Tensor Graph Convolutional Networks for Prediction on Dynamic Graphs
Authors: Osman Asif Malik
Shashanka Ubaru
Lior Horesh
Misha E. Kilmer
Haim Avron
Venue:  (2020)
Keywords: graph convolutional networks
graph learning
dynamic graphs
edge classification
tensors

Abstract: Many irregular domains such as social networks, financial transactions, neuron connections, and natural language structures are represented as graphs. In recent years, a variety of  graph neural networks (GNNs) have been successfully applied for representation learning and prediction on such graphs. However, in many of the applications, the underlying graph changes over time and existing GNNs are inadequate for handling such dynamic graphs. In this paper we propose a novel technique for learning embeddings of dynamic graphs based on a tensor algebra framework. Our method extends the popular graph convolutional network (GCN) for learning representations of dynamic graphs using the recently proposed tensor M-product technique. Theoretical results that establish the connection between the proposed tensor approach and spectral convolution of tensors are developed. Numerical experiments on real datasets demonstrate the usefulness of the proposed method for an edge classification task on dynamic graphs.

Under review as a conference paper at ICLR 2020
TENSOR GRAPH CONVOLUTIONAL NETWORKS FOR PREDICTION ON DYNAMIC GRAPHS
Anonymous authors Paper under double-blind review
ABSTRACT
Many irregular domains such as social networks, financial transactions, neuron connections, and natural language structures are represented as graphs. recent years, variety of graph neural networks (GNNs) have been successfully applied for representation learning and prediction on such graphs. However, in many of the applications, the underlying graph changes over time and existing GNNs are inadequate for handling such dynamic graphs. In this paper we propose a novel tensor algebra
technique for learning embeddings of dynamic graphs based framework. Our method extends the popular graph convolutional network (GCN) for learning representations of dynamic graphs using the recently proposed tensor M-product technique. Theoretical results that establish the connection between the proposed tensor approach and spectral convolution of tensors are developed. Numerical experiments on real datasets demonstrate the usefulness of the pro- posed method for an edge classification task on dynamic graphs.
INTRODUCTION
Graphs popular data structures used to effectively represent interactions and structural relation- ships between entities in structured data domains. Inspired by the success of deep neural networks for learning representations in the image and language domains, recently, application of neural net- works for graph representation learning has attracted much interest. A number of graph neural net- work (GNN) architectures have been explored in the contemporary literature for a variety of graph related tasks and applications (Hamilton et al., 2017: Seo et al., 2018: Chen et al., 2018; Zhou et al., 2018; Wu et al., 2019). Methods based on graph convolution filters which extend convolutional neu- ral networks (CNNs) to irregular graph domains are popular (Bruna et al., 2013; Defferrard et al., 2016; Kipf and Welling, 2016). Most of these GNN models operate on a given, static graph.
In many real-world applications, the underlining graph changes over time, and learning representa- tions of such dynamic graphs is essential. Examples include analyzing social networks (Berger-Wor Wolf and Saia, 2006), predicting collaboration in citation networks (Leskovec et al., 2005), detecting fraud and crime in financial networks (Weber et al., 2018; Pareja et al., 2019), traffic control (Zhao et al., 2019), and understanding neuronal activities in the brain (De Vico Fallani et al., 2014). In such dynamic settings, the temporal interdependence in the graph connections and features also play a substantial role. However, efficient GNN methods that handle time varying graphs and that capture the temporal correlations are lacking.
By dynamic graph, we mean a sequence of graphs (V, A,, x()), t (1,2,...,,Ty with a fixed set V of N nodes, adjacency matrices A (c) E N and graph feature matrices (() E gN where x(!) E F is the feature vector consisting of F features associated with node at time t. The graphs can be weighted, and directed or undirected. They can also have additional properties like (time varying) node and edge classes, which would be stored in separate structure. Suppose we only observe the first T' < T graphs in the sequence. The goal of our method is to use these observations to predict some property of the remaining T - T" graphs. In this paper, we use it for edge classification. Other potential applications are node classification and edge/link prediction.
In recent years, tensor constructs have been explored effectively process high- sigh-dimensional data, in order to better leverage the multidimensional structure of such data (Kolda and Bader, 2009). Tensor based approaches have been shown to perform well in many image and video processing ap-

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Adjacency tensor
Time
Embedding
A(
Grank bucke Link prodiction
TensorGCN
Hdge clarstiiatiion Node claseiication
X(1
Dynamic graph
Feature Tensor
Figure I: TensorGCN approach.
plications (Hao al., 2013; Kilmer al., 2013; Martin al., 2013; Zhang al., 2014; Zhang and Aeron, 2016; Lu et al., 2016; Newman et al., 2018). A number tensor based neural networks have extract and learn multi-dimensional representations, e.g. methods based
also been investigated on tensor decomposition (Phan and Cichocki, 2010), tensor-trains (Novikov et al., 2015; Stouden- mire and Schwab, 2016), and tensor factorized neural network (Chien and Bao, 2017). Recently, new tensor framework called the tensor M- M-product framework (Braman, 2010; Kilmer and Martin, 2011: Kernfeld al., 2015) was proposed that extends matrix based theory high-dimensional architectures.
