Graph Convolutional Networks (GCNs) have become a crucial tool on learning representations of graph vertices. The main challenge of adapting GCNs on large-scale graphs is the scalability issue that it incurs heavy cost both in computation and memory due to the uncontrollable neighborhood expansion across layers. In this paper, we accelerate the training of GCNs through developing an adaptive layer-wise sampling method. By constructing the network layer by layer in a top-down passway, we sample the lower layer conditioned on the top one, where the sampled neighborhoods are shared by different parent nodes and the over expansion is avoided owing to the fixed-size sampling. More importantly, the proposed sampler is adaptive and applicable for explicit variance reduction, which in turn enhances the training of our method. Furthermore, we propose a novel and economical approach to promote the message passing over distant nodes by applying skip connections. Intensive experiments on several benchmarks verify the effectiveness of our method regarding the classification accuracy while enjoying faster convergence speed.
More recently, two kinds of sampling-based methods including GraphSAGE @cite_8 and FastGCN @cite_10 were developed for fast representation learning on graphs. To be specific, GraphSAGE computes node representations by sampling neighborhoods of each node and then performing a specific aggregator for information fusion. The FastGCN model interprets graph convolutions as integral transforms of embedding functions and samples the nodes in each layer independently. While our method is closely related to these methods, we develop a different sampling strategy in this paper. Compared to GraphSAGE that is node-wise, our method is based on layer-wise sampling as all neighborhoods are sampled as altogether, and thus can allow neighborhood sharing as illustrated in Figure . In contrast to FastGCN that constructs each layer independently, our model is capable of capturing the between-layer connections as the lower layer is sampled conditionally on the top one. We detail the comparisons in . Another related work is the control-variate-based method by @cite_5 . However, the sampling process of this method is node-wise, and the historical activations of nodes are required.
@cite_5 @cite_10 @cite_8
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Graph Convolutional Networks (GCNs) have become a crucial tool on learning representations of graph vertices. The main challenge of adapting GCNs on large-scale graphs is the scalability issue that it incurs heavy cost both in computation and memory due to the uncontrollable neighborhood expansion across layers. In this paper, we accelerate the training of GCNs through developing an adaptive layer-wise sampling method. By constructing the network layer by layer in a top-down passway, we sample the lower layer conditioned on the top one, where the sampled neighborhoods are shared by different parent nodes and the over expansion is avoided owing to the fixed-size sampling. More importantly, the proposed sampler is adaptive and applicable for explicit variance reduction, which in turn enhances the training of our method. Furthermore, we propose a novel and economical approach to promote the message passing over distant nodes by applying skip connections. Intensive experiments on several benchmarks verify the effectiveness of our method regarding the classification accuracy while enjoying faster convergence speed. The graph convolutional networks (GCN) recently proposed by Kipf and Welling are an effective graph model for semi-supervised learning. Such a model, however, is transductive in nature because parameters are learned through convolutions with both training and test data. Moreover, the recursive neighborhood expansion across layers poses time and memory challenges for training with large, dense graphs. To relax the requirement of simultaneous availability of test data, we interpret graph convolutions as integral transforms of embedding functions under probability measures. Such an interpretation allows for the use of Monte Carlo approaches to consistently estimate the integrals, which in turn leads to a batched training scheme as we propose in this work---FastGCN. Enhanced with importance sampling, FastGCN not only is efficient for training but also generalizes well for inference. We show a comprehensive set of experiments to demonstrate its effectiveness compared with GCN and related models. In particular, training is orders of magnitude more efficient while predictions remain comparably accurate. This paper deals with active sampling of graph nodes representing training data for binary classification. The graph may be given or constructed using similarity measures among nodal features. Leveraging the graph for classification builds on the premise that labels across neighboring nodes are correlated according to a categorical Markov random field (MRF). This model is further relaxed to a Gaussian (G)MRF with labels taking continuous values—an approximation that not only mitigates the combinatorial complexity of the categorical model, but also offers optimal unbiased soft predictors of the unlabeled nodes. The proposed sampling strategy is based on querying the node whose label disclosure is expected to inflict the largest change on the GMRF, and in this sense it is the most informative on average. Connections are established to other sampling methods including uncertainty sampling, variance minimization, and sampling based on the @math optimality criterion. A simple yet effective heuristic is also introduced for increasing the exploration capabilities of the sampler, and reducing bias of the resultant classifier, by adjusting the confidence on the model label predictions. The novel sampling strategies are based on quantities that are readily available without the need for model retraining, rendering them computationally efficient and scalable to large graphs. Numerical tests using synthetic and real data demonstrate that the proposed methods achieve accuracy that is comparable or superior to the state of the art even at reduced runtime. Mapping the Internet generally consists in sampling the network from a limited set of sources by using "traceroute"-like probes. This methodology, akin to the merging of different spanning trees to a set of destinations, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. Here we explore these biases and provide a statistical analysis of their origin. We derive a mean-field analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular we find that the edge and vertex detection probability is depending on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with scale-free topology. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in different network models. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. The numerical study also allows the identification of intervals of the exploration parameters that optimize the fraction of nodes and edges discovered in the sampled graph. This finding might hint the steps toward more efficient mapping strategies.