon
HGNN
79.39 1 1.36
72.45 + 1.16
86.44 + 0.44
39.80 1 2.77
44.32 5.44
ACHA/HGNNT
79.14 + 1.02
72.42 + L.42
86.41 + 0.36
41.01 3.55
44.59 6.77
HCHA w/ Attention
58.20 1 2.53
68.44 1 1.27
79.90 1 1.70
28.75
2.82
35.68
6.96
HNHN
76.36 + 1.92
72.64 + 1.57
86.90 + 0.30
29.92 t 1.88
30.01 - 12.56
HyperGCN
78.45 + 1.26
71.28 + 0.82
82.84 + 8.67
30.81 + 2.80
36.76 + 5.87
UniGCNII
78.81 1 1.05
73.05 1 2.21
88.25 1 0.40
40.81 1: 2.76
41.62 1 5.28
LEGCN
74.74 + 1.25
72.74 + 0.86
88.12 + 0.74
30.16 t 2.28
35.41 + 5.76
HyperND
79.20 + 1.14
72.62 + 1.49
86.68 + 0.43
29.23 + 1.80
35.41 + 5.62
AllDeepSets
76.88 + 1.80
70.83 + 1.63
88.75 + 0.33
29.92 1.88
36.76 1 7.01
AllSet Transformer
78.58 + 1.47
73.08 - 1.20
88.72 = 0.37
41.58 + 5.13
45.41 8.43
ED-HINN
80.31 1 1.35
73.70 1 1.38
89.03 0.53
30.85 2.87
42.97 7.37
PhenomNN
82.29 + 1.42
75.10 + 1.59
88.07 0.48
31.09
3.83
37.03 7.21
GeDi-HNN
84.04 1 1.15
75.68 + 1.04
0.51 88.67 0.58
49.27
3.17
5.28
GeDi-HNN w/o directions
78.85 + 1.75
74.08 + 1.15
46.88
3.04
48.38 6.55
Method
Telegram
Texas
Wisconsin
Cornell
WikiCS
HGNN
59.42 1 6.04
71.08 1 7.32
75.69 4.64
70.81
4.73
77.95 5.69
HCHA/HGNN
52.12 + 3.32
71.35 = 6.77
73.53 - 5.41
70.81 5.06
76.50 5.07
HCHA w/ attcution
37.69 1 2.86
72.97 1 6.50
70.59 6.40
73.51 4.73
11.67 4.79
HNHN
50.77 8.27
81.18 3.72
74.32
5.14
26.47 18.1
HyperGCN
55.77 + 3.95
65.95 = 9.03
70.98 5.05
68.39
6.87
75.80 6.16
UniGCNII
55.58 1 5.01
84.17 1 5.44
86.47 5.02
76.76
5.13
83.24 1.07
LEGCN
45.19 + 5.15
79.19 = 4.78
84.51 + 5.35
73.78 + 6.12
78.73 + 1.19
HDperND
43.65 + 4.35
81.62 + 6.60
85.10 + 4.45
74.87 + 4.60
72.28 + 3.14
AllDeepSets
38.46 1 6.08
82.97 1 5.85
84.51 + 5.43
78.11 1 3.70
83.00 + 1.10
AllSet Transformer
57.12 1 5.21
80.27 5.56
81.96 6.26
76.47 5.41
83.37 1 3.77
ED-HNN
54.42 + 6.01
83.78 + 7.64
86.27 + 2.45
77.84
5.67
82.12 1.57
PhenomNN
54.61 + 4.72
84.59 - 5.41
86.28 4.62
76.49
5.56
80.07 0.61
GeDi-HNN
75.01 1 4.96
84.59 1 4.78
88.43 1 3.31
2.79 77.83
80.54
82.23 1.47
GeDi-HNN w/o directions
64.80 + 6.60
83.51 = 4.51
86.66 = 4.96
4.65
82.52 1.19
The accuracy obtained across the different methods and datasets is reported Table 1 The results show that, across the whole dataset, GeDi-HNN achieves an average additive performance improvement over the pert--prfo competitor of approximately 4.20 percentage points. In terms of Relative Percentage Difference (RPD)) T we have average RPD improvement of 7.06%. The most significant improvement is observed on Telegran where GcDi-HNN achieves an average RPD improvement of approximately 23. 23.19% and an average additive improvement of 15.59 percentage points w.r.t. the best competitor from the literature (HGNN). Overall, GeDi-HNN ranks first on out of 10 datasets and fourth on the 10th dataset, where it
TThe RPD of values Pl, the percentage ratio of their difference their average, Le., IP P/l/ P
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achieves an accuracy of 82.23, which is only 1.14 percentage points less than the best one recorded in the experiment.