In this paper, we propose novel tensor variant of the popular graph convolutional network (GCN) architecture (Kipf and Welling, 2016), which we call TensorGCN. It captures correlation over time by leveraging the tensor M-product framework. The flexibility and matrix mimeticability of the framework, help adapt the GCN architecture to tensor space. Figure illustrates our method at high level: First, the time varying adjacency matrices ( and feature matrices dynamic graph are aggregated into an adjacency tensor and a feature tensor, respectively. These tensors are then fed into our TensorGCN, which computes an embedding that can be used for variety of tasks, such as link prediction, and edge and node classification. GCN architectures are motivated by graph convolution filtering, i.e., applying filters/functions to the graph Laplacian (in turn its eigenvalues) (Bruna et al., 2013), and we establish similar connection between TensorGCN and spectral filtering of tensors. Experimental results on real datasets illustrate the performance of our method for the edge classification task on dynamic graphs. Elements of our method can also be used preprocessing step for other dynamic graph methods.
X() of the
2 RELATED WORK
The idea of using graph convolution based on the spectral graph theory for GNNs was first intro- duced by Bruna et al. (2013). Defferrard et al. (2016) then proposed Chebnet, where the spectral filter approximated by Chebyshev polynomials in order make it faster and localized. Kipf and Welling (2016) presented the simplified GCN, a degree-one polynomial approximation of Chebnet, in order speed up computation further and improve the performance. There are many other works that deal with GNNs when the graph and features are fixed/static; see the review papers by Zhou al. (2018) and Wu et al. (2019) and references therein. These methods cannot be directly applied to the dynamic setting we consider. Seo et al. (2018) devised the Graph Convolutional Recurrent Network for graphs with time varying features. However, this method assumes that the edges are fixed over time, and is not applicable in our setting. Wang et al. (2018) proposed method called neural network (NN) approach that applies convolution operations on static
EdgeConv, which graphs in dynamic fashion. Their approach is not applicable when the graph itself is dynamic. Zhao et al. (2019) develop a temporal GCN method called T-GCN, which they apply for traffic prediction. Their method assumes the graph remains fixed over time, and only the features vary.
The set of methods most relevant to our setting of learning embeddings of dynamic graphs use com- binations of GNNs and recurrent architectures (RNN), to capture the graph structure and handle time dynamics, respectively. The approach in Manessi et al. (2019) uses Long Short- Memory (LSTMM, recurrent network, in order handle time variations along with GNNs. They design
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architectures for semi-supervised node classification and for supervised graph classification. Pareja et al. (2019) presented a variant of GCN called EvolveGCN, where Gated Recurrent Units (GRU) and LSTMs are coupled with GCN to handle dynamic graphs. This paper is currently the state- of-the-art. However, their approach is based on a heuristic RNN/GRU mechanism, which is not theoretically viable, and does not harness tensor algebraic framework incorporate time varying information. Newman et al. (2018) present a tensor NN which utilizes the M-product tensor frame- work. Their approach can be applied to image and other high-dimensional data that lie on regular grids, and differs from ours since we consider data on dynamic graphs.
3 TENSOR M-PRODUCT FRAMEWORK
Here, we cover the necessary preliminaries on tensors and the M-product framework. For a more general introduction tensors, we refer the reader to the review paper by Kolda and Bader (2009). In this paper, fensor is a three-dimensional array of real numbers denoted by boldface Fuler script letters, e.g. E Txxx Matrices are denoted by bold uppercase letters, e.g. X; vectors are denoted by bold lowercase letter, e.g. x; and scalars are denoted by lowercase letters, e.g. z. An element at position (i. in a tensor is denotes by subscripts, e.g. Xejz. with similar notation for elements of matrices and vectors. A colon will denote all elements along that dimension; X,; denotes the ith row of the matrix X, and X.:k denotes the kth frontal slice of X. The vectors X.j: called the tubes of X.
The framework we consider relies on a new definition of the product of two tensors, called the M-product (Braman, 2010; Kilmer and Martin, 2011; Kilmer al., 2013; Kernfeld et al., 2015). A distinguishing feature of this framework is that the M-product of two three-dimensional tensors also three-dimensional, which not the case for e.g. tensor contractions (Bishop and Goldberg, 2012). It allows one to elegantly generalize many classical numerical methods from linear algebra, and has been applied e.g. neural networks (Newman et al., 2018), imaging (Kilmer et al., 2013; Martin et al., 2013; Semerci et al., 2014), facial recognition (Hao et al., 2013), and tensor completion and denoising (Zhang et al., 2014; Zhang and Aeron, 2016; Lu al., 2016). Although the framework was originally developed for thre-ddimeniooss tensors, which is sufficient for our purposes, it has been extended to handle tensors of dimension greater than three (Martin al., 2013). The following definitions 3.1-3.3 describe the M-product.
