Graph convolutional networks (GCNs) have recently received wide attentions, due to their successful applications in different graph tasks and different domains. Training GCNs for a large graph, however, is still a challenge. Original full-batch GCN training requires calculating the representation of all the nodes in the graph per GCN layer, which brings in high computation and memory costs. To alleviate this issue, several sampling-based methods have been proposed to train GCNs on a subset of nodes. Among them, the node-wise neighbor-sampling method recursively samples a fixed number of neighbor nodes, and thus its computation cost suffers from exponential growing neighbor size; while the layer-wise importance-sampling method discards the neighbor-dependent constraints, and thus the nodes sampled across layer suffer from sparse connection problem. To deal with the above two problems, we propose a new effective sampling algorithm called LAyer-Dependent ImportancE Sampling (LADIES). Based on the sampled nodes in the upper layer, LADIES selects their neighborhood nodes, constructs a bipartite subgraph and computes the importance probability accordingly. Then, it samples a fixed number of nodes by the calculated probability, and recursively conducts such procedure per layer to construct the whole computation graph. We prove theoretically and experimentally, that our proposed sampling algorithm outperforms the previous sampling methods in terms of both time and memory costs. Furthermore, LADIES is shown to have better generalization accuracy than original full-batch GCN, due to its stochastic nature.
Graph neural networks (GNNs) have been widely used in representation learning on graphs and achieved state-of-the-art performance in tasks such as node classification and link prediction. However, most existing GNNs are designed to learn node representations on the fixed and homogeneous graphs. The limitations especially become problematic when learning representations on a misspecified graph or a heterogeneous graph that consists of various types of nodes and edges. In this paper, we propose Graph Transformer Networks (GTNs) that are capable of generating new graph structures, which involve identifying useful connections between unconnected nodes on the original graph, while learning effective node representation on the new graphs in an end-to-end fashion. Graph Transformer layer, a core layer of GTNs, learns a soft selection of edge types and composite relations for generating useful multi-hop connections so-called meta-paths. Our experiments show that GTNs learn new graph structures, based on data and tasks without domain knowledge, and yield powerful node representation via convolution on the new graphs. Without domain-specific graph preprocessing, GTNs achieved the best performance in all three benchmark node classification tasks against the state-of-the-art methods that require pre-defined meta-paths from domain knowledge.
Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g. spectral clustering) without requiring any changes to the latter or affecting its computational complexity. Our implementation is available online.
Interest has been rising lately towards methods representing data in non-Euclidean spaces, e.g. hyperbolic or spherical, that provide specific inductive biases useful for certain real-world data properties, e.g. scale-free, hierarchical or cyclical. However, the popular graph neural networks are currently limited in modeling data only via Euclidean geometry and associated vector space operations. Here, we bridge this gap by proposing mathematically grounded generalizations of graph convolutional networks (GCN) to (products of) constant curvature spaces. We do this by i) introducing a unified formalism that can interpolate smoothly between all geometries of constant curvature, ii) leveraging gyro-barycentric coordinates that generalize the classic Euclidean concept of the center of mass. Our class of models smoothly recover their Euclidean counterparts when the curvature goes to zero from either side. Empirically, we outperform Euclidean GCNs in the tasks of node classification and distortion minimization for symbolic data exhibiting non-Euclidean behavior, according to their discrete curvature.
Seminal works on graph neural networks have primarily targeted semi-supervised node classification problems with few observed labels and high-dimensional signals. With the development of graph networks, this setup has become a de facto benchmark for a significant body of research. Interestingly, several works have recently shown that in this particular setting, graph neural networks do not perform much better than predefined low-pass filters followed by a linear classifier. However, when learning from little data in a high-dimensional space, it is not surprising that simple and heavily regularized methods are near-optimal. In this paper, we show empirically that in settings with fewer features and more training data, more complex graph networks significantly outperform simple models, and propose a few insights towards the proper choice of graph network architectures. We finally outline the importance of using sufficiently diverse benchmarks (including lower dimensional signals as well) when designing and studying new types of graph neural networks.