Table D also presents the results of an ablation study aimed at demonstrating that a significant portion of the superior performance of GeDi-HNN attributable to the Generalized Directed Laplacian ΣN rather than to the network's architecture. this study, we compare GeDi-HNN GeDi-HNN w/o directions version which employs the undirected hypergraph Laplacian A proposed in Zhou al. (2006) (which disregards hyperedge directions) instead LN. As shown in Table 1 GeDi-HNN outperforms GeDi-HNN w/o directions with an RPD improvement of 4.90% (an additive difference of 3.35 percentage points) on average across out of 10 datasets and achieves nearly identical performance on the 10th dataset (WikiCS), where the difference between the two versions is negligible (of, additively, only 0.29 percentage points). The largest improvement, observed on Telegram, is of RPD of 14.61% (an additive difference of 10.21 percentage points).
These results underscore the importance of incorporating the directionality of the hyperedges and suggest that, thanks to the Generalized Directed Laplacian, GeDi-HNN effectively captures and utilizes this information.
5.2 Node Classification Task on Synthetic Datasets
To emphasize the importance of leveraging the hypergraph's directions in identifying the class of each node, conduct set of experiments on synthetic datasets specifically designed to exhibit a high degree of correlation between node classes and hyperedge directions.
Drawing inspiration from the methodology proposed in Zhang et al. (2021b), we rely on collection of datasets which are generated as lollows. First, the set V of vertices is partitioned into e equally-sized classes Ci...., C. with uniform probability. Subsequently, for each class C, 1, intra-class undirected hyperedges created, each with containing vertices of the same class also sampled with uniform probability. Similarly, for each pair classes C₁ and C with < J. to inter- class directed hyperedges are created. The head and tail sets are sampled from C₁ and Cy, respectively, with uniform probability, and both have cardinality uniformly sampled from (haiin-- ... hmaxx) ) Figure 2 portrays the directional flow the hyperedges induce among the classes of a synthetic dataset with 5 classes. Notice how the flow is directed from class C. to a class C₁ only if <
Flow
cardinality uniformly sampled from
1
3
2
Figure 2: Schematic representation of the direction of the flow induced by the hyperedges in a synthetic dataset with 5 classes (c (e= 5).
Using this methodology, we generate three distinct datasets with parameters n 500, e 5, Amin 3, hmax 10,
30, and an increasing number of inter-class hyperedges Lo 10, 30, 50. For these datasets, we implement a 50% 225% split for training, validation, and testing, ivspectively. The experiments are conducted using 10-fold cross-validation.
The experimental results, summarized in Table 2 indicate a substantial performance difference between GeDi-HNN and the other 11 baselines, which increases the higher the value of T, is. On average, GeDi-HNN achieves an accuracy that surpasses the best competitor from the litcrature with an RPD improvement of 65.31% (an additive improvement of 35.47 percentage points). The additive difference compared to the second-best performer is of up to 39.19 percentage points.
We conclude with the ablation study where GcDi-HNN is compared to GeDi-HNN w/o directions. The results reveal a substantial accuracy difference of 40.48 percentage points (on average) between GeDi-HNN and GeDi-HNN w/o directions Notably; as the number inter-class hyperedges (I,) increases, the performance of GeDi-HNN improves, while that of GeDi-HNN w/o directions doclines. This finding underscores the significant contribution of our proposed Generalized Directed Laplacian the superior performance of GeDi-HNN.
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Table 2: Mean accuracy and standard deviation obtained on the node classification tasks on the synthetic datasets.
Method
1. 10
7, 30
1, 50
HGNN
30.02 + 5.99
31.52
4.20
32.40 + 3.36
HCHA/HGNX
33.60 + 4.76
36.96 = 4.60
39.04 = 2.66
HCHA w/ Attention
17.28 2.42
18.64 =
20.44 + 2.24
HNHN
19.28 + 4.16
20.16 = 3.88
19.28 = 2.86
HyperGCN
21.04 3.99
21.28 3.11
17.84 + 3.33
UniGCNII
20.80 + 3.94
21.52 = 3.72
20.40 = 4.67
LEGCN
17.84 + 1.31
19.76 + 5.27
19.84 + 4.04
HyperND
18.16 3.11
18.40 3.85
18.16 + 4.06
AllDeepSets
18.32 + 4.12
19.20 = 4.33
18.72 = 4.40
AllSetTransfrmmn
19.44 + 4.42
18.96 + 4.30
22.72 + 5.06
ED-HNN
19.12 3.32
21.12 3.56
18.96 + 3.33
PhenomNN
20.88 + 4.24
21.20 + 4.30
20.32 + 4.62
GeDi-HNN
65.92 1 3.98
71.84 3.31
78.24 1 5.64
GeDi-HNN w/o directions
36.72 + 5.84
29.68 = 6.78
28.16 = 10.65
6 Conclusion
We introduced GeDi-HNN, the first spectral HNN capable of handling hypergraphs with both undirected and directed edges. GeDi-HNN is built upon novel complex-valued Laplacian matrix, the Generalized Directed Laplacian, which is a Hermitian matrix that employs complex-number number representation of the hyperedge directions. This approach naturally generalizes several previously proposed Laplacians for both graphs and hypergraphs. Our proposal enables the seamless integration of directionality HNNs, which is crucial for accurately modeling various real-world phenomena involving asymmetric high-order interactions. Our proposed GeDi-HNN model utilizes this new Laplacian matrix to perform spectral convolutions hypergraphs featuring both undirected and directed hyperedges.
