Kernels for structured data are commonly obtained by decomposing objects into their parts and adding up the similarities between all pairs of parts measured by a base kernel. Assignment kernels are based on an optimal bijection between the parts and have proven to be an effective alternative to the established convolution kernels. We explore how the base kernel can be learned as part of the classification problem. We build on the theory of valid assignment kernels derived from hierarchies defined on the parts. We show that the weights of this hierarchy can be optimized via multiple kernel learning. We apply this result to learn vertex similarities for the Weisfeiler-Lehman optimal assignment kernel for graph classification. We present first experimental results which demonstrate the feasibility and effectiveness of the approach.
This paper studies learning the representations of whole graphs in both unsupervised and semi-supervised scenarios. Graph-level representations are critical in a variety of real-world applications such as predicting the properties of molecules and community analysis in social networks. Traditional graph kernel based methods are simple, yet effective for obtaining fixed-length representations for graphs but they suffer from poor generalization due to hand-crafted designs. There are also some recent methods based on language models (e.g. graph2vec) but they tend to only consider certain substructures (e.g. subtrees) as graph representatives. Inspired by recent progress of unsupervised representation learning, in this paper we proposed a novel method called InfoGraph for learning graph-level representations. We maximize the mutual information between the graph-level representation and the representations of substructures of different scales (e.g., nodes, edges, triangles). By doing so, the graph-level representations encode aspects of the data that are shared across different scales of substructures. Furthermore, we further propose InfoGraph*, an extension of InfoGraph for semi-supervised scenarios. InfoGraph* maximizes the mutual information between unsupervised graph representations learned by InfoGraph and the representations learned by existing supervised methods. As a result, the supervised encoder learns from unlabeled data while preserving the latent semantic space favored by the current supervised task. Experimental results on the tasks of graph classification and molecular property prediction show that InfoGraph is superior to state-of-the-art baselines and InfoGraph* can achieve performance competitive with state-of-the-art semi-supervised models.
Graph neural networks denote a group of neural network models introduced for the representation learning tasks on graph data specifically. Graph neural networks have been demonstrated to be effective for capturing network structure information, and the learned representations can achieve the state-of-the-art performance on node and graph classification tasks. Besides the different application scenarios, the architectures of graph neural network models also depend on the studied graph types a lot. Graph data studied in research can be generally categorized into two main types, i.e., small graphs vs. giant networks, which differ from each other a lot in the size, instance number and label annotation. Several different types of graph neural network models have been introduced for learning the representations from such different types of graphs already. In this paper, for these two different types of graph data, we will introduce the graph neural networks introduced in recent years. To be more specific, the graph neural networks introduced in this paper include IsoNN, SDBN, LF&ER, GCN, GAT, DifNN, GNL, GraphSage and seGEN. Among these graph neural network models, IsoNN, SDBN and LF&ER are initially proposed for small graphs and the remaining ones are initially proposed for giant networks instead. The readers are also suggested to refer to these papers for detailed information when reading this tutorial paper.
Deep learning models have achieved huge success in numerous fields, such as computer vision and natural language processing. However, unlike such fields, it is hard to apply traditional deep learning models on the graph data due to the 'node-orderless' property. Normally, adjacency matrices will cast an artificial and random node-order on the graphs, which renders the performance of deep models on graph classification tasks extremely erratic, and the representations learned by such models lack clear interpretability. To eliminate the unnecessary node-order constraint, we propose a novel model named Isomorphic Neural Network (IsoNN), which learns the graph representation by extracting its isomorphic features via the graph matching between input graph and templates. IsoNN has two main components: graph isomorphic feature extraction component and classification component. The graph isomorphic feature extraction component utilizes a set of subgraph templates as the kernel variables to learn the possible subgraph patterns existing in the input graph and then computes the isomorphic features. A set of permutation matrices is used in the component to break the node-order brought by the matrix representation. Three fully-connected layers are used as the classification component in IsoNN. Extensive experiments are conducted on benchmark datasets, the experimental results can demonstrate the effectiveness of ISONN, especially compared with both classic and state-of-the-art graph classification methods.
We propose a new graph kernel for graph classification and comparison using Ollivier Ricci curvature. The Ricci curvature of an edge in a graph describes the connectivity in the local neighborhood. An edge in a densely connected neighborhood has positive curvature and an edge serving as a local bridge has negative curvature. We use the edge curvature distribution to form a graph kernel which is then used to compare and cluster graphs. The curvature kernel uses purely the graph topology and thereby works for settings when node attributes are not available.
Graph neural networks (GNNs) have emerged recently as a powerful architecture for learning node and graph representations. Standard GNNs have the same expressive power as the Weisfeiler-Leman test of graph isomorphism in terms of distinguishing non-isomorphic graphs. However, it was recently shown that this test cannot identify fundamental graph properties such as connectivity and triangle freeness. We show that GNNs also suffer from the same limitation. To address this limitation, we propose a more expressive architecture, k-hop GNNs, which updates a node's representation by aggregating information not only from its direct neighbors, but from its k-hop neighborhood. We show that the proposed architecture can identify fundamental graph properties. We evaluate the proposed architecture on standard node classification and graph classification datasets. Our experimental evaluation confirms our theoretical findings since the proposed model achieves performance better or comparable to standard GNNs and to state-of-the-art algorithms.