Graph neural networks (GNNs) are a powerful tool to learn representations on graphs by iteratively aggregating features from node neighbourhoods. Many variant models have been proposed, but there is limited understanding on both how to compare different architectures and how to construct GNNs systematically. Here, we propose a hierarchy of GNNs based on their aggregation regions. We derive theoretical results about the discriminative power and feature representation capabilities of each class. Then, we show how this framework can be utilized to systematically construct arbitrarily powerful GNNs. As an example, we construct a simple architecture that exceeds the expressiveness of the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theory on both synthetic and real-world benchmarks, and demonstrate our example's theoretical power translates to strong results on node classification, graph classification, and graph regression tasks.
Graph convolutional neural networks (GCNN) have numerous applications in different graph based learning tasks. Although the techniques obtain impressive results, they often fall short in accounting for the uncertainty associated with the underlying graph structure. In the recently proposed Bayesian GCNN (BGCN) framework, this issue is tackled by viewing the observed graph as a sample from a parametric random graph model and targeting joint inference of the graph and the GCNN weights. In this paper, we introduce an alternative generative model for graphs based on copying nodes and incorporate it within the BGCN framework. Our approach has the benefit that it uses information provided by the node features and training labels in the graph topology inference. Experiments show that the proposed algorithm compares favorably to the state-of-the-art in benchmark node classification tasks.
In this paper, we focus on learning low-dimensional embeddings for nodes in graph-structured data. To achieve this, we propose Caps2NE -- a new unsupervised embedding model leveraging a network of two capsule layers. Caps2NE induces a routing process to aggregate feature vectors of context neighbors of a given target node at the first capsule layer, then feed these features into the second capsule layer to infer a plausible embedding for the target node. Experimental results show that our proposed Caps2NE obtains state-of-the-art performances on benchmark datasets for the node classification task. Our code is available at: \url{https://github.com/daiquocnguyen/Caps2NE}.
Graph Neural Networks (GNNs) are efficient approaches to process graph-structured data. Modelling long-distance node relations is essential for GNN training and applications. However, conventional GNNs suffer from bad performance in modelling long-distance node relations due to limited-layer information propagation. Existing studies focus on building deep GNN architectures, which face the over-smoothing issue and cannot model node relations in particularly long distance. To address this issue, we propose to model long-distance node relations by simply relying on shallow GNN architectures with two solutions: (1) Implicitly modelling by learning to predict node pair relations (2) Explicitly modelling by adding edges between nodes that potentially have the same label. To combine our two solutions, we propose a model-agnostic training framework named HighwayGraph, which overcomes the challenge of insufficient labeled nodes by sampling node pairs from the training set and adopting the self-training method. Extensive experimental results show that our HighwayGraph achieves consistent and significant improvements over four representative GNNs on three benchmark datasets.
Graph Convolution Network (GCN) has been recognized as one of the most effective graph models for semi-supervised learning, but it extracts merely the first-order or few-order neighborhood information through information propagation, which suffers performance drop-off for deeper structure. Existing approaches that deal with the higher-order neighbors tend to take advantage of adjacency matrix power. In this paper, we assume a seemly trivial condition that the higher-order neighborhood information may be similar to that of the first-order neighbors. Accordingly, we present an unsupervised approach to describe such similarities and learn the weight matrices of higher-order neighbors automatically through Lasso that minimizes the feature loss between the first-order and higher-order neighbors, based on which we formulate the new convolutional filter for GCN to learn the better node representations. Our model, called higher-order weighted GCN(HWGCN), has achieved the state-of-the-art results on a number of node classification tasks over Cora, Citeseer and Pubmed datasets.
Graph Convolutional Networks (GCNs) have recently been shown to be quite successful in modeling graph-structured data. However, the primary focus has been on handling simple undirected graphs. Multi-relational graphs are a more general and prevalent form of graphs where each edge has a label and direction associated with it. Most of the existing approaches to handle such graphs suffer from over-parameterization and are restricted to learning representations of nodes only. In this paper, we propose CompGCN, a novel Graph Convolutional framework which jointly embeds both nodes and relations in a relational graph. CompGCN leverages a variety of entity-relation composition operations from Knowledge Graph Embedding techniques and scales with the number of relations. It also generalizes several of the existing multi-relational GCN methods. We evaluate our proposed method on multiple tasks such as node classification, link prediction, and graph classification, and achieve demonstrably superior results. We make the source code of CompGCN available to foster reproducible research.