Definition 3,1 (M-transform). Let M E R be mixing matrix. The M-transform of a tensor XC E XxxT is denoted by X X3 M E RXXJXX and defined elementwise as
(XC X3 M) ije
del
We say that X X3 M is in the transformed space. Note that if M is invertible, then (X X3 M) X3
M X. Consequently, X X3 M is the inverse M-transform of X. The definition in (1) may also be written in matrix form as X X3 M fold(M un)old(3)), where the unfold operation takes the tubes of X and stack them as columns into a T x IJ matrix, and fold (nfold))l)) X. Appendix A provides illustrations of how the M-transform works.
Definition 3.2 (Facewise product). Let X E IXXXT and y E TR J be two tensors. The facewise product, denote by X A y E IXXX, is defined facewise as (X A y)
de' y..
Definition 3.3 (M-product). Let X E RIXJXT and y E R be two tensors, and let M E RTxT xI' be an invertible matrix. The M-product, denoted by X y E R/xXxT is defined as
X y def ((X xa M) ( X3 M)) Xa M-1
In the original formulation of the M-product, M was chosen to be the Discrete Fourier Transform (DFT) matrix, which allows efficient computation using the Fast Fourier Transform (FFT) (Braman, 2010; Kilmer and Martin, 2011; Kilmer et al., 2013). The framework was later extended for arbitrary invertible M (e.g. discrete cosine and wavelet transforms) (Kernfeld et al., 2015). A benefit of the tensor M M-product framework is that many standard matrix concepts straightforward manner. Definitions 3.4-3.7 extend the matrix concepts of diagonality, identity, transpose and orthogonality to tensors (Braman, 2010; Kilmer al., 2013).
be generalized in
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Definition 3.4 (f-diagonal). A tensor X RNXNX is said to be f-diagonal if each frontal slice X.. diagonal.
Definition 3.5 (Identity tensor). Let I TRNNNX be defined facewise as I.. = I, where I is the matrix identity. The M-product identity tensor RNxxNx then defined as def X3 M-
Definition 3.6 (Tensor transpose). The transpose of tensor X is defined as X del X3 M- where y = (X X3 M) for each E [T....,T}.
Definition 3.7 (Orthogonal tensor). A tensor X RNNNNT is said be orthogonal
X X J.
Leveraging these concepts, tensor cigendecoompositinn now be defined (Braman, 2010; Kilmer et al., 2013):
Definition 3.8 (Tensor cigendccomposition). Let X e NxXx be tensor and assume that each frontal slice (XC X3 M)::. is symmetric. We can then eigendecompose these as (X X3 M):;
Q...... D.... :gt where Q. E TRNN is orthogonal and DD:. E NRN is diagonal (see e.g. Theo- rem 8.1.1 ; Golub and Van Loan (2013)). The tensor eigendeconposition of X is then defined as
x x QT, where Q Q X3 M-1 is orthogonal, and 'D D M- if f-diagonal.
4 TENSOR DYNAMIC GRAPH EMBEDDING
Our approach is inspired by the first order GCN by Kipf and Welling (2016) for static graphs, owed simplicity and effectiveness. For a graph with adjacency matrix A and feature matrix X, GCN layer takes the form Y (AXW), where
def D 1112 (A I)D 1)D-1/2 1/2
D is diagonal with D 1+ A.j. I the matrix identity, W is matrix to be learned when training the NN, and is an activation function, e.g., ReLU, Our approach translates this a tensor model by utilizing the M-product framework. We first introduce tensor activation function â which operates in the transformed space.
Definition 4.1. Let A RXXxT be tensor and elementwise activation function. We define the activation function ē as ((A)
def (A X3 M) X M-
We can now define our proposed dynamic graph embedding. Let A E RNNxN be a tensor with frontal slices A = A (t), where Á is the normalization of A (1) Morcover, let X I NxFT be tensor with frontal slices Xx:, (t), Finally, let W FxF'x be weight tensor. We define our dynamic graph embedding as y A + X W E Nx'xT, This computation can also be repeated in multiple layers. For example, 2-layer formulation would be of the form
y A ( w())) W1))
One important consideration is how choose the matrix M which defines the M-product. For time-varying graphs, we choose M to be lower triangular and banded so that each frontal slice (A X3 M)::: is linear combination of the adjacency matrices A (=m=x(( A.:, where refer to b as the "bandwidth" of M. This choice ensures that each frontal slice (A X3 M).. only contains information from current and past graphs that are close temporally. Specifically, the entries of M are set to
Mu def (in( if max(1, b+ k
otherwise,
which implies that Σg x M for each t. Another possibility is treat M as parameter matrix
to be learned from the data.