Extensive computational experiments on both real-world and synthetic datasets have demonstrated the superior performance of GeDi-HNN in 12 out of 13 experiments compared to comprehensive representative group of state-of-thh-art methods for the node classification task. These findings underscore the importance of incorporating directional information within GeDi-HNN's convolution operator. Specifically, GeDi-HNN consistently outperforms existing models across various datasets, achieving an average rercnive- difference improvement of 7% real-world dataset (with maximun improvement of 23.19%) and of 65.3% synthetic datasets. The superiority of our method is particularly evident in the experiments synthetic hypergraphs. These results highlight the potential of adopting GeDi-HNN to significantly enhance the modeling of complex, directed interactions within a hypergraph to the benefit of the hypergraph-learinng task at hand.
Broader Impact Statement
All the data used are publicly available for research purposes and do not contain personally identifiable information or offensive content (see Appendix A for more details). The methods presented here have an impact on society comparable to other graph neural network algorithms.
Acknowledgments
This work is supported by PNRR MUR Project Cod. PE0000013 "Future Artificial Intelligence Research (hereafter FAIR)' CUP J53C22003010006
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A Code Repository and Licensing
The code written for this research work is available at httter//gghhbboccoo///saaa/////2 and freely distributed under the Apache 2.0 license 0
The Texas, Wisconsin, Cornell, WikiCS, and Telegram datasets were obtained from the PyTorch Geometric Signed Directed (He et al. 2022b) library (distributed under the MIT license). The Cora, Citeseer, and PubMed datasets are available al https://lings.orrr hwt.:/orr/datttttaeeeee The email-Eu and email-Enron datasets
are available at Inttps://ww. cs cornell wawpaarb/daaeeee data/
The code for the baselines used in the experimental analysis is available at https://giiii https///ggthuu..cccaaapa--hCC http:///t...oo/ooyc////////onnll under the MTT license
ED-HNN and
B Properties of Our Proposed Laplacian LN
This section contains the proofs of the theorems, corollaries, propositions, and lemma reported in the main paper.
Theorem 1. If H. an undirected hypergraph, ΣN A and QN = QN.
Proof. Since H = (V, E) is undirected hypergraph, B is binary and only lakes values 0 and 1 (rather than being ternary and taking values 0, -i, which is the case in general). In particular, for each edge e e E we
have Buc = 1 if either E (e) T(e) and Bu 0 otherwise. Consequently, the directed incidence matrix B is identical to the non-directed incidence matrix B, i.e., B = B. Thus, by construction, LN A and QN QN-
Corollary 1. If H is an undirected 2-uniform hypergraph, LN bN and ON _QN-
Proof. Since H is an undirected 2-uniform hypergraph, follows that:
BwB
Do A
D=1
=
Based this, we can rewrite QN as follows:
ON Dj BW D. B3a D.
D
B
W
B" D'
1
D (D. + A) D.
2
1
I+ D AAD
I+D.*
2
1
(I+AN)
2
1
-Q..
2
This proves the second part of the result. Since ON QQQ and, due to equation 5 L 1 - _ it
follows that L - Qx = Ly.
Theorem 2. H is a directed 2-uniform hypergraph with no antiparallel edges, we have En = 1N with A, = -
hhttps://ww. www.he/wwppcccccc.cc http://wwwcccorrollceeesse//aaae
http://chhoselllcnss.cccaa///// hamn
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Proof. Since H is directed 2-uniform hypergraph without antiparallel edges, it follows that:
BWB
D2 H
Dee = LL
Since has no digons, the assumption A, A A implies D D.. Thus, we can rewrite EN as follows:
N
D x BW Dr
13- De
I D t B
W
B D.
D (D, + HD) Dj
2
[ D D.
2 NN
Corollary If H a directed 2-uniform unweighted hyperyruph with no antiporallel edges, we have (g) with and A, = A A+ LN - L
Proof. Since H is a directed 2-uniform unweighted hypergraph, A E (0, 1]mxn. Thus, as shown by Fiorini
et al. 2023), with g 1/2 we have L = L(9), Since Theorem states that Lx L it follows that
1 LR IN 2
1 1
Thcorem 3. LN and @N are diagonalizable with real eigenvalues.
