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Title: Tensor Graph Convolutional Networks for Prediction on Dynamic Graphs

Paper content:
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.
Graphs are popular data structures used to effectively represent interactions and structural relationships 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 networks for graph representation learning has attracted much interest.
A number of graph neural network (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; Zhou et al., 2018; Wu et al., 2019) .
Methods based on graph convolution filters which extend convolutional neural 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 representations of such dynamic graphs is essential.
Examples include analyzing social networks (Berger-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 (t) , X (t) ), t ∈ {1, 2, . . . , T }, with a fixed set V of N nodes, adjacency matrices A (t) ∈ R N ×N , and graph feature matrices X (t) ∈ R N ×F where X (t) n: ∈ R F is the feature vector consisting of F features associated with node n 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 a 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 to effectively process high-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- plications Martin et al., 2013; Zhang et al., 2014; Zhang and Aeron, 2016; Lu et al., 2016; Newman et al., 2018) .
A number of tensor based neural networks have also been investigated to extract and learn multi-dimensional representations, e.g. methods based on tensor decomposition (Phan and Cichocki, 2010), tensor-trains (Novikov et al., 2015; Stoudenmire and Schwab, 2016) , and tensor factorized neural network (Chien and Bao, 2017) .
Recently, a new tensor framework called the tensor M-product framework (Braman, 2010; Kilmer and Martin, 2011; Kernfeld et al., 2015) was proposed that extends matrix based theory to high-dimensional architectures.
In this paper, we propose a 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 us adapt the GCN architecture to tensor space.
Figure 1 illustrates our method at a high level: First, the time varying adjacency matrices A (t) and feature matrices X (t) of the 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 a 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 a 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 as a preprocessing step for other dynamic graph methods.
We have presented a novel approach for dynamic graph embedding which leverages the tensor Mproduct framework.
We used it for edge classification in experiments on four real datasets, where it performed competitively compared to state-of-the-art 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.

assistant: text: We propose a novel tensor based method for graph convolutional networks on dynamic graphs