Graph representation learning resurges as a trending research subject owing to the widespread use of deep learning for Euclidean data, which inspire various creative designs of neural networks in the non-Euclidean domain, particularly graphs. With the success of these graph neural networks (GNN) in the static setting, we approach further practical scenarios where the graph dynamically evolves. For this case, combining the GNN with a recurrent neural network (RNN, broadly speaking) is a natural idea. Existing approaches typically learn one single graph model for all the graphs, by using the RNN to capture the dynamism of the output node embeddings and to implicitly regulate the graph model. In this work, we propose a different approach, coined EvolveGCN, that uses the RNN to evolve the graph model itself over time. This model adaptation approach is model oriented rather than node oriented, and hence is advantageous in the flexibility on the input. For example, in the extreme case, the model can handle at a new time step, a completely new set of nodes whose historical information is unknown, because the dynamism has been carried over to the GNN parameters. We evaluate the proposed approach on tasks including node classification, edge classification, and link prediction. The experimental results indicate a generally higher performance of EvolveGCN compared with related approaches.
Methods for dynamic graphs are often extensions of those for a static one, with an additional focus on the temporal dimension and update schemes. For example, in matrix factorization-based approaches @cite_17 @cite_30 , node embeddings come from the (generalized) eigenvectors of the graph Laplacian matrix. Hence, DANE @cite_3 updates the eigenvectors efficiently based on the prior ones, rather than computing them from scratch for each new graph. The dominant advantage of such methods is the computational efficiency.
@cite_30 @cite_3 @cite_17
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Embedding large graphs in low dimensional spaces has recently attracted significant interest due to its wide applications such as graph visualization, link prediction and node classification. Existing methods focus on computing the embedding for static graphs. However, many graphs in practical applications are dynamic and evolve constantly over time. Naively applying existing embedding algorithms to each snapshot of dynamic graphs independently usually leads to unsatisfactory performance in terms of stability, flexibility and efficiency. In this work, we present an efficient algorithm DynGEM based on recent advances in deep autoencoders for graph embeddings, to address this problem. The major advantages of DynGEM include: (1) the embedding is stable over time, (2) it can handle growing dynamic graphs, and (3) it has better running time than using static embedding methods on each snapshot of a dynamic graph. We test DynGEM on a variety of tasks including graph visualization, graph reconstruction, link prediction and anomaly detection (on both synthetic and real datasets). Experimental results demonstrate the superior stability and scalability of our approach. Abstract Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map everywhere on the low-dimensional image of the forward map. We demonstrate that the scale-free cubic RBF kernel performs better than the Gaussian kernel: it does not suffer from ill-conditioning, and does not require the choice of a scale. The proposed construction is shown to be similar to the Nystrom extension of the eigenvectors of the symmetric normalized graph Laplacian matrix. Based on this observation, we provide a new interpretation of the Nystrom extension with suggestions for improvement. This paper exploits the properties of the commute time between nodes of a graph for the purposes of clustering and embedding and explores its applications to image segmentation and multibody motion tracking. Our starting point is the lazy random walk on the graph, which is determined by the heat kernel of the graph and can be computed from the spectrum of the graph Laplacian. We characterize the random walk using the commute time (that is, the expected time taken for a random walk to travel between two nodes and return) and show how this quantity may be computed from the Laplacian spectrum using the discrete Green's function. Our motivation is that the commute time can be anticipated to be a more robust measure of the proximity of data than the raw proximity matrix. In this paper, we explore two applications of the commute time. The first is to develop a method for image segmentation using the eigenvector corresponding to the smallest eigenvalue of the commute time matrix. We show that our commute time segmentation method has the property of enhancing the intragroup coherence while weakening intergroup coherence and is superior to the normalized cut. The second application is to develop a robust multibody motion tracking method using an embedding based on the commute time. Our embedding procedure preserves commute time and is closely akin to kernel PCA, the Laplacian eigenmap, and the diffusion map. We illustrate the results on both synthetic image sequences and real-world video sequences and compare our results with several alternative methods. The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used for spectral clustering and community detection. However, in real-life applications the number of clusters or communities (say, @math ) is generally unknown a-priori. Consequently, the majority of the existing methods either choose @math heuristically or they repeat the clustering method with different choices of @math and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the @math -th eigenpairs of the Laplacian matrix given a collection of all the @math smallest eigenpairs. Our proposed method adapts the Laplacian matrix such that the batch eigenvalue decomposition problem transforms into an efficient sequential leading eigenpair computation problem. As a practical application, we consider user-guided spectral clustering. Specifically, we demonstrate that users can utilize the proposed incremental method for effective eigenpair computation and determining the desired number of clusters based on multiple clustering metrics.