In order to avoid over- parameterization and improve the performance, choose the weight tensor W (at each layer), such that each of the frontal slices of W in the transformed domain remains the same, i.e., (W X3 M).t- (W X3 M) Vt, f'. In other words, the parameters in each layer
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shared and learned over all the training instances. This reduces the number of parameters to be learned significantly.
An embedding y RNNXFXT can now be used for various prediction tasks, like link prediction, and edge and node classification. In Section 5, apply method for edge classification by using a model similar to that used by Pareja et al. (2019): Given an edge between nodes m and n at time the predictive model
p(m, t) del softmax(U[(} X3 M) (y X3 M)
where (y X3 mi E RF and ( X3 M) E IRF are row vectors, U E RCX2F is a weight matrix, and C the number of classes. Note that the embedding is first M-transformed before the matrix U is applied to the appropriate feature vectors. This, combined with the fact that the tensor activation functions are applied elementwise in the transformed domain, allow us to avoid ever needing to apply the inverse M-transform. This approach reduces the computational cost, and has been found to improve performance in the edge classification task.
4.1 THEORETICAL MOTIVATION FOR TENSORGCN
Here, present the results that establish the connection between the proposed TensorGCN and spectral convolution of tensors, in particular spectral filtering and approximation on dynamic graphs. This is analogous to the graph convolution based spectral graph theory in the GNNs by Bruna et al. (2013), Defferrard et al. (2016), and Kipf and Welling (2016). All proofs are provided in Appendix D.
Let L E N be a form of tensor Laplacian defined as L A. Throughout the remainder of this subsection, we will assume that the adjacency matrices A (9) are symmetric.
Proposition 4.2. The tensor & has an eigendecomposition & = 2 D QT.
Much like the spectrum of normalized graph Laplacian contained in [0, 2] (Shuman al., 2013), the tensor spectrum of & satisfies a similar property.
Proposition 4.3 (Spectral bound). The entries D D X3 M lie in [0,
Following the work by Kilmer et al. (2013), tree-dimensional tensors in R. MxxT can be viewed as operators on x T matrices, with those matrices "twisted" into tensors in NxIT With this in mind, we define a tensor variant of the graph Fourier transform.
Definition 4.4 (Tensor-tube M-product). Let X E IXXT and E RIXIXT Analogously to the definition of the matrix-scalar product, we define X * 0 via (XC 0) X,y. * 0.
Definition 4.5 (Tensor graph Fourier transform). Let XC E NxFT be tensor. We define tensor graph Fourier transform F as F(X) Q *.C XxFX
This is analogous to the definition of the matrix graph Fourier transform. This defines a convolution like operation for tensors similar to spectral graph convolution (Shuman et al., 2013; Bruna et al., 2013). Each lateral slice X.j: is expressible in terms of the set (Q.::) n as follows:
N
X. = Q X
Σ
.m: * (QT X
where each (Q * X M1:: E xxxxXT can be considered a tubal scalar. In fact, the lateral slices Q.n form a basis for the set NIxX with product *; see Appendix D for further details.
Definition 4.6 (Tensor spectral graph filtering). Given a signal E RNxxxT and a function g : RlxixT + B Rxxxxx, we define the tensor spectral graph filtering of X with respect to g
Xgl
del Q g( (D)
XX,
where
der (mmmm):) mn:)
if m n,
g
00
if m + n.
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In order to avoid the computation of an eigendecomposition, Defferrard al. (2016) use polyno- mial to approximate the filter function. We take a similar approach, and approximate g with an M-product polynomial. For this approximation to make sense, impose additional structure g.
Assumption 4.7. Assume that
R xxIx7
7 R x is defined
s def f(V
X3 M) X3 M
where is defined elementwise as f(V X3 M) def ( xs M)111) with each f(u IR R continuous.
Proposition 4.8. Suppose g satisfies Assumption 4.7. For any E > there exists an integer K and a set (0))) =k=1 C RIxxxx such that
K (((D) - D 0 <E,
k=0
where || is the tensor Frobenius norm, and where Di J.
D * * is the M-product of
instances D, with the convention that D
As in the work of Defferrard et al. (2016), a tensor polynomial approximation allows us to approxi- mate X in (2) without computing the eigendecomposition of L:
K
K Σ c ((()) A==
Xm 2 * (D) 2 * *0 2
D sk ((h))
2 +X
* X.
(3)
All that is necessary is to compute tensor powers of L. We can also define tensor polynomial analogs the Chebyshev polynomials and the approximation (3) terms of those instead of the tensor monomials D* This not necessary for the purposes of this paper. Instead, we note that if degree-one approximation is used, the computation in (3) becomes
0 + (g A) ( X. Xyite ~ (9 (()) + + (()) X (J
Setting 0 (0) = - (2)) which is analogous to the parameter choice made in the degree-one
approximation by Kipf and Welling (2016), we get
Xm ~ * O.