Proof. This follows from the fact that the two matrices are, by construction, Hermitian. Theorem 4. ON is positive semidefinite.
Proof.
x INZ::==** (D.
4 DD' Bw &
Do & W
(D. B De x1
Do D w B
(I. w B D.
1
Theorem 5. Let x a ib C", with a, b R The 2- 2-Dirichlet energy function | x Lxx of x induced by En the following quadratic form:
w(e)
Σ
au
av
bu
by
2)
+
1
2 EEE
5(c)
/du
Vd.
Vdu
Vdo
UEE
Un
by
2
Uv
ba Vdua
LuEH()),VET(e))
Vdu
Vdd
Vdl
an
by
2
+
av
INEMIO) vole
2 )²)
bm
(12)
/ du
Vdl
dd
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where is the indicator function.
Proof.
w(e) B(u,e) 2uz
x² LNN
ō(e) vd(u)y d(v)
NEV
K.EV RE
w(e) B(u, (v, e) 5(e) vd(u), d(n) Luz
VEV
EEE WVEVV
w(e)
Σ B(u,e)B(v.e)*
Fu
+ B(e, B(u,e) In
2 Tu
EEE 8(e)
vd(
Vd(u]/d(u))
VVV
u.vEViaee
((e)
Fip Fi d(u)
8(e)
d(v)
CEF F
w(e)
Σ
B(u,e)B(u,
JaJ vd(u)/d(o)
+ B(u, B(u,)* vd(u)/d(u)
EEE ((e)
w(e))
Σ
Fy d(u)
Te d(v)
B(u, B(U,e))
F. dd(
B(u,e)B(u,c)* F
8(e)
vd(u) ddu
u Vugge
Let us analyze the three possible cases for the summand.
Case 1.a: u E H(e) A (eee) B(u, e) 1, B(u, e) 1. We have ((v,e)) ((,,)) =1.
Case 1.b: E T(e) Av T(e) +> B(u,e) -i, B(,,)) = -i. We have B(u,e) = B(u,e)((u,e)" =
(-i)() 1.
In both cases, we have:
Le Ty
I x
Fv
Iy
Iv
d(u) d(v)
vdlu)/d(u))
Vdlu)
d(lu)
Vd(()
Vi(u)
Vilu))
Letting Fu au + ib, and v. au idr; we have:
au ag
2
bn
by } Vdla
)a
+
Vdu
vohy
Vdn
Case 2.a: u ((e) Av E T(e) 4 B(u,e) = 1, (,B(v, E) =i. We have B(u,e)B(o,ee) = (1)(-i) = and
B(u,e)((u,e) = (-i) -i.
Thus:
Tu Fy Vd(u) d(u)
d(u))
d(v)
ddr) d(u)
Let Zn a. + ib. and F. a + iby, then we have:
au
by )²
2
+ +
ar
by
Vdn
Voz)
Vd
Vd
Case 2.b: u E T(e) A E (e) e B(u,c) = -, B(v, e) 1. We have B(u,e) = (-i)(1)* = -i
((u,e) = (1)(-i)* i. We have:
x
T
+
du)
d(v)
/ d(u) d(v) /d(c)
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Let Zu au ibu and Ty dv + ibg then we have:
(va.
au
by 2
Gr
by vdy
+
+
vdll
Vd
The final equation reported the statement of the theorem is oblained by combining the four cases we just analyzed.
Corollary 3. En is positive semidefinite.
Proof. Since Ln is Hermitian, it can be diagonalized UAU for some U E Caxn and A E R where A is diagonal and real. We have x LNx = x =UUAU*x Ay with U. Since A is diagonal, we have y yAAy = Evev λuve Thanks to Theorem 5 the quadratic form x² LNN associated with LN is a sum of squares and, hence, nonnegative. Combined with x² LNLxx- Duev Aubie we deduce X 2 0 for all u E E.V. ]
Corollary Amax >max((N) and Amax (QN)
Proof. Amax (Ln) S 1 holds if and only if LN I = 0. Since ΣN I QN holds by definition, need to prove -QN 3 0, which holds true due to Theorem
Similarly, (QN) < 1 holds if and only if QN - I x 0. Since QN =I I- LN holds by definition, we need to prove -In = O, which holds true due Corollary 3
Proposition 1. The convolution operator obtained from equation 1 by letting L. = T.N with porameters og 01 coincides with the one obtained by letting L = QN with parameters 8% = 8g + 02, 6 = -01.
Proof. Consider the two operators 0ol + 0 LN and 0, 01ÔN. Since iN I QN, ; the first operator reads: 0od + 01(1 2N). This is rewritten as (6o + 01)1 01 QN.. By operating the choice 0 = 9o + 0 and 01 = -0, the sccond operator is obtained.
C Complexity of GeDi-HNN
The detailed calculations for the (inference) complexity of GeDi-HNN are as follows.