Recently, Graph Convolutional Networks (GCNs) and their variants have been receiving many research interests for learning graph-related tasks. While the GCNs have been successfully applied to this problem, some caveats inherited from classical deep learning still remain as open research topics in the context of the node classification problem. One such inherited caveat is that GCNs only consider the nodes that are a few propagations away from the labeled nodes to classify them. However, taking only a few propagation steps away nodes into account defeats the purpose of using the graph topological information in the GCNs. To remedy this problem, the-state-of-the-art methods leverage the network diffusion approaches, namely personalized page rank and its variants, to fully account for the graph topology, {\em after} they use the Neural Networks in the GCNs. However, these approaches overlook the fact that the network diffusion methods favour high degree nodes in the graph, resulting in the propagation of labels to unlabeled centralized, hub, nodes. To address this biasing hub nodes problem, in this paper, we propose to utilize a dimensionality reduction technique conjugate with personalized page rank so that we can both take advantage from graph topology and resolve the hub node favouring problem for GCNs. Here, our approach opens a new holistic road for message passing phase of GCNs by suggesting the usage of other proximity matrices instead of well-known Laplacian. Testing on two real-world networks that are commonly used in benchmarking GCNs' performance for the node classification context, we systematically evaluate the performance of the proposed methodology and show that our approach outperforms existing methods for wide ranges of parameter values with very limited deep learning training {\em epochs}.
Many real world networks are very large and constantly change over time. These dynamic networks exist in various domains such as social networks, traffic networks and biological interactions. To handle large dynamic networks in downstream applications such as link prediction and anomaly detection, it is essential for such networks to be transferred into a low dimensional space. Recently, network embedding, a technique that converts a large graph into a low-dimensional representation, has become increasingly popular due to its strength in preserving the structure of a network. Efficient dynamic network embedding, however, has not yet been fully explored. In this paper, we present a dynamic network embedding method that integrates the history of nodes over time into the current state of nodes. The key contribution of our work is 1) generating dynamic network embedding by combining both dynamic and static node information 2) tracking history of neighbors of nodes using LSTM 3) significantly decreasing the time and memory by training an autoencoder LSTM model using temporal walks rather than adjacency matrices of graphs which are the common practice. We evaluate our method in multiple applications such as anomaly detection, link prediction and node classification in datasets from various domains.
Graph representation learning is of paramount importance for a variety of graph analytical tasks, ranging from node classification to community detection. Recently, graph convolutional networks (GCNs) have been successfully applied for graph representation learning. These GCNs generate node representation by aggregating features from the neighborhoods, which follows the "neighborhood aggregation" scheme. In spite of having achieved promising performance on various tasks, existing GCN-based models have difficulty in well capturing complicated non-linearity of graph data. In this paper, we first theoretically prove that coefficients of the neighborhood interacting terms are relatively small in current models, which explains why GCNs barely outperforms linear models. Then, in order to better capture the complicated non-linearity of graph data, we present a novel GraphAIR framework which models the neighborhood interaction in addition to neighborhood aggregation. Comprehensive experiments conducted on benchmark tasks including node classification and link prediction using public datasets demonstrate the effectiveness of the proposed method.
Deep generative models (DGMs) have achieved remarkable advances. Semi-supervised variational auto-encoders (SVAE) as a classical DGM offer a principled framework to effectively generalize from small labelled data to large unlabelled ones, but it is difficult to incorporate rich unstructured relationships within the multiple heterogeneous entities. In this paper, to deal with the problem, we present a semi-supervised co-embedding model for attributed networks (SCAN) based on the generalized SVAE for heterogeneous data, which collaboratively learns low-dimensional vector representations of both nodes and attributes for partially labelled attributed networks semi-supervisedly. The node and attribute embeddings obtained in a unified manner by our SCAN can benefit for capturing not only the proximities between nodes but also the affinities between nodes and attributes. Moreover, our model also trains a discriminative network to learn the label predictive distribution of nodes. Experimental results on real-world networks demonstrate that our model yields excellent performance in a number of applications such as attribute inference, user profiling and node classification compared to the state-of-the-art baselines.
