Vectorial Graph Convolutional Networks
Authors: ZhongYu Li
Geng Zhao
Hao Ning
Venue: Submitted to ICLR 2023 (2023)
Keywords: GNN
GCN

Abstract:    Graph Convolutional Networks (GCN) have drawn considerable attention recently due to their outstanding performance in processing graph-structured data. However, GCNs still limited to the undirected graph because they theoretically require a symmetric matrix as the basis for the Laplacian transform. This causes the isotropic problem of the operator and reduced sensitivity in response to different information. In order to solve the problem, we generalize the spectral convolution operator to directed graphs by field extension, which improves the edge representations from scalars to vectors. Therefore, it brings in the concept of direction. That is to say, and even homogeneous information can become distinguishable by its differences in directions.In this paper, we propose the Vectorial Graph Convolutional Network(VecGCN) and the experimental evidence showing the advantages of a variety of directed graph node classification and link prediction tasks. 

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VECTORIAL GRAPH CONVOLUTIONAL NETWORK
Anonymous authors Paper under double-blind review
ABSTRACT
Graph Convolutional Networks (GCN) have drawn considerable atten- tion recently due to their outstanding performance in processing graph- structured data. However, GCNs still limited to the undirectod graph be- cause they theoretically require symmetric matrix as the basis for the Laplacian transform. This causes the isotropic problem of the operator and reduced sensitivity in response to different information. In order to solve the problem, we generalize the spectral convolution operator to directed graphs by field extension, which improves the edge representations from scalars to vectors. Therefore, it brings in the concept of direction. That is to say, and even homogencous information can become distinguishable by its differences in directions. In this paper, we propose the Vectorial Graph Convolutional Network((eeGGNN) and the experimental evidence showing the advantages of a variely of directed graph node classification and link prediction tasks.
1 INTRODUCTION
A graph is ubiquitous data structure where entities are vertices and edges their pairwisc relationships. Most. Graph Neural Networks((GNNNe) fall into one of two categories; spectral
Defferrard et. al. (2016); Kipf& Welling (2016) or spatial networks Hamilton et al. (2017a);
Veličković et al. (2017); Backstrom & Leskoved (2011). Spatial approaches are based on
localized averaging operator with learnable weights that iteratively traverse the entire graph. Spectral approaches based on eigen-decom positionsioooooo _f graph Laplacian and smooth
(2020); Wu et al. (2020). The application 2012) to quantum chemistry Liao et al. domains ranging from social networks Chen et, al
those signals through Fourier transform Zhon al
(2019) and text classification Yao et et.allllllll............ (2019) etc. One of the key techniques is Graph
Convolutional Networks (GCNs) Defferrard et al (2016); Kipf & Welling (2016); Xu et al. (2018a), it's the variant of Convolutional Neural Networks (CNNs) Mallat (2016) on graphs, that learns the representations from both vertices and endesees It particularly important to
apply representations to downstream lasks Hamilton el. al. (2017b), e.g., node classification
and link prediction Hu et al. (2020).
However, the vast majority of these researches are based on undirected graphs, and even the original graphs are naturally directed. This phenomepon phieemmmmnn will take the risk of discard- ing potentially important information Kawamoto et al. (2018); Zhang et al. (2021). For example, you may have heard of celebrity, but he/she doesn't 't know you. From the GATs" perspective, it is casy to understand that the attention values from node to node j and node j node are not necessarily equal, which means the information is not symmetric on the edges.
Adjacency matrix is the topological edge set. Unless graph G is undirected is not symmetric. Unfortunately. GCNs, are developed from spectral theory Kipf & Welling (2016); CannnattlyyyCyNs....a
Xt al (2018a); Gilmer et al. (2017) and limited to symmetric convolutional kernels Beaini et al. (2021), the object matrix of the kernels needs to be positive semi-definite and symmetric bocause the decomposition
a such matrix is orthogonal that can be taken as Fourier transform basis. It, in turn, requires the graph to be undirected to satisfy the above
two conditions, or the eigenvalues of A can not be solved in the real number field. Thus,
extending spectral methods to directed graphs is not straight Zhang et al. (2021)

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Therefore, one of the key challenges lies in defining symmetric adjacency matrix directed graph.
Recently, there has heen surge of interest in directed GCNs Tong et al (2020ba3 Monti et al, (2018); Beaini et al, (2021): Zhang et al, (2021). These studies have proposed different approaches to solve the problem. However, the original purpose of constructing directed GCNs is to keep more information from the graph. While these previous studics view the direction a one-dimensional scalar, is supposed to be vector that shares the same dimension with node vector. From a Principal Component Analysis(PCA) perspective Abdi & Williams (2010), The n-dimensional node vectors decomposed into k principal components (k I n), GCNs preserved the 1st component on edge and directed GCNs preserved the 1st and 2st components. This shows that some of the information lost.
To address these issues, we proposed VecGCN. We overcome the symmetric problem and the information loss problem at the same time by using Field Extension, which is the main research object of field theory in abstract algebra. The basic idea is to start from base field and somehow construct a "larger" field that contains it. And we construct a high dimensional field according to distances between nodes. Firstly, the distance matrix is symmetric. Secondly, a high dimensional field does not cause information loss.