The Generalized Directed Taplacian n is constructed following equation in time Onnm), where the factor m is due to the need for computing the product between two rows of B to calculate each entry of LN. After Ln has been computed, the convolution matrix Ý e C"xn is constructed in time O(n²). Note that such construction is carried out entirely pre-processing and is not required inference time.
2. Each of the convolutional layers of GeDi-HNN requires O(n nc² +nc) O(n2c+ clementary operations across 3 steps. Let X be the input matrix to layer / 1,...,f. L. The operations that are carried out are the following ones.
(a) LN multiplied by the node-feature matrix X E Caxe, obtaining ph E Cxxe in time Onnec (we assume matrix multiplications takes cubic time);
(b) The matrices plo = - = x and pll are multiplied by the weight matrices Og. θ1 E Rexe (respectively), obtaining the intermediate matrices plu piu e Cnxe in time O(nc²)
(c) The matrices plo and p'n are additioned in time O(nc) obtain pla,
(d) The activation function & is applied component-wise to pla in time O(nc), resulting in the output matrix X E Caxe of the 1-th convolutional layer.
3. The unwind operator transforms X (the output of the last convolutional layer E) into the matrix
U Rax2e in linear time O(nc).
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4. Call Ua-1 the input matrix to each linear layer of index = ,....., S. The application of the s-th linear layer to U+-1 E C" requires multiplying U==1 by weight matrix M, E C xe' (where e is the number of channels from which and into which the feature vector of each node is projected). This is done in time (
In the last linear layer of index S, the input matrix US-1 E Raxd' is projected into the output matrix O e gaxd in time dnn'd).
The application of the Softmax activation function takes linear time O(nd).
We deduce an overall complexity of Oll(nec+ + na²) + ne + (S 1)(nc²)) ne' + nd) which, letting maxfe, d' d), coincides with ((((n²)) + (e+ () (())))
D Further Details on the Datasets
We test GeDi-HNN on ten real-world dataset. Cora, Citeseer, and PubMed (Zhang et al. 2022); email-Eu,
and email-Enron [ensoo et al., 2018); Texas, Wisconsin, and Cornell (Pei al. 2020); ; WikiCS (Mernyei
and Cangea 2020 and Telegram (Bovet and Grindrod 2020
Cora, Citeseer, and PubMed are citation networks with node labels based on paper topics. In these citation nelworks, the nodes represent papers, their relationships denole citations of one paper by another, and the node features are the ofr-of-words representation of papers.
Email-Enron and email-Eu are two email datascts- one from communications exchanged between Enron employees (Kliml and Yang 2004) and the other from European research institution (Paranjape el al. 2017 The nodes are email addresses and their relationships are of sender-receiver type. Since no node labeling is present in these two datasets, we define the node labels (node classes) using the Spinglass algorithm (Reichardt and Bornholdt 2006).
Texas, Wisconsin, and Cornell are WebKB data sets extracted from the CMU World Wide Knowledge Base (Web->XB) project WebKB is a webpage data set collected from computer science departments of various universities by Carnegie Mellon University. In these networks, the nodes represent web pages, and the relationship are hyperlinks between them. The node features the bag-of-words representation of the web pages. The web pages are manually classified into the five categories: student, project, course, staff, and faculty.
WikiCS is a directed network whose nodes correspond to Computer Science articles, and the relationships are on hyperlinks. This network has 10 classes representing different branches of the field.
Telegram models an influence network built on top of interactions among distinct users who propagate ideologies of political nature.
The statistic of these ten real-world datasets and of the synthelic datasets generale are summarized in Tables 3 and
E
Experiment Details
Hardware. The experiments were conducted on 2 different machines:
1. An Intel(R) Xeon(R) Gold 6326 CPU @ 2.90GHz with 380 GB RAM, equipped with an NVIDIA Ampere A100 40GB.
A 12th Gen Int)l Core(TM) 19-12900KF CPU @ 3.20GHz CPU with 64 GB RAM, equipped with an NVIDIA RTX 4090 GPU.
hhtt:://wwi cs. cmu. aadu/af///e cac. cec...mm..eeee htm..e//pppojettteeee/////////ee
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Published in Transactions on Machine Learning Research (10/2024)
Table 3: Statistics of the real-world datasets
Data set
# node
# hyperedges
# classes
average
e
Cora
2708
1579
7
3.03
Citeseer
3312
1079
6
3.20
Pubmed
19717
7963
4.35
email-Eu
986
873
10
38.01
email-Enron
143
128
7
20.03
Telegram
245
185
48.04
Texas
183
40
5
4.45
Wisconsin
251
65
5
4.77
Cornell
183
41
5
3.88
WikiCS
11701
6827
10
42.08
Table 4: Statistics of the synthetic datasets
Data set
# node
# hyperedges
# classes
average |e]
Io 10
500
250
5
9.05
I. 30
500
450
5
10.79
I. 50
500
650
5
11.63
Model Scttings. We trained every learning model considered in this paper for up to 500 epochs. We adopted learning rate of 5 10-3 and employed the optimization algorithm Adam with weight decays equal to 5 10-4 (in order to avoid overfitting). For all the models that adopt the classification layer, we set it to 2.