(4)
we let X contain F signals, i.e., NXPX and apply F" filters, (4) becomes
Xiie 2 A X E NxF'XT
where @ E FXX'T XT This is precisely our embedding model, with O replaced by a learnable
parameter tensor W.
NUMERICAL EXPERIMENTS
Here, we present results for edge classification on four datasets'; The Bitcoin Alpha and OTC transaction datasets (Kumar et al., 2016), the Reddit body hyperlink dataset (Kumar et al., 2018), and a chess results dataset (Kunegis, 2013). The bitcoin datasets consist of transaction histories for users on two different platforms. Each node is a user, and each directed edge indicates a transaction and labeled with an integer between 10 and which indicates the senders trust for the receiver. We convert these labels to two classes: positive (trustworthy) and negative (untrustworthy). The Reddit dataset build from hyperlinks from one subreddit to another. Each node represents a subreddit, and each directed edge is interaction which is labeled with -1 for hostile interaction +1 for a friendly interaction. We only consider those subreddits which have a total of 20 interactions or more. In the chess dataset, each node is player, and each directed edge represents match with the source node being the white player and the target node being the black player. Each edge is labeled -1 for a black victory, 0 for draw, and +1 for white victory. Table summarizes the statistics for the different datasets.
We provide links the datasets in Appendix

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Table 1:
Dataset statistics.
Dataset
Nodes
Edges
Graphs (7")
Time window length
Classes
Bitcoin OTC
6,005
35,569
135
14 days
2
Bitcoin Alpha
7,604
24,173
135
14 days
2
Reddit
3,818
163,008
86
14 days
2
Chess
7,301
64,958
100
31 days
3
The data temporally partitioned into T graphs, with each graph containing data from particular time window. Both T" and the time window length can vary between datasets. For each node- pair (n, t) these graphs, we compute the number of outgoing and incoming edges and use these two numbers as features. The adjacency tensor A is then constructed as described in Section 4. The frontal slices of A are divided into Suain training slices, Sval validation slices, and Sist testing slices, which come sequentially after each other; see Figure and Table 2.
N
N
N
Testing Validation
Testing Validation
Testing Validation
Training
Training
Training
Figure 2: Partitioning of A into training, validation and testing data.
Table 2: Partitioning and performance metric for each dataset.
Partitioning
Dataset
Suain
S
S
Performance metric
Bitcoin OTC
95
20
20
Fl score
Bitcoin Alpha
95
20
20
Fl score
Reddit
66
10
10
Fl score
Chess
80
10
10
Accuracy
Since the adjacency matrices corresponding to graphs are very sparse for these datasets, we apply the same technique Pareja et al. (2019) and add the entries of each frontal slice A to the following 1-1 frontal slices A.t.... ................................ (-)++--)) where we refer as the "edge life." " Note that this only affects A, and that the added edges are not treated as real edges in the classification problem.
The bitcoin and Reddit datasets are heavily skewed, with about 90% of edges labeled positively, and the remaining labeled negatively. Since the negative instances are more interesting to identify (e.g. to prevent financial fraud or online hostility), we use the score to evaluate the experiments on these datasets, treating the negative edges the ones we want to identify. The classes are more well-balanced the chess dataset, so we use accuracy to evaluate those experiments.
We choose to use an embedding Yusin (1:Smm)) (.: W for training. When computing the embeddings for the validation and testing data, we still need Spain frontal slices of A, which get by using sliding window of slices. This is illustrated in Figure 2, where the green, blue and red blocks show the frontal slices used when computing the embeddings for the training, vali- dation and testing data, respectively. The embeddings for the validation and testing data are y
A:: 1:Su * (S. respectively. Preliminary experiments with 2-layer architectures did not show convincing improve- ments in performance. We believe this is due to the fact that the datasets only have two features, and that a l-layer architecture therefore is sufficient for extracting relevant information in the data. For training, we use the cross entropy loss function:
W and y A.: Sentsu-SSetttii S.i+ * Sp W,
loss Σ
Σ
Σ cll
aaf(m, t) log(p(m, t)c).
(5)
(M.)EE,
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Table 3: Results without symmetrizing adjacency matrices. A higher value is better.
Dataset
Method
Bitcoin OTC
Bitcoin Alpha
Reddit
Chess
WD-GCN
0.2062
0.1920
0.2337
0,4311
EvolveGCNN
0.3284
0.1609
0.2012
0.4351
GCN
0.3317
0.2100
0.1805
0.4342
TensorGCN (Proposal)
0.3529
0.2331
0.2028
0.4708
Table 4: Results when using symmetrized adjacency matrices, A higher value is better.