The main contributions are summarized follows:
1. Replace the adjacency matrix with the distance matrix. The advantages of replac- ing the adjacency matrix with the distance matrix include two main aspects. On the one hand, the distance matrix is symmetric. On the other hand, the topology of the distance matrix is the same that of the adjacency matrix. If the distance between two adjacent nodes is 1, then the distance matrix and the adjacency matrix are equal. Therefore, we can consider the distance matrix as a generalization of the adjacency matrix. Not only that, but the distance matrix is also satisfied by the theory of GCN.
2. The concept of direction is proposed.
Since the adjacency matrix is binary, 0 indicates that there is no edge between two nodes, while is the opposite. Therefore the adjacency matrix does not imply the concept of direction, which makes the model unable to distinguish the importance of neighboring nodes, which manifests isotropy. The above problem can be solved by improving the elements of the adjacency matrix from scalars vectors using the field extension method.
3. Propose the VecGCN. Our extensive experiments on a series of datasets clearly show that VecGCN's performance exceeds most other methods.
RELATED WORK
Most graph neural network structures can be categorized as eithr sseeetherrsppeerallll or spatial Neigh= borhoods in spatial nelwsrks net such Veličković al. (2017); Hamilton et all (2017a); At- wood & Towsley (2016): Duvenaud et al. (2015) are well-defined even when their adjacency matrices are not symmetric. Although spatial methods typically have natural extensions to directed graphs, they may ignore important information in the directed graph, as we discussed before. Spectral approaches also suffer from this problem. In this section, we review related work on constructing neural networks for directed graphs, and describe the development, the problem detail as well as the various solutions. We refer the reader to Wu et al. (2020); Zhang et al. (2020) for more background information.
2.1 NOTATIONS AND PRELIMINARIES.
Given a simple and connected undirected graph G = (V, E) with n nodes and m edges, Let A denote the adjacency matrix and D the diagonal degree matrix. Spectral-based GCNs are based on the Laplacian matrix, and the graph Laplacian matrix is defined as D-A. The normalized format of Laplacian matrix is defined as Logm = = +D & I-D-+ 4, where 7 is identity matrix that has same shape with A. Taggin is matrix representation
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of graph in graph theory and can be used to find many useful properties. It is symmetric positive semi-definite matrix. With these properties, the eigen-decomposition of normalized Laplacian matrix Lays write UAU Here A is diagonal matrix of the eigenvalues of Lays, and U E Ruxn a unitary matrix that consists of the eigenvectors of Lyys- Let X Ra denote the representation of node feature matrix, that is, each node in I th layer associated with d-dimensional leature vector X = [a],.. ,,,].. While is the input node feature and is only used in the input layer. The graph convolution operation between signal defined as go(L)*x (A)(A) 2r, where the feature vector x E X is the signal and (A) the spectral filter, and 0 corresponds to a vector of spectral filter coefficienss. Finally, Each hidden layer [ is assigned with learnable parameters W and a( an activation function such as ReLU, and (,,)) represents the inner product between any two vectors.
2.2 VANILLA GCNS
GCN generalization of the Fourier transform on the graph. Bruna al. (2013) proposed first generation of GCN, written as:
X =UU x*)
Since the computational complexity is O(n)), Defferrard et al. (2016) (referred to as Cheb- Net) proposed the second generation of GCN, and model outputs can be approximated by
the K- - th order polynomial of Laplacians by Chebyshev Polynomials, written as:
K
X
1=0
The derivation can be found in B.1 Kipf & Welling (016)) further simplified the model,
they set K = 1 and use the amnormalization trick, replaces Leys by a normalized version - 1 AD (D (D+1) + +) obtain:
x = =[Lx'w")
According the above equation, we know that the vanilla GCN multi-layer neural network that receives and update node features across the graph.
2.3 LIMITITION OF UNDIRECTED GCNs
Figure 1: Exammpes of ChebNet's outputs a circulant graph. Note that the graph convolutional operators are isotropic on some local areas due to rotational invariance. Hence, it is difficult to define directions.
Since L.is. symmetric matrix, spectral-based GCNs are limited to applying to undirected graphs Wu et al. (2020). Unlike convolutional neural networks, GCNs' convolutional op- erators symmetric. One way to apply GCNs to directed graph is to relax the graph
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structure by trivially adding edges to symmetrize the adjacency matrices Kipf & Welling (2016); Wu et al, (2019). However, it loses the distinctive structure.c the directed graph and misleads the message passing scheme with incorrect weights Wang et al. (2020). Almost all relationships in nature are irreversible, such as time-series relationships, parent-child rela- tionships, virus transmission relationships, and thermal conductivity relationships. Besides, it is not capable to learn from complex local structureMonti et al (2018))) inheritance re- lationship Kampffinyer
al. (2019) and second-order proximity Tong et al. (2020b). The above problems arc mainly duc to the of anisotropy in the GCN convolution operator of GCNMonti et al. (2018); Beaini et al. (2021) as shown in Fig~ 1.
2.4 THREE TYPES OF EDGE VECTORS
To the best of our knowledge, the edge direction in GNNs are divided into four types:
1. Direction of information flow. Information flow is a series samples of nodes, Since GNNs are based on the Mcssage Passing(MP) techniqueGilmer et al. (2017). that allows nodes repeatedly aggregate and update their own representations. Different sampling methods cause MP to produce different aggregation results. Therefore, these sampling methods generate generatee information flows and significantly alfect the models' performances Tong al. (2020b).