Graph-based semi-supervised learning has been shown to be one of the most effective approaches for classification tasks from a wide range of domains, such as image classification and text classification, as they can exploit the connectivity patterns between labeled and unlabeled samples to improve learning performance. In this work, we advance this effective learning paradigm towards a scenario where labeled data are severely limited. More specifically, we address the problem of graph-based semi-supervised learning in the presence of severely limited labeled samples, and propose a new framework, called {\em Shoestring}, that improves the learning performance through semantic transfer from these very few labeled samples to large numbers of unlabeled samples. In particular, our framework learns a metric space in which classification can be performed by computing the similarity to centroid embedding of each class. {\em Shoestring} is trained in an end-to-end fashion to learn to leverage the semantic knowledge of limited labeled samples as well as their connectivity patterns with large numbers of unlabeled samples simultaneously. By combining {\em Shoestring} with graph convolutional networks, label propagation and their recent label-efficient variations (IGCN and GLP), we are able to achieve state-of-the-art node classification performance in the presence of very few labeled samples. In addition, we demonstrate the effectiveness of our framework on image classification tasks in the few-shot learning regime, with significant gains on miniImageNet ($2.57\%\sim3.59\%$) and tieredImageNet ($1.05\%\sim2.70\%$).
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs. We derive GCN operations in the hyperboloid model of hyperbolic space and map Euclidean input features to embeddings in hyperbolic spaces with different trainable curvature at each layer. Experiments demonstrate that HGCN learns embeddings that preserve hierarchical structure, and leads to improved performance when compared to Euclidean analogs, even with very low dimensional embeddings: compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to 63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node classification, also improving state-of-the art on the Pubmed dataset.
Graph convolutional neural networks (GCNN) have been successfully applied to many different graph based learning tasks including node and graph classification, matrix completion, and learning of node embeddings. Despite their impressive performance, the techniques have a limited capability to incorporate the uncertainty in the underlined graph structure. In order to address this issue, a Bayesian GCNN (BGCN) framework was recently proposed. In this framework, the observed graph is considered to be a random realization from a parametric random graph model and the joint Bayesian inference of the graph and GCNN weights is performed. In this paper, we propose a non-parametric generative model for graphs and incorporate it within the BGCN framework. In addition to the observed graph, our approach effectively uses the node features and training labels in the posterior inference of graphs and attains superior or comparable performance in benchmark node classification tasks.
Graph neural networks (GNNs) have shown great power in learning on attributed graphs. However, it is still a challenge for GNNs to utilize information faraway from the source node. Moreover, general GNNs require graph attributes as input, so they cannot be appled to plain graphs. In the paper, we propose new models named G-GNNs (Global information for GNNs) to address the above limitations. First, the global structure and attribute features for each node are obtained via unsupervised pre-training, which preserve the global information associated to the node. Then, using the global features and the raw network attributes, we propose a parallel framework of GNNs to learn different aspects from these features. The proposed learning methods can be applied to both plain graphs and attributed graphs. Extensive experiments have shown that G-GNNs can outperform other state-of-the-art models on three standard evaluation graphs. Specially, our methods establish new benchmark records on Cora (84.31\%) and Pubmed (80.95\%) when learning on attributed graphs.
We propose a novel spectral convolutional neural network (CNN) model on graph structured data, namely Distributed Feedback-Looped Networks (DFNets). This model is incorporated with a robust class of spectral graph filters, called feedback-looped filters, to provide better localization on vertices, while still attaining fast convergence and linear memory requirements. Theoretically, feedback-looped filters can guarantee convergence w.r.t. a specified error bound, and be applied universally to any graph without knowing its structure. Furthermore, the propagation rule of this model can diversify features from the preceding layers to produce strong gradient flows. We have evaluated our model using two benchmark tasks: semi-supervised document classification on citation networks and semi-supervised entity classification on a knowledge graph. The experimental results show that our model considerably outperforms the state-of-the-art methods in both benchmark tasks over all datasets.