Dataset
Method
Bitcoin OTC
Bitcoin Alpha
Reddit
Chess
WD-GCN
0.1009
0.1319
0.2173
0.4321
EvolveGCN
0.0913
0.2273
0.1942
0,4091
GCN
0.0769
0.1538
0.1966
0.4369
TensorGCN (Proposal)
0.3103
0.2207
0.2071
0.4713
where f(m, n,,) RC is one-hot vector encoding the true class of the edge (m, n) time t, and E RC is vector summing to which contains the weight of each class. Since the bitcoin and Reddit datasets are so skewed, we weigh the minority class more heavily in the loss function for those datasets, and treat as hyperparameter; see Appendix C for details.
The experiments are implemented in PyTorrh with some preprocessing done in Matlab. Our code is available at [url redacted for review]. In the experiments, we use an edge life of I = 10, a bandwidth = 20, and Fx 6 output features. Since the graphs in the considered datasets are directed, we also investigate the impact of symmetrizing the adjacency matrices, where the symmetrized version of an adjacency matrix is defined as Asym def + 12(A+A))
We compare our method with three other methods. The first one is variant of the WD-GCN by Manessi et al. (2019), which they specify in Equation (8a) of their paper. For the LSTM layer in their description, method more comparable to ours which uses 6 output features. For the final layer, we use the same prediction model as that used by Pareja et al. (2019) for edge classification. The second method is a I l-layer variant of EvolveGCN-H by Pareja et al. (2019). The third method is simple baseline which uses I-layer version of the GCN by Kipf and Welling (2016). It uses the W for all temporal graphs. Both EvolveGCN-H and the baseline GCN use 6 output features as well.
6 output features instead of N. This is to avoid overfitting and make the
weight matrix
Table shows the results when the adjacency matrices have not been symmetrized. In this case, our method outperforms the other methods on the two bitcoin datasets and the chess dataset, with WD-GCN performing best on the Reddit dataset. Table 4 shows the results for when the adjacency matrices have been symmetrized. Our method outperforms the other methods on the Bitcoin OTC dataset and the chess dataset, and performs similarly but slightly worse than the best performing methods on the Bitcoin Alpha and Reddit datasets. Overall, it seems like symmetrizing the adjacency matrices leads to lower performance.
6 CONCLUSION
We have presented novel approach for dynamic graph embedding which leverages the tensor M- product framework. We used it for edge classification in experiments four real datasets, where it performed competitively compared to state-of-the- methods. Future research directions include further developing the theoretical guarantees for the method, investigating optimal structure and learning of the transform matrix M. using the method for other prediction tasks, and investigating how to utilize deeper architectures for dynamic graph learning.
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Ling Zhao, Yujiao Song, Chao Zhang, Yu Liu, Pu Wang, Tao Lin, Min Deng, and Haifeng Li. T- GCN: Temporal Graph Convolutional Network for Traffic Prediction. IEEE Transactions on Intelligent Transportation Systems, 2019.
Jie Zhou, Ganqu Cui, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, and Maosong Sun. Graph neural 2018.
networks: A review of methods and applications. arXiv preprint

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Appendices
A ILLUSTRATION OF THE M-TRANSFORM
We provide some illustrations that show how the M-transform in Definition 3.1 works. Recall that X X3 M fold(M un)) The matrix X first unfolded into matrix, as illustrated in Figure 3. This unfolded tensor is then multiplied from the left by the matrix M, as illustrated in Figure 4; the figure also illustrates the banded lower triangular structure M. Finally, the output
matrix is folded back into tensor. The fold operation is defined to be the inverse of the unfold operation.
unfold
Figure Illustration of unfold operation applied to x tensor.
M
unfolded tensor
output
Figure 4: Illustration of matrix product between M and the unfolded tensor.
B LINKS TO DATASETS
The Bitcoin Alpha dataset available at
https:/ //asss///aapp stanford. edu http://oo-ssomm-bbiooooo-oo----- html.
The Bitcoin OTC dataset is available at
https: /tsts::/ssap.. snap. s st tanford. edu/ addu/daaa/sooa socc www.--siiinniiiiiniiiieeooe-eeee html.
The Reddit dataset is available at https: : / anap. stanford. edu/ htt///ata//ee Rew.eeeddtttyeeereeeeeeeeeeeeeee html.
Note that we use the dataset with hyperlinks in the body of the posts.
The chess dataset available at http: / uni ani-kola -oookoollee -ookoooezzzz.ooo.. hwt.ee/w....eeeeere/eeeeeeeeeeee http://wee...oooooooeeeeeeeeeeae
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C FURTHER DETAILS ON THE EXPERIMENT SETUP
When partitioning the data into graphs, as described Section 5, if there are multiple data points corresponding an edge (m,n) for given time step we only add that edge once to the cor- responding graph and set the label equal to the sum of the labels of the different data points. For example, if bitcoin user m makes three transactions to n during time step with ratings 10, 2, 11, then we add single edge (m, n.) to graph t with label 10 + 2 1 11.