We adopted a hyperparameler optimization procedure to identify the best set of parameters for each model. In particular, the hyperparameter values are:
For AllDeepSots and ED-HNN, the number of basic block chosen in 2,,488,, the number MLPs per block in {1,2), the dimension of the hidden MLP (i.e., the number of filters) in {64, 128, 256, 512), and the classifier hidden dimension in {64, 128, 256).
For Transforreee the number basic block chosen in {2, 8), the number of MLPs per block in {1,2}, the dimension of the hidden MLP in (64, 128, 256, 512), the classifier hidden dimension in (64, 128, 256), and the number of heads in (1,4,8).
For UniGCNII, HGNN, HNHN, HCHA/HGNNY LEGCN, and HCHA with the attention mechanism, the number of basic blocks is chosen {2, 4, 8) and the hidden dimension of the MLP layer (64, 128, 256, 512).
For HyperGCN, the number of basic blocks is chosen in (2,4,8).
For HyperND, the classifier hidden dimension is chosen in (64, 128, 256).
For PhenomNN, the number of basic blocks is chosen in {2, 4, 83. We select four different settings:
1. AS 0.1, \ = 0.1 and prop step= 8,
2. Ao = 0, \ 50 and prop step= 16,
3. Ao 1, A1 and prop step= 16,
4. O, 20 and prop step= 16.
For GeDi-HNN and GeDi-HNN w/o directionality, the mumber convolutional layers chosen in (1,2,3), the number of filters in (64, 128, 256, 512), and the classifier hidden dimension (64, 128, 256). We tested GeDi-HNN both with the input feature matrix X Caxe where R(X) 3(X) o O and with 3(X)= =
19

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Published in Transactions on Machine Learning Research (10/2024)
Node Features. For Cora, Citeseer, PubMed, Texas, Wisconsin, Cornell, WikiCS, and Telegram, we retain the datasets' original features. For email-Eu, email-Enron, and the synthetic datasets, the feature vectors are generated using the vertex degree of each node.
F From a Directed Hypergraph to the Generalized Directed Laplacian
To illustrate the representation of directed hypergraph in our Generalized Directed Laplacian, consider a directed hypergraph H = (V, E) with V = {ui, 12 va, 44, us} and E = [ei,e2]: The incidence relationships are defined as follows: 01, 2/2 E (()), v3 T(e)), d4, 8 E (e2), and U1, U2 E T(e2). The hyperedges have unit weights (i.c., W = I). The hyperedge cardinalitics are de, 3 and 6,2 4.
For this hypergraph, we construct our Generalized Directed Laplacian using the following matrices: the incidence matrix B, its conjugate transpose Br, the vertex degree matrix D., and the hyperedge degree matrix De.
2
0
0
2
i 1
3 De
B -i 0
B
D.
0
0
0
4
0
Based on these matrices, we build QN as follows:
0.29
0.29
i0.24
10.18
-00.18
0.29
0.29
i0.24
10.18
-0.18
QN
-00.24
-0..4
0.33
0
10.18
i0.18
0
0.25
0.25
i0.18
i0.18
0
0.25
0.25
and then our Generalized Directed Laplacian:
0.71
0.29
-0024
10.18
i0.18
-0.29
0.71
i00.24
i0.18
i0.18
ΣN
i0.24
10.24
0.66
0
110.18
-i0.18
0
0.75
00.25
10.18
-10.18
0
-0.25
0.75
By inspecting IN, one can observe that it encodes the elements of the hypergraph in the following way:
The presence of nodes belonging to the same head or tail set, i.c., U1 E H(e)), ea. E H(e)), and U1 E T(e2), is encoded in the real part. Specifically, (ΣN)rre = (Ix)eam 0.29 and (ΣN) (LN)ege, -0255.