2. Edge embedding and representation. Many graph data in the real world contain types.of podes and edges, and these graphs are called heterogeneous graphs Gasteiger et al. (2019). Typical heterogeneous graphs include molecular graphs, academic graphs, and recommendation graphs. In order to distinguish different types of edge intormaatoneeebbeeddinng methods are used to represent different edges WWan et al 2): Schlichtkrull et al. (2018); Zhang et al. (2019). This approach brings in addirional learnable mordles based on the GCN model. Further, another approach Pei et al. (2020); Klicpera et al. (2020) constructs a continuous space underlying the graph by combining dillerent types of edge representations.
typess
3. Spectral direction. By constructing the Hermitian Laplacian matrix Zhang et al. (2021), the complex eigenvalues are generated after decomposition.. The real part of an eigenvalue indicates the presence of an edge, and the imaginary part indicates the direction of the edge. Since the obtained directions are unlearnable scalars, it is also necessary to bring learnable modules to improve the model's performance.
4. Spatial direction. The Spatial approach is the most intuitive where the edge vector is defined as the difference between two adjacent nodes Beaini et al. (2021) Eij =
u, uy where i * j.
Our proposed VecGCN belongs to the Spatial approach. The advantage of this approach is that it very natural to understand the meaning of the direction, the same as our intuition in daily life. In addition, the design of VecGCN is simple and hardly changes the structure of vanilla GCN.
Based vanilla GCN, define the vector field the graph and prove the generalization of the method on an n-dimensional grid, which provides the basis for subsequent analysis. The vector in the vector field is divided into directiop length TeeihhThhhhhhee -ifferrfeeence between the two nodes defines the direction. Inspired by GATVeličković et al. (2017), we define the length the importance of an edge, so we again design a normalization function to calculate the significance of each edge.
lenghh
3 METHOD
Since GCNs are based on the explicit assumption of an undirected graph, this leads to a drawback that the Laplacian operator is rotationally symmetricB.5 For example, order of neighbor nodes does not change the final aggregation result, which is isotropicMonti et al. (2018), making it no preferred direction
the graph. In order solve the above problems, we propose the VecGCN, and the key idea is to expand the definition domain of the adjacency matrix using the field expansion method.
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3.1 PERMUTATION INVARIANCE OF DISTANCE FUNCTION
It is not easy to keep the symmetry of the adjacency matrix. However, we notice that the distance is the same between arbitrary two adjacency node and node j, written as d(i.j) = d(j.i), i), where d is a L norm distance function defines in Euclidean space, it tells us the distance function permutation invariant. What more, the distance matrix is symmetric. Moreover, the topology of the distance matrix and the adjacency matrix are the same because the adjacency nodes define both. Therefore, in this paper, we replace the adjacency matrix with the distance matrix and note it as F, which is the first step in constructing VecGCN.
3.2 FIELD EXTENSION
With the help Field Extension..4 distance matrix can be extend to high dimension, R + R". In this paper, we set the distance matrix to 64-dimension, note 1 and [F = 64. Since is the basis of Fe4 _et_we we can view F64 as a linear combination of keep symmetric
F, more details are provided in Appendix B.2. That is to say, distance matrix in high dimensions. This the second step in constructing VecGCN.
3.3 VECGCN
%
step
step 2.
step 3.
step 4.
Figure The four steps of VecGCN are calculate the node reprosentation, the edge direction, the local importance degree of cach edge, and update the node representation, respectively.
Now have high dimensional form of distance matrix J to replace the adjacent matrix in vanilla GCNs. Overall speaking, in order to adapt the distance matrix F66,, we also need to upgrade both the learnable parameter W and the input node feature to 64 dimensions, in which the learnable parameter W is easy to modify by adding only one additional dimension, and the modification of the input feature h requires a linear layer mapping to 64-dimension. In addition, while keeping the structure of GCN unchanged, it is necessary to use the partition function to normalize edge vectors.
First, a linear transformation is applied to the node feature to map it to higher dimensional space Eq
Xo (hh) +
(1)
Where á a linear transform that operates on the input feature space, and b is a basis terme.
Second, calculate the edge vectors.
ctyy = x; Iz
(2)
btyy is defined on the Euclidean space, it is natural and easy to understand the definition of an edge vector in terms of the difference between two adjacent nodes. The superscript /
'The and the bias term only included in this section and will be omitted in later sections.
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in B.10 represents the fiylr. Besides, the distance is shift-invariant, and its visual explanation can found A1.
Third, compute the length of ely, which is the L norm |le,yll2, it indicates the strength of the interrelationship between node i and j. Then normalize the vector length with the partition function Zig in Eq so that ||e/lll l22 E [0, 1]. Basically, |lelyll ll2 is expected to be small if node and j are close or similar each other.
Zi = Σ le/yl12
JEN(S)
Fourth, agg in Eqr the importance coelficient that measures the relatedness between node node with respect to layer I, and inversely proportional the length of the edge vector. D; is smoothing value that indicates the degree of node i.
||//gla 1/D,
d =
(4)
Zzj
Due to the value domain of a is in [0, 1 1/D; B.3 we know that E [0, 1], it tells us that the smaller the degree a node, the less significant is, i.e,, a node with a minor degree can be ignored.
Subsequently, according to Eq and Fard, the new edge vector can be updated by e og x ety. For example, in Fig-2 the lengths of the three adjacent edges of node u are 0.3, 0.2, 0.1. These edges' lengths after normalization Eq equals 0.50, 0.33, 0.16 and corresponding importance coefficients are 0.17, 0.11, 0.05.