For training, we run gradient descent with learning rate of 0.01 and momentum of 0.9 for 10,000 iterations. For each 100 iterations, we compute and store the performance of the model on the validation data. As mentioned as hyperparameter in the bitcoin and Reddit experiments. Since these datasets all have two edge classes, let ag and æj be the weights of the minority (negative) and majority (positive) classes, respectively. Since these parameters add to I, we have 0i = do- For all methods, we repeat the bitcoin and Reddit experiments once for each dy E {0.75, 0.76........... 0.95}. For each model and dataset, we then find the best stored performance of the model the validation data across all co values. We then treat the corresponding model as the trained model, and report its performance on the testing data in Tables 3 and 4. The results for the chess experiment computed in the same way, but only for a single vector a = /3, 1/3, 1 1/3]-
the loss function (5) treated Section the weight vector
D ADDITIONAL RESULTS AND PROOFS
Throughout this section, || will denote the Frobenius norm (i.e., the square root of the sum of the elements squared) of matrix or tensor, and || ||2 will denote the matrix spectral norm.
We first provide a few further results that clarify the algebraic properties of the M-product. Let RIxixT denote the of 1 x T tensors. Similarly, let R-- x.Nxxx denote the set of N 1 x T tensors. Under the M- M-product framework, the set R 1xIxT play role similar to that played by scalars matrix algebra. With this in mind, the set NxxxT can be seen as length N vector consisting of tubal elements of length T. Propositions D.1 and D.2 make this more precise.
Proposition D.1 (Proposition 4.2 in Kernfeld et al. (2015)). The set IxxxxT with product *, which a commutative ring with identity.
denoted by (+, (4, IIxixT),
Proposition D.2 (Theorem 4.1 in Kernfeld al. (2015)). The set IR NXIXT with product which is denoted by (+, RR XxIxX)) a free module over the ring (+, (+,R1xIXT))
free module is similar a vector space. Like vector space, has basis. Proposition D.3 shows that the lateral slices of Q in the tensor eigendecomposition form a basis for (*, RxIXT),, similarly how the eigenvectors in a matrix eigendecomposition form a basis.
Proposition D.3. The lateral slices 2.n: NxxxT of Q in Definition 3.8 form basis for (x, N XxIXT".
Proof. Let E NxIXT Note that
N 2.n: Vni:s
X = =
+X
n=1
where
de Q
* X E NxIxT So the lateral slices of Q are generating set for (x, xxxxxxI..
Now suppose
Σ O.n 8.1: 0.
for some NxIXT Then = Q S. and consequently
(Q X3 M) ASS X3 M).
Since each frontal face of Q X3 M is invertible matrix, this implies that each frontal face of Xa M is zero, and hence = 0. So the lateral slices of Q are also linearly independent in (x, NXIIX).
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Under review as a conference paper at ICLR 2020
D.1 PROOFS PROPOSITIONS IN THE MAIN TEXT
Proof of Proposition 4.2. Since each adjacency matrix A (0) and each I:r is symmetric, each frontal slice Lut is also symmetric. Consequently,
(L X3 M)i;: Lyj; X3 M L ja: M (L M)
so each frontal slice of L X3 M is symmetric, and therefore L has an eigendecomppssiton.
Proof of Proposition 4.3, Each A.. has a spectrum contained in [-1, 1]. Since A., symmetric, follows that !.A S 1. Consequently,
T
IIA X3 M).eel2= = Σ M A.:
My Allاle S 1,
where we used the fact that Ej Mayl 1. So since the frontal slices (A X3 M)::. are symmetric, they each have spectrum in 1-1,1] 1]. It follows that each frontal slice
(C X3 M) = =I - (A X3 M)::.
has spectrum contained in [0, 2], which means that the entries of DD all lie in [0, 2]. Lemma D.4. Let X MxNxT and let M TxT be invertible. Then
|D||| s || |22110 X3 Ml
Proof. We have
||]]| = [[X X3 M) X3 MI = |M 1 unfold(3 X3 M)]l
MM |all (n X3 M) = || ||21| X3 Ml],
where the inequality a well-known relation that holds for all matrices.
Proof of Proposition 4.8. By Weierstrass approximation theorem, there exists an integer K and
(0 (k)
1==1 CRI RlxIxT such that for all & (T.
K
sup (x)
(R)) 1ir
E
x
<
=E(0,2]
M
IVVT
K=U
Let 0 def 0 (k)
X3 M- Note that if m + n, then
D * ((k)
Σ
mnn X3 M- ((() = (g(D) mm::
man:
k=O
since D
X3 M is f-diagonal. So
K
N
t g(k) 1
Σ
nn:: Σ
D nn: * ((k)
g
n=1
0
K
1
M
(D)
Dnn X3 M
DD X3 M) k)
nn: 4
1=
((0((D) X3 M)nnt) Σ D x3 M) min 0 (2)
|M
<
where the first inequality follows from Lemma D.4. Taking square roots completes the proof.