The directed hyperedges are encoded via the imaginary part. For example, considering nodes e and U3, we have (I.N) = (T.N) -00.44
3. The absence of relationship between pair of nodes is encoded by Specifically, (Tx))egs (IN)wns = and (IN) (Ix))sss
4. The 'self-loop information" (a measure of how strongly the feature of a node depends on its current value within the convolution operator) is encoded by the diagonal of Lx-
20


Reviews:
[{"review_id": "zLrDdRjrhp", "reviewer": "Reviewer_GHzt", "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\n1] The manuscript is well-written and your ideas are clearly described. \n\n2] The synthetic dataset generator defined in Section 5.2 is rather nice in my opinion. I suppose this is what a 'Directed Hypergraph Stochastic Blockmodel' may look like and I quite like it. \n\n3] Considering 11 baseline methods is extensive and provides strong evidence in favour of your model. \n\n\nWeaknesses\n\n1] The baselines were all run on undirected graphs and seemingly no effort was made to construct a directed baseline. \n\n2] It seems to me that several aspects need to be further specified and resolved for publications (see my Requested Changes).", "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": "In the submitted manuscript, the authors propose a spectral hypergraph neural network (GeDi-HNN). They do so by proposing and theoretically studying a new graph shift operator: the generalised directed Laplacian. GeDi-HNN is demonstrated to outperform 11 baseline models on 10 datasets and in a simulation study.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1] In the last equation of Section 2 you have the following term $\\text{sign}(|A-A^T|)$. To me it seems that this term should always be one since you are taking the sign of an absolute value. Could you please elaborate on the meaning and benefit of this term? If the notation $|\\cdot|$ refers to the matrix determinant here, then it may be valuable to define the dimension of the different terms in this equation involving the element-wise product. \n\n2] In the discussion of Corollary 3 you mention that the positive semidefiniteness of $L_N$ is sufficient for its eigenvalues to be interpreted as graph frequencies. To me, this does not seem to be a valid conclusion to draw. It seems to me that for the eigenvalues to be interpreted as graph frequencies one also needs to consider the associated eigenvectors. Shuman et al. [1] for example did this when they recorded the zero-crossings of the Laplacian eigenvectors in their Figure 3. If you want to keep this conclusion in your paper, then a similar study may be helpful to you. Since it seems to me that the eigenvalues of any positive semidefinite graph shift operator cannot be interpreted as graph frequencies in general.\n\n3] In Equation (11), where you define your GeDi-HNN you appear to be accounting for self-loops in the graph twice. Once explicitly via the term $IX\\Theta_0$ and a second time implicitly in the term $L_NX\\Theta_1$ since $L_N=I-Q_N.$ May it not be simpler (and better performing) to use $Q_N$ instead of $L_N$ in your model definition to only account for self-loops once with a single trainable weight matrix? My worry is that two separate trainable weights on self-loops destabilise your training process. It would be great if you could run some ablation studies on this topic. \n\n4] Could you please report the sensitivity of your results to different initialisations for $\\theta_0$ and $\\theta_1$? Since these are trainable parameters with significant impact on your graph representation, I could imagine you being relatively sensitive to an advantageous initialisation of these.   \n\n5] Some of the 11 baselines that you consider in Section 5.1 may trivially extend to handle directed graphs. However, you simply consider their performance on the undirected graphs. Could you confirm that taking directed graphs as input is impossible for the 11 baselines? And if not, may it not make for a fairer model comparison to also consider the performance of these models on the directed graphs?\n\n6] Minor Changes:\n\n- In the defnition of $\\bar{D}_s$ you use the notation $e$, which I did not see previously defined. Could you please define this notation or point me to where you already defined it previously?\n\n- In Theorem 2 you discuss antiparallel edges, which I did not see previously defined. Could you please add their definition?\n\n- In Theorem 2 you define $A_s = A+ A^T$. In Section 2 you use the same notation: $A_s = 1/2(A+ A^T).$ It may be marginally nicer to be consistent with the notation $A_s.$\n\n- In Section 5.2 you write that you partition the set of vertices into equally-sized classes \"with uniform probability\". Since the sets are equally-sized and no edges are drawn at this stage, the sampling step may be redundant. I recommend removing the formulation with \"with uniform probability\" here. \n\n- In Appendix F, the top right two entries of $B$ should be $-i$ instead of $i$ if I understood correctly. I furthermore noticed that the font for the imaginary number $i$ is different here in comparison to the main paper, e.g., Section 2. \n\n\n[1] Shuman, D., S. Narang, Pascal Frossard, Antonio Ortega, and P. Vanderghenyst. \"The Emerging Field of Signal Processing on Graphs.\" IEEE Signal Proc. Magazine (2013).", "broader_impact_concerns": "I do not have any concerns about this work that would necessitate a broader impact statement.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "VGpomOmOda", "reviewer": "Reviewer_r6ZH", "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\n(+) In theory, the design of the new Laplacian operator seems reasonable. The discussion on its connection to prior Laplacian operators is extensive.\n\n(+) In practice, the authors demonstrate superior learning performance for hypergraph node classification tasks.\n\n(+) The authors also demonstrate the importance of the direction in hyperedges for learning tasks via experiments on both synthetic and real-world datasets.\n\nWeaknesses:\n\n(-) (minor) The paper reads a bit not concise, especially the discussion on the prior Laplacian operators. \n\n(-) (minor) There are a few issues for the experiments, see comments below.