Fifth, under the message passing scheme, the update of the node vector can be determined by Eq
1+1 = Σ dyeeyaly
(5)
JEN(()
In summary, as shown in Eq the way cach layer is updated can be written as:
x = =[Ta'e/x'w')
(6)
From Eqr we know that VecGCN is beneficial to address the isotropy directed graph- based GCNs. There are two main reasons: the one hand, the inclusion of edge vectors makes the importance of neighbor nodes v to the central node u change, and prior to that, the importance is the same 2.2, since the edge vectors all 1; On the other hand, the order of neighboring nodes cannot be changed during the information aggregation phase, because eyJay * Gsyask where o k and j, k Mi), this shows that VecGCN is anisotropic. The pscudo-code for VocGCN detailed in A.2
EXPERIMENTS
In this section, we will verify our proposed VecGCN on datasots: Cora, Citesoer, Pubmed, Texas, Wisconsin, and Cornell(An overview sunmary of characteristies of the datasets is given in l)ble-
Cora, Citeseer, and Pubmed are popular citation networks with node labels correspond- ing to scientific subareas and downloaded automatically through the official Deep Graph Library(DGL. program. In these networks, nodes represent papers, and edges denote cita- tions one paper by another. Node features are the bag-of-words representation of papers, and the node label is the academic topic of a paper.
Phttps///sthuub.cm///m///b/

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Table Datasets statistics
Dataset
Cora
Citescer
Pubmed
Cornell
Texas
Wisconsin
# Nodes
2708
3327
19717
183
183
251
# Edges
5429
4732
44338
295
309
499
# Features 1433
3703
500
1703
1703
1703
# Classes
3
WebKB is a webpage dataset collected from computer science departments various uni- versities by Carnegie Mellon University and available from the official PyTorch Geomet- ric(PyG)e y program. We use the three sub-datasets of Cornell, Texas, and Wisconsin, where nodes represent web pages, and edges are hyperlinks between them. Node features are the bag-of-words representation of web pages. The web pages are manually classified into five categories, student, project, course, staff, and faculty.
To guarantee the correctness of our proposed method, we conduct experiments with 12 different GNN models A.1 in the following experiments with pytorch-1.104, hardware is an 8-core CPU with 16G Memory, and the GPU is GeForce RTX 2080 (11GB).
4.1 TRAINING AND IMPLEMENTATION DETAILS.
4.1.1 NODE CLASSIFICATION
We performed node classification in a semi-supervised setting, Due to the small graph size, bobiiingvaaldattinn test split for Cornell, Texas, Wisconsi.. For Cora, CiteSeer, and Pubmed. we use the same split as Kipf & Welling (2016). Because VecGCN an extension of Kipf & Welling (2016) in higher dimensional space, we take only hyperparameter d, which is the dimension of the feature, fixed at 64. The appendix A.5 provides more hyper-parameter details.
we use a 60%/20%% 202%
4.1.2 LINK EXISTENCE PREDICTION
Given arbitrary two nodes u and v, the link existence prediction model asked to predict if euv E E, by scoring their inner product of representations za and z, as the likelihood of connectivity, written as score(euv) = (Zurze). We applied negative sampling to VecGON. Training a link prediction model involves comparing the scores between nodes connected by an edge against the scores between an arbitrary pair of nodes. For example, given an edge connecting u and v, we encourage the score between node and to be higher than the score between node u and a sampled node from an arbitrary noise distribution which is v' & (u). Like the section 1.1.1 above, randomly split the edge-set into parts at each beginning of the epochs: 80% for training, 15% of edges for testing and 5% for validation. During splitting, the connectivity was maintained. The appendix provides more
~ Pa (u)
details.
4.1.3 THE IMPACT OF DIMENSION ON VEGGON
The dimension is the key hyperparameter in VecGCN, which determines the size of the edge direction vector. We tested the effect of dimension d model performance on two tasks, node classification and link prediction, and three datasels, Cora, Citeseer, and Pubmed. In these experiments, we set the dimension 23 21, 25 26 27, and 28, respectively.
4.2 RESULTS
As shown in Table we that VecGCN performs well across all tasks, implying that the symmetrization- based approach can benefit citation networks in the context of node classi-
http///tta//pttrreeeeee geometric
'https:/ http:/yytooocoor..oo
P. (v) indicates the nodes that not u's neighbours

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fication tasks. This matches our intuition, and if given paper on the topic of reinforcement learning, then it is likely to cite or to be cited by other reinforcement learning papers.
Node classification accuracy (mea %). The best results are in bold. Because the best scores are distributed over several models, make this analysis more quantitative, we compute the absolute difference in the accuracy of cach method from that of the top performing method (%) on cach data sot and take the average as "inferior index", the closer to O, the better this indicator is. (Digraph and MagNet are out of memory on the pubmed dataset because they use the PyG framework.)