14


Reviews:
[{"review_id": "rygrOcyJ5r", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary: this work uses tensor methods to improve graph convolution for dynamic graph, where the nodes are fixed and the edges are changing. Specifically, it uses the M-product technique to develop the operations of sequence of matrices that analog to these operations of matrices. In the M-product notations, everything seems to be as neat as matrix operations. The works also shows decent supremacy on edge classification tasks.\n\n\nComments: this paper is mathematically interesting. It is well-written in general, but the definitions are dense and hard to follow.\n\nIt will be better to give some examples of M-product. For example, what these operations will be if we choose M to be the identity matrix?\n\nM-transfer is a tensor contraction, right?\n\nIt seems if you do the operations of the sequence of matrix, there is no need to do iterations like RNN. I am interested in how this will influence the runtime and memory cost.\n\nThe M matrix is defined as a lower triangle matrix such as (A \\times M)_::t depends on A^(1:t). Is it possible to formulate M such that (A \\times M)_::t will depend heavily on A^(t), and less on the farther matices? such that we encode some Markov property?\n\nDoes there exist some condition when this method will be equivalent to RNN?\n\n\nDecision: I feel this work novel and interesting in general. I would like to weakly accept it.\n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "Bkx1Ww41qH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents a M-product based temporal GCNs to handle dynamic graphs. Experiments on four real datasets are performed to verify the effectiveness of the proposed model.\n\nOverall, I think this paper make a few contributions to advocate tensor M-product. However, there are several big issues as listed below. Given the current status, I could not accept the paper.\n\nPros:\n\n1, The generalization brought by M-product seems to be general as it includes quite a few graph convolution elements for 3D tensors in a natural way.\n\n2, The experimental setup is reasonable. Datasets are collected from practical problems and of moderately large scale.\n\n3, The paper is clearly written and easy to follow.\n\nCons & Questions: \n\n1, My first concern is that M-product formulation does not bring any new insights as people have already used some of the key elements in practice for a long time. For example, the M-transform is just applying 1x1 convolution to multi-channel image. Slice-wise matrix multiplication is also common in practice.\n\n2, Moreover, I think there are several challenges in the M-product formulation which prevent the technique from being practical.\n\n(1) Sharing M such that frontal slices of the transformed signal are the same, i.e., each row of M share the same vector, limits the model capacity significantly. If there is no sharing mechanism, then the model learned on one sequence of graphs could not be applied to another sequence of graphs given two sequences have different lengths. \n\n(2) If you learn M from data, how could you ensure that M is invertible? In the paragraph before section 4.1, an edge classification formulation is proposed where the inverse M-transform is abandoned. However, if in practice, you do not need the inverse transform, then do those theoretical properties still hold and what is the meaning of introducing such M-product formulation?\n\n3, A few temporal GCN baselines are neither compared or discussed, e.g., [1]. \n\n4, Could you explain why all the other GCN variants performs significantly worse with a symmetrized adjacency matrix compared to using the asymmetric one? \n\n[1] Li, Y., Yu, R., Shahabi, C. and Liu, Y., 2017. Diffusion convolutional recurrent neural network: Data-driven traffic forecasting. arXiv preprint arXiv:1707.01926.\n\n======================================================================================================\n\nAfter I read authors' reply and other reviewers' comments, I would like to keep my original rating as the issues have not been properly addressed. I agree with the Reviewer #4 that the theoretical results are a bit artificial and trivial. ", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "B1eIYqtL9B", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposed a new type of graph embedding technique for dynamic graphs based on tensor representation (node x feature x time).  Experiments on edge classification demonstrate improved prediction accuracy. \n\n+ Clear writing with tensor notations and explanation is well-structured \n+ Improved prediction results on 3/4 real-world dynamic graph datasets\n\n- The theoretical results are a bit artificial. The tensor eigendecomposition used in this paper and [Kilmer and Martin] is for slices of the tensor, similarly for FFT and convolution. The technique is a trivial generalization from matrix results. \n- The paper is missing a large body of baselines, both from the network science community (non-deep learning methods) and from this community (diffusion convolutional RNNs, graph attention networks, etc). \n- The method doesn't scale well, especially for graphs with long-term dynamics. It would be good to show the scaling behavior of the proposed model. ", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a tensor-based extension to graph convolutional networks for prediction over dynamic graphs. 

The proposed model is reasonable and achieves promising empirical results. After discussion, it is agreed that while the problem of handling dynamic graphs is interesting and challenging, the proposed tensor method lacks novelty, the theoretical analysis is artificial, and the empirical study does not cover enough benchmarks. 

The current version of the paper is not ready for publication. Addressing the issues above could lead to a strong publication in the future. 
Decision: Reject