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Register a new userThe Power of Graph Convolutional Networks to Distinguish Random Graph Models: Short Version
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Graph convolutional networks (GCNs) are a widely used method for graph representation learning. We investigate the power of GCNs, as a function of their number of layers, to distinguish between different random graph models on the basis of the embeddings of their sample graphs. In particular, the graph models that we consider arise from graphons, which are the most general possible parameterizations of infinite exchangeable graph models and which are the central objects of study in the theory of dense graph limits. We exhibit an infinite class of graphons that are well-separated in terms of cut distance and are indistinguishable by a GCN with nonlinear activation functions coming from a certain broad class if its depth is at least logarithmic in the size of the sample graph. These results theoretically match empirical observations of several prior works. Finally, we show a converse result that for pairs of graphons satisfying a degree profile separation property, a very simple GCN architecture suffices for distinguishability. To prove our results, we exploit a connection to random walks on graphs.
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Graph convolutional networks (GCNs) are a widely used method for graph representation learning. To elucidate the capabilities and limitations of GCNs, we investigate their power, as a function of their number of layers, to distinguish between different random graph models (corresponding to different class-conditional distributions in a classification problem) on the basis of the embeddings of their sample graphs. In particular, the graph models that we consider arise from graphons, which are the most general possible parameterizations of infinite exchangeable graph models and which are the central objects of study in the theory of dense graph limits. We give a precise characterization of the set of pairs of graphons that are indistinguishable by a GCN with nonlinear activation functions coming from a certain broad class if its depth is at least logarithmic in the size of the sample graph. This characterization is in terms of a degree profile closeness property. Outside this class, a very simple GCN architecture suffices for distinguishability. We then exhibit a concrete, infinite class of graphons arising from stochastic block models that are well-separated in terms of cut distance and are indistinguishable by a GCN. These results theoretically match empirical observations of several prior works. To prove our results, we exploit a connection to random walks on graphs. Finally, we give empirical results on synthetic and real graph classification datasets, indicating that indistinguishable graph distributions arise in practice.
This paper introduces a generalization of Convolutional Neural Networks (CNNs) from low-dimensional grid data, such as images, to graph-structured data. We propose a novel spatial convolution utilizing a random walk to uncover the relations within the input, analogous to the way the standard convolution uses the spatial neighborhood of a pixel on the grid. The convolution has an intuitive interpretation, is efficient and scalable and can also be used on data with varying graph structure. Furthermore, this generalization can be applied to many standard regression or classification problems, by learning the the underlying graph. We empirically demonstrate the performance of the proposed CNN on MNIST, and challenge the state-of-the-art on Merck molecular activity data set.
Graph neural networks have become one of the most important techniques to solve machine learning problems on graph-structured data. Recent work on vertex classification proposed deep and distributed learning models to achieve high performance and scalability. However, we find that the feature vectors of benchmark datasets are already quite informative for the classification task, and the graph structure only provides a means to denoise the data. In this paper, we develop a theoretical framework based on graph signal processing for analyzing graph neural networks. Our results indicate that graph neural networks only perform low-pass filtering on feature vectors and do not have the non-linear manifold learning property. We further investigate their resilience to feature noise and propose some insights on GCN-based graph neural network design.
We introduce a family of multilayer graph kernels and establish new links between graph convolutional neural networks and kernel methods. Our approach generalizes convolutional kernel networks to graph-structured data, by representing graphs as a sequence of kernel feature maps, where each node carries information about local graph substructures. On the one hand, the kernel point of view offers an unsupervised, expressive, and easy-to-regularize data representation, which is useful when limited samples are available. On the other hand, our model can also be trained end-to-end on large-scale data, leading to new types of graph convolutional neural networks. We show that our method achieves competitive performance on several graph classification benchmarks, while offering simple model interpretation. Our code is freely available at https://github.com/claying/GCKN.
We present a method that processes 3D point clouds by performing graph convolution operations across shapes. In this manner, point descriptors are learned by allowing interaction and propagation of feature representations within a shape collection. To enable this form of non-local, cross-shape graph convolution, our method learns a pairwise point attention mechanism indicating the degree of interaction between points on different shapes. Our method also learns to create a graph over shapes of an input collection whose edges connect shapes deemed as useful for performing cross-shape convolution. The edges are also equipped with learned weights indicating the compatibility of each shape pair for cross-shape convolution. Our experiments demonstrate that this interaction and propagation of point representations across shapes make them more discriminative. In particular, our results show significantly improved performance for 3D point cloud semantic segmentation compared to conventional approaches, especially in cases with the limited number of training examples.
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