Table 2:
Model
Cora
Citeseer
Pubmed
Cornell
Texas
Wisconsin
inferior
GCN
81.1=0.2
70.3+0.4
79.4=0.1
59.0
58.723.8
55.9
-1.88
JKnet
86.1+1.5
70.9
76,611.2
57.355.0
61.126.2
52.815.7
-14.8
Spectral
SGC
79.3 = 0.0 71.910.1
78.9 + 0.0
58.613.4
56.414.3
51.3166.4
-6.2
GCNII
82.2+0.1
67.8+0.2
78.1+0.1
74.2+6.5
69.246.6
70.3+4.8
-8.6
ChebNet
81.2=0.5
69.8+0.2
74.4+0.3
79.8+5.0
79.247.5
81.646.3
-4.6
GAT
82.3+0.3 70.0
78.5+0.2
57.6+4.9
61.1#5.0
54.1+4.2
-15.0
SAGE
82.1#0.4 70.3
78.2#0.4
80.0
84.3+5.5
83.1+4.8
-2.6
Spatial
APPNP
82.4+0.7 70.4+1.1
79.3+0.6
58.7+44.0
57.044.8
51.847.4
-15.6
GIN
78.1+2.00 63.3+...5
78.9+0.1
57.9+5.7
65.2+6.5
58.245.1
-15.3
DGCN
81.3+1.4 66.34200
76.9
1.9
67.3443
71.7+7.4
65.5+4.7
-14.2
Directed Digraph
79.4+1.8 62.6+22.2
66.8+6.2
64.948.1
59.6+3.8
-16.1
MagNet
79.8
67.5
84.3+7.0
83.316.1
85.713.2
-2.7
Ours
VecGCN
83.1+0.3
71.4+0.4
79.1
85.746.3
84.5+5.1
84.1+3.7
-0.9
For the node classification task,
task, we added the jump-connection JKnet et al (2018)) the
multi-layer stack GCNIIChen al. (2020), and the low-complexity SGCWu et al. (2019), for detailed descriptions refer to AppendixA.11 The purpose of doing is to expand the scope of comparison and highlight VecGCN features and advantages. From the above tableTable- see that generalization of vanilla. GCN, VecGCN has achieved almost compre- hensive surpass, which shows that VecGCN inherits and carries forward the advantages of GCN. Similarly, the edge vector computation module of VecGCN is also inspired by GAT, which achieves a comprehensive surpass, showing that VecGCN inherits and carries forward the advantages of GAT. In addition, we also see that other directed methods perform rela- tively poorly on the WebKB network, probably because these graphs are small and have few training samples. Therefore, this does not indicate that those models that perform poorly are inferior. In adn,tion, because of the uneven distribution of optimal scores, (o make the analysis more quantitative. We use "inferior" Lo calculate the difference between each model and the optimal model by calculating the absolute difference between the classification curacy of each method and that of the best performing method (percentage), and averaging over the six data sets. In this case, lower score is better, and method with score of 0 indicates that the method is the best- best-performing method on cach datasct. As shown in Table- VecGCN achieved overall best result of -0.9%.
(a) GCN on Cora
(b) VecGCN on Cora
(c) GCN on Citeseer (d) VecGCN on Citeseer
Figure 3: t-SNE Analysis: results from visual analysis of the distribution of node
representations.
Moreover, we give the visualizationFig- of VecGON on the Cora and Citeseer datasets
after dimension reduction on node vectorsVan der Maaten & Hinton (2008). We see
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that VecGCN has better clustering tendency than discriminative power as the Weisfeiler- Leaieer-LLhaa(((W (WL) test,Xu et al. (2018a), we argue that this is mainly owed to the fact that VecGCN brings in the edge directional vectors.
GNNccam have sizeable
In link existence classification task, we achieve the best performance on all datasets in Ta- ble-g. Due to the clear clustering tendency of VecGCN, which makes node representations more distinguishable, the discrimination between edge directions is more significant. This matches our intuition and demonstrates that VecGCN is natural extension of GCN.
Link existence classification accuracy (mean %). In addition, the spectral domain approach has many jump connections that directly the information of the adjacency matrix, which is equivalent to label leakage in link existence prediction, and therefore not considered for inclusion in this table. (Digraph and MagNet are out of memory on the pubmed dataset because they use the PyG framework.)
Table 3:
Model GCN
Cora
Citeseer
Pubmed
Cornell
Texas
Wisconsin
7..0+10.3
65.44:088
68.9+0.3
62.941.6
63.34:2.6
63.7+22.3
Spectral
ChebNet
70.9
68.210.3
71.0=0.4
70.913.2
71.613.6
72.4+2.3
GAT
72.3:+0.9
68.240.4
68.440.8
64.7+1.7
64.4+3.2
65.5+2.4
SAGE
63.8100.8
62.840.6
65.110.1
67.843.0
66.7422.6
66.4+3.4
Spatial
APPNP
75.0+10.6
71.310.8
72.6+10.9
67.24:2.3
67.0
68.1
GIN
75.4110.4
69.3+0..7
72.0+0.8
68.242.6
67.5+3.6
69.3+1.66
DGCN
80.2+0.4
78.240.5
79.3+0.6
77.943.4
80.2+4.55
80.8+2.0
Directed
Digraph
80.0+0.2
80.3+0.9
78.7+:2.5
79.2+22.8
80.9+-2.6
MagNet
80.7
77.910.7
79.5
79.3
80.8+22.9
Ours
VecGON
82.611.0
81.810.2
82.1
80.815.2 81.115.1
81.245.1
Finally, we test the effect of dimensions on the model performances. As shown in Fig- A2
higher dimension is not better, and the model's performance will be saturated. The emergence of this phenomenon will be studied in future works.