\n\n===\n\nOverall I enjoy reading the manuscript. I feel the authors make a nice extension of the Laplacian operator to directed non-uniform hypergraphs. The experiments, especially those on the synthetic dataset, clearly demonstrate that the direction of hyperedges can be very helpful in the learning tasks. \n\nThere are some (minor) issues I found for the experiments, which may be resolved to strengthen the manuscript further. Firstly, the 10 real-world datasets are all undirected when they are initially introduced in the literature. The authors additionally construct a directed version of these datasets which technically speaking is not an apple-to-apple comparison. Nevertheless, the authors have made some effort to address this issue by testing their model with the original dataset as well. The comparison between GeDi-HNN versus its w/o directions counterpart at least demonstrates the importance of the direction in the hyperedges. I would suggest the authors further investigate (probably in future work) what datasets or real-world applications that inherently have these directions in the hyperedges. Then running it with GeDi-HNN versus ignoring the direction + standard HNN would better align with the claim made by the authors.\n\nSecond, I wonder why even applying GeDi-HNN w/o directions can already have better performance compared to standard HNN in general? Does that mean GeDi-HNN used in the experiments by default have more learnable weights? This makes me question if the number of learnable parameters of all methods is set to be similar so that it would be a fair comparison. I hope the authors can clarify this point.\n\nFinally, there are some additional comments:\n\n- The generic Laplacian matrix $\\mathcal{L}$ is undefined for its first appearance. It should be at least explained briefly like $\\mathcal{L}$ determined by the adjacency matrix.\n\n- Why do we need $L_N = I-Q_N$ and $\\Delta = I-Q_N$ when introducing what Zhou et al. (2006) did?", "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 study the problem of learning on directed hypergraphs. They proposed a generalized directed hypergraph Laplacian operator and designed a hypergraph neural network corresponding to it. In theory, they show some desired properties of the proposed Laplacian operator and why it generalizes prior Laplacian operators. In practice, they conduct extensive experiments on both synthetic and real-world datasets to demonstrate the importance of both the direction of hyperedges and the new Laplacian operator for directed hypergraph learning.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please address my comments regarding the weaknesses if possible. Nevertheless, I am also fine with the manuscript in its current form.", "broader_impact_concerns": "I do not foresee any broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "4JOguONIhj", "reviewer": "Reviewer_hfM7", "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 idea is intuitive and sound.\n\nThe authors go in great lengths to position their Laplacian matrix with the literature e.g. the theorems linking it to the hypergraph Laplacian and the interpretation of Magnetic Laplacian with imaginary parts. This helps to motivate the method and clarify the contribution.\n\nThe outcome of the experiments clearly underlines the usefulness of considering the direction.\n\nWeaknesses:\n\nAlthough the definition in eq. 9 is clear and the examples in Appendix F are clear and useful, the inclusion of a scheme could facilitate conveying the message in the main paper.\n\nTo my understanding, part of the experiments on real-world datasets are based on regular undirected graphs that are enriched with hyperedges (e.g. citeseer), hence it would make sense if they are compared with regular GNNs as well (since the hyperedges are not integral to the dataset).", "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 paper introduces a novel message passing technique for hypergraphs that allows to process both directed and undirected hyperedges simultaneously. \nThe method relies on a novel definition of the hypergraph Laplacian matrix that is proven to generalize previously introduced Laplacian matrices for undirected hypergraphs.\nSpecifically, the methodology is based on defining a complex-valued incidence matrix which allows the respective laplacian to capture the directed hyperedges in its imaginary part. \nFurthermore, the authors establish a relationship between the undirected hypergraph Laplacian and verify that the matrix can indeed undergo spectral decomposition.\nFinally, they propose a graph neural network based on a message passing on the proposed Laplacian matrix and concatenate the real and imaginary parts of the activations for the final output layer.\nThe method is evaluated by experiments on well-known datasets, indicating the superiority of the method compared to the literature of hypergraph NNs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "As mentioned above, the paper could benefit from a comparison with simple GNNs on the regular datasets, to quantify the overall impact of adding hyperedges in this setting where they do not occur naturally.\n\nMoreover, the authors could comment on any potential scalability issues that the method could encounter if it were tested with larger hypergraphs (e.g. DBLP and MAG datasets from https://www.cs.cornell.edu/~arb/data/).", "broader_impact_concerns": "There is no evident ethical concern about the paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper was reviewed by three expert reviewers. All the reviewers acknowledged the technical contribution of the work. Some concerns were raised mainly about the fairness of the experiments and the baselines (e.g., whether they can be generalized to handle directed hypergraphs). The authors provided clarifications in their response, and all reviewers eventually recommended acceptance of the paper. I also agree with the reviewers and feel that the paper is ready for publication at TMLR.
Decision: Accept (poster)