The graph convolution network (GCN) is a widely-used facility to realize graph-based semi-supervised learning, which usually integrates node features and graph topologic information to build learning models. However, as for multi-label learning tasks, the supervision part of GCN simply minimizes the cross-entropy loss between the last layer outputs and the ground-truth label distribution, which tends to lose some useful information such as label correlations, so that prevents from obtaining high performance. In this paper, we pro-pose a novel GCN-based semi-supervised learning approach for multi-label classification, namely ML-GCN. ML-GCN first uses a GCN to embed the node features and graph topologic information. Then, it randomly generates a label matrix, where each row (i.e., label vector) represents a kind of labels. The dimension of the label vector is the same as that of the node vector before the last convolution operation of GCN. That is, all labels and nodes are embedded in a uniform vector space. Finally, during the ML-GCN model training, label vectors and node vectors are concatenated to serve as the inputs of the relaxed skip-gram model to detect the node-label correlation as well as the label-label correlation. Experimental results on several graph classification datasets show that the proposed ML-GCN outperforms four state-of-the-art methods.
Attention operators have been widely applied in various fields, including computer vision, natural language processing, and network embedding learning. Attention operators on graph data enables learnable weights when aggregating information from neighboring nodes. However, graph attention operators (GAOs) consume excessive computational resources, preventing their applications on large graphs. In addition, GAOs belong to the family of soft attention, instead of hard attention, which has been shown to yield better performance. In this work, we propose novel hard graph attention operator (hGAO) and channel-wise graph attention operator (cGAO). hGAO uses the hard attention mechanism by attending to only important nodes. Compared to GAO, hGAO improves performance and saves computational cost by only attending to important nodes. To further reduce the requirements on computational resources, we propose the cGAO that performs attention operations along channels. cGAO avoids the dependency on the adjacency matrix, leading to dramatic reductions in computational resource requirements. Experimental results demonstrate that our proposed deep models with the new operators achieve consistently better performance. Comparison results also indicates that hGAO achieves significantly better performance than GAO on both node and graph embedding tasks. Efficiency comparison shows that our cGAO leads to dramatic savings in computational resources, making them applicable to large graphs.
Recent advances in Graph Convolutional Networks (GCNs) have led to state-of-the-art performance on various graph-related tasks. However, most existing GCN models do not explicitly identify whether all the aggregated neighbors are valuable to the learning tasks, which may harm the learning performance. In this paper, we consider the problem of node classification and propose the Label-Aware Graph Convolutional Network (LAGCN) framework which can directly identify valuable neighbors to enhance the performance of existing GCN models. Our contribution is three-fold. First, we propose a label-aware edge classifier that can filter distracting neighbors and add valuable neighbors for each node to refine the original graph into a label-aware~(LA) graph. Existing GCN models can directly learn from the LA graph to improve the performance without changing their model architectures. Second, we introduce the concept of positive ratio to evaluate the density of valuable neighbors in the LA graph. Theoretical analysis reveals that using the edge classifier to increase the positive ratio can improve the learning performance of existing GCN models. Third, we conduct extensive node classification experiments on benchmark datasets. The results verify that LAGCN can improve the performance of existing GCN models considerably, in terms of node classification.
Graph Neural Networks (GNNs) are powerful to learn the representation of graph-structured data. Most of the GNNs use the message-passing scheme, where the embedding of a node is iteratively updated by aggregating the information of its neighbors. To achieve a better expressive capability of node influences, attention mechanism has grown to be popular to assign trainable weights to the nodes in aggregation. Though the attention-based GNNs have achieved remarkable results in various tasks, a clear understanding of their discriminative capacities is missing. In this work, we present a theoretical analysis of the representational properties of the GNN that adopts the attention mechanism as an aggregator. Our analysis determines all cases when those attention-based GNNs can always fail to distinguish certain distinct structures. Those cases appear due to the ignorance of cardinality information in attention-based aggregation. To improve the performance of attention-based GNNs, we propose cardinality preserved attention (CPA) models that can be applied to any kind of attention mechanisms. Our experiments on node and graph classification confirm our theoretical analysis and show the competitive performance of our CPA models.
With the advent of data science, the analysis of network or graph data has become a very timely research problem. A variety of recent works have been proposed to generalize neural networks to graphs, either from a spectral graph theory or a spatial perspective. The majority of these works however focus on adapting the convolution operator to graph representation. At the same time, the pooling operator also plays an important role in distilling multiscale and hierarchical representations but it has been mostly overlooked so far. In this paper, we propose a parameter-free pooling operator, called iPool, that permits to retain the most informative features in arbitrary graphs. With the argument that informative nodes dominantly characterize graph signals, we propose a criterion to evaluate the amount of information of each node given its neighbors, and theoretically demonstrate its relationship to neighborhood conditional entropy. This new criterion determines how nodes are selected and coarsened graphs are constructed in the pooling layer. The resulting hierarchical structure yields an effective isomorphism-invariant representation of networked data in arbitrary topologies. The proposed strategy is evaluated in terms of graph classification on a collection of public graph datasets, including bioinformatics and social networks, and achieves state-of-the-art performance on most of the datasets.
Spectral clustering (SC) is a popular clustering technique to find strongly connected communities on a graph. SC can be used in Graph Neural Networks (GNNs) to implement pooling operations that aggregate nodes belonging to the same cluster. However, the eigendecomposition of the Laplacian is expensive and, since clustering results are graph-specific, pooling methods based on SC must perform a new optimization for each new sample. In this paper, we propose a graph clustering approach that addresses these limitations of SC. We formulate a continuous relaxation of the normalized minCUT problem and train a GNN to compute cluster assignments that minimize this objective. Our GNN-based implementation is differentiable, does not require to compute the spectral decomposition, and learns a clustering function that can be quickly evaluated on out-of-sample graphs. From the proposed clustering method, we design a graph pooling operator that overcomes