CONCLUSION
In this paper, we propose VecGCN solve the problem of the lack of anisotropy in GNNss From intuition, we define the edge vectors in Euclidean space. This approach to defining di- rection at the physical level naturally extends the usability of vanilla GCN. Typical examples include different atoms in a chemical molecule passing messages to cach other, relational data crystal lattices, and magnetic anisotrop-
in social networks, all kinds of field propagation icity in metals and alloys. VecGCN is a new approach to designing spectral-based GCN layers to fit directed graphs without requiring feature vectors. Moreover, this approach can be applied to any GCN model and is compatible with real problems to the maximun extent. Supported by a series of theoretical and empirical results, we believe that our approach will generate new ideas for the study of GCN.
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Dominique Beaini, Saro Passaro, Vincent Létournean, Will Hamilton, Gabriele Corso, and Pietro Liò. Directional graph networks. International Conference on Machine Learning, pp. 748 758. PMLR, 2021.

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Joan Bruna, Wojeiech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. arXiv preprint arXi::3312, 1312.666, 2013.
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Johannes Gasteiger, Janek GroB, and Stephan Günnemann. Directional message passing for molecular graphs. In International Conference on Learning Representations, 2019.
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11

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Under review as a conference paper at ICLR 2023
A APPENDIX DETAILS
A.1 LIST OF METHOD ABBREVIATIONS
Spectral
Spatial
Directional
1. GCN
Kipf
&
1. GAT
Veličkovid
DGCN
Zilly et al.
Welling
2016))
ct al.
(2017)
(2177)
2. .KKnet.
Xu et al.
(2018b)
2. SAGE
Hamilton
Digraph 2020aa)
(Tong al.
3. SGC (Wu et al.
et al.
(201 7a
(2019)
APPNP
Klicpera
MagNet et al. (2021)
Zhang
4. GCNT Chen et al. (2020)
et al.
(2018
5. Cheb Defferrard
4. GIN
Xuu et al.
4. VecGCN per)
(this pa-
al. (2016)
2018al (018
A.2 PSEUDO CODE
Algorithm 1 An algorithm with caption
ileration t- 0 L L=D++1] 1)- (A+ I)(D +1--
Xo = <(h) +
while iter S 200 do
dg T; zj
lcll 12
a/j = (1 - - lct, (12) D/D
c/ =
x = a' x
iter iter +1
end while
A.3 SHIFT INVARIANT
Figure A1: The picture's red points change only in coordinates but not in content.
A.4 ACCURACY IN DIFFERENT DIMENSIONS
A.5 HYPER-PARAMETERS DETAILS
B APPENDIX MATHEMATICAL PROOFS
B.1 VANILLA GCNs
x (A)U
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Under review as a conference paper at ICLR 2023
Node Clrsifiction of Crr
14
Lrk bickainn Proli poimna
01.0
0110
02.
22.35
5150
2
53.55
00.
05.
3020
Figure A2: Effect of dimensions(mean %). The vertical axis indicates the accuracy and the horizontal axis the dimension.
Table Al: Hyperparameters for Baseline GNNs
Hyperparameters
Models
Ir
weight decay dropout hidden layers
others
GCN
0.01
5E-4
0.5
2
GCNTT
0.01
0.01 /5F44
0.6
64
64
x= 0.5, a=0.2
Spectral
SGC
0.2
5F-6
0
JKnet
0.005
0.5
32
5
cat
ChebNet
0,01
5E-4
0
32
2
K=2,num hop-2
GAT
0.01
5E-4
0.6
2
num heads-8
Spatial
SAGE
0.001
5E-4
0.5
256
3
mean
APPNP
0.01
5E-4
0.5
64
2
a = 0.1
GIN
0.01
5E-6
0.6
64
5
mean
DGCN
0.001
IE-8
0.5
64
5
cat
Digraph
0.01
IE-4
0.5
128
2
Q=0.1
Directed
MagNet
0.005
5E-4
0.5
2
K 0=1.0=0225
VecGCN
0.005
5E4
0.5
2
dimension =64
where (A) Σx= a,A,.
Ugo(A)UT = U U
TTATTT
1=0
Σ
UUUAUT
(B.1)
Σ
T
B.2 LINEAR COMBINATION
Proof. The eigenvalues in the extended field are linear combinations the eigenvalues in its basis F
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Under review as a conference paper at ICLR 2023
G
G can be viewed as a linear space on F G. {br Bli E I}, where I is the index and B is the basis of G on F, that is to say, every element in G is a linear combination of F-basis {bs E Bli E Т}. Written as = b, Sis where g E G, b B, feF.
Thus, has a group of F basisis
let Mp = UA be the dccomposition of matrix M in F.
G BF
Mc BMF
(B.2)
=BUAUT
= -UBAUT
From Eq
B.2 we know that, the eigenvalues of are linear combinations of basis on
F
B.3
VALUE DOMAIN OF a
Proof. Equ
Value Domain of
Σ (EN(()
(1 (1-e6)) (1.)
Σ 1-
Σ
ez//2
jEN(()
JEN())
(B.3)
= D -1
bring Eq B.3 into Eq- we have:
Σ JEN(()
oij (D, (D/-1)/D,
(B.4)
=1 -1-11 1/D.
a. E 10.1-1/D.]= [0
B.4 FIELD EXTENSION
A subfield G of field F is subset G C F that G is field with respect the field operations inherited from F. Then F is an extension field of G, and is the basis field of F. Such a field extension is denoted F/G.
In each field expansion, the subfield can be regarded as a vector space with the base field as the coefficient field. with filed expansion F F/G, considering the elements in F as vectors and the elements in G as the corresponding coefficients. Under such definition, F is an G-vector space. The dimension of the vector space is called the degree of the filed expansion, which is generally denoted as [F: G].
For example, the field of complex numbers C is an extension field of the field of real numbers R. Clearly then, C/R is a field extension and we have [C R] = 2 because {1, is basis, such extension C/R. finite.
B.5 ROTATION INVARIANT
Proof. Laplacian operator is rotational invariant.
The Laplace operator in GNN is equivalent to that under discrete conditions, where prove that the Laplace operator under continuous conditions is rolationally invariant.
let:
14

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Under review as a conference paper at ICLR 2023
z' cos y/ sin
(B.5)
y x' sin 1f cos
(B.6)
where (2,y) are the original coordinates and (a', y²) are the rotation coordinates.
8f
f
f2f
+
0
01
0s
+
01'
8y
0y'
0
Of dr
0f dy
Of dr
0f dy
+
+
Oz
dr dr
Dy dx
dy
dx Oy
dy dry
0
0f
0f
0 0f
0f
COS 0
+
sin
+
sin
cos
Ex
dr
by
Oy
dx
dy
0s
0f
0
0f
05
Oy
cOs
0
+
sin
COS 1
sin A
8r
Oz
dy
Ox
dy
dr
Oy
Ox
(B.7)
0f
0f
Ox
0
0f
0f
+
sin
0
+
cos
+
sin
0
+
cos
dr
Oz
dy
Oy
Oy
dr
dy
Oy
0
0s
05
0
05
05
COs
+
sin
COs 0
+
cos
0
+
sin 0
sin
dr
Oz
Oy
By
d
Oy
0f
0f
0 05
0f
+ dr
sin
0
+
cos
(- sin
0)
+
sin
+
cos
1
cos 0
Oz
dy
Oy
Or
0y
0 0f
os
du dx dy 0y
f + f2
0x 0y7²
B.6 PROOF ON THE RIGENVECTORS OF THE SKAW-SYMMETRIC MATRIX PURE IMAGINARY NUMBERS
Proof. The eigenvectors of the skew-symmetric matrix are pure imaginary numbers. M is a real skew-symmetric matrix of order T':
Let À be the eigenvalue of M, and YE R
s.t. ME = Xg
Multiply both sides by the conjugate transpose of &, and have:
<TME = A
(B.8)
Take the conjugate transpose of both sides: &T ME = XET Similarly,
TM { AET
(B.9)
M T -M
bring M -Min into B.9
=
(B.10)
15

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Under review as a conference paper at ICLR 2023
Add
B.10) to B.8
(A+ (A+ 0
E. E # - real (A) =0
16


Reviews:
[{"review_id": "TcCnl8l7HO", "reviewer": "Reviewer_T4f3", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper deals with the problem of the GCN architecture being limited to undirected graphs due to the requirement of the symmetric matrix as the basis for the Laplacian transformation. To overcome this limitation the authors propose (i) replacing the adjacency matrix with the distance matrix and (ii) extending the elements of the adjacency matrix from scalars to vectors using the field extension method to include the concept of direction. They evaluate the proposed model on six benchmark datasets for node classification and link prediction tasks and compare the performance to several baselines.", "summary_of_the_review": "In my view, the paper seems to be proposing an incremental contribution for the graph representation learning community. Due to reasons discussed above (related to limited novelty, poor presentation, and incomplete experimental evaluation), I think the work in its current state is not quite ready for publication, thus I cannot recommend acceptance.", "strength_and_weaknesses": "Strengths:\n\n- The paper deals with a limitation of spectral GNNs which cannot be directly applied to directed graphs. The proposed method is relatively simple which a good thing in my opinion, however, as discussed below, there are several issues that the authors need to address.\n\n- The proposed approach seems to perform relatively well in the two considered tasks. In the task of link prediction, it outperforms all the baselines on all datasets, while in node classification, it is the best-performing approach on 2 out of the 6 datasets.\n\nWeaknesses:\n\n- Most existing message passing graph neural networks can directly handle directed graphs (since they operate on the adjacency matrix which is not assumed to be symmetric), while there have also been proposed several models for that kind of graphs. Thus, the authors should explain in more details what makes the proposed approach appropriate for processing directed graphs and what are its advantages over previous models. Extensive experimental analysis and mathematical proofs of the soundness of the proposed method are rather limited.\n\n- Several details of the proposed approach are not properly explained in the paper. For instance:\n\t- In subsection 3.1, how exactly is the distance between two nodes defined? In my understanding, it is the Euclidean distance between the nodes' representations. By replacing the adjacency matrix with the matrix of distances between nodes, the graph's structural information is lost. I would like the authors to comment on that. Furthermore, the matrix of distances is likely to be dense (distances are positive in most cases). This is another limitation of the method since it implies that the proposed approach cannot be applied to very large graphs since the distance matrix might not fit in memory.\n\t- It is not clear to me why the scalar elements of the distance matrix are mapped to a high dimensional space. I suggest the authors provide clear explanations.\n\n- In the experimental evaluation, several benchmark datasets are not considered (especially larger ones such as ogbn-arxiv, squirrel, actor, Amazon Computers, Amazon Photo for the node classification task). It is thus not clear whether the proposed approach can be applied to large datasets.\n\n- The writing is not very clear. In fact, the paper leaves the reader with a feeling that this was an initial draft of the work. I found it relatively difficult to follow the analysis due to multiple grammatical and syntactic mistakes, while the vocabulary used is in many cases poor. Therefore, the considered manuscript lacks clarity and I suggest the authors work on improving that.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "PF9B7mF-ZTE", "reviewer": "Reviewer_opRy", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper focuses on the problem that graph convolutional networks are limited to the undirected graph due to theoretically needing the symmetric matrix for the Laplacian transform. To tackle the problem, this paper generalizes the spectral convolution operator to directed graphs by field extension, overcoming the symmetric problem as well as the information loss problem. Some experimental results show the effectiveness of the proposed VecGCN on several directed graph node classification and link prediction tasks.", "summary_of_the_review": "In summary, this paper gives an alternative construction of GCNs on directed graph. The main concerns lie in the technical novelty and some experiment analyses. I vote reject for this paper.", "strength_and_weaknesses": "**Strength:**\n\nThis paper has some advantages as follows:\n- This paper focuses on a fundamental and important research problem namely how to generalize the spectral convolution operator to directed graphs. \n\n- The paper is clearly written and the proposed method is easy to understand. Figure 2 helps readers to better understand the calculation mechanisms which is good.\n\n- The experiments on link classification tasks well verify the effectiveness of the proposed VecGCN, which achieves consistent improvements against baselines.\n\n**Weaknesses:**\n\nHowever, I still have some concerns:\n- The main concern is about the technical novelty. I think the technique of replacing the adjacency matrix with the distance matrix has been explored in the literature, e.g. DMPNN [1] and the directed graph neural networks have also been proposed in the community, e.g. R-GCN [2]. Moreover, I do not agree that “GCNs still limited to the undirected graph”. In a narrow sense, this condition of an undirected graph  is only required by the spectral interpretation  of GCNs. The general framework --  message passing paradigm can deal with the directed graph smoothly.\n\n- Technically, I think that using distance matrix to replace the adjacency matrix just evades the asymmetry problem of directed graph, rather than solving it. I can't see any theoretical justification of this replacement. \n\n- The time complexity as well as the empirical efficiency of the proposed VecGCN is not well discussed, since relatively complex techniques are proposed in the model.\n\n- I also have concerns about the experiments. The improvements of the VecGCN seem not significant compared with baselines in node classification tasks. Furthermore, it would be better to consider Open Graph Benchmark in the experiments for more convincing results.\n\n[1] Yang, Kevin, et al. \"Analyzing learned molecular representations for property prediction.\" Journal of chemical information and modeling 59.8 (2019): 3370-3388.\n\n[2] Schlichtkrull, Michael, et al. \"Modeling relational data with graph convolutional networks.\" European semantic web conference. Springer, Cham, 2018.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Ov9ld5Jxb-_", "reviewer": "Reviewer_DTdX", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes a new variant of the GCN model, namely VecGCN, based on replacing the adjacency matrix as the distance matrix. The main idea of VecGCN is to solve the requirement for using a symmetric matrix in GCNs as the basis of Fourier transformation. The authors propose to define edge vectors with the help of filed extension on graphs to denote the direction. They further use the edge vectors to compute the weights of edges for aggregation in a GAT-like style.\nExperiments on both node classification and link prediction tasks over six graph benchmark datasets show that VecGCN could perform on par with many spectral-based GNNs.\n", "summary_of_the_review": "My concerns are mainly from two aspects:\n* The current version seems to be incomplete and the theoretical analysis seems to be redundant. \n* The experimental analysis is insufficient and can be further improved to make it more convincing.\n", "strength_and_weaknesses": "Strengths:\n* The idea of generalizing GCNs from undirected graphs to directed ones is of great importance in some practical scenarios.\n* The use of field extension to regard the graph as the basis of its extension to define the edge vector space is interesting.\n\nWeaknesses:\n* This paper is a bit loosely written as it contains many typos, grammar errors, and incomplete sentences. I suggest the authors do thorough proofreading during the rebuttal. For examples: Eq~B.10 -> Eq~2 in line 1, page 6; adjacent matrix -> adjacency matrix in line 1 of Sec. 3.3. Sections A.3, A.4 and A.5 are all empty.\n* The novelty of this paper is limited, especially the theoretical parts. The theoretical proof of B.5 that the Laplacian operator is rotational invariant is relatively a well-known property of the Laplacian operator. The theoretical proof of B.6 that the eigenvectors of the skew-symmetric matrix are pure imaginary numbers seems to be disconnected from the main context as well as the design of VecGCN, which is strange to be included in the Appendix.\n* For experiments, the concerns include:\n    - The chosen benchmarks are all undirected graphs, while VecGCN is designed with a clear motivation for directed graphs. It is unclear why the performance of VecGCN on undirected graphs could demonstrate its effectiveness of VecGCN. Moreover, VecGCN performs worse on 4 out of 6 datasets in the node classification task, which makes VecGCN even weak.\n    - Most graphs in the benchmarks seem to be small. The scalability of VecGCN is unclear and needs to be demonstrated with some massive graphs such as benchmarks from OGB. \n    - For the link prediction task, some GNNs that were deliberately designed for this task are missing, such as [1-2]. The authors might include them as baselines for comparison for better demonstration. Moreover, the link direction prediction task is more suitable and should be included.\n\n[1] Link prediction based on graph neural networks, NeurIPS 2018\n\n[2] Line graph neural networks for link prediction, TPAMI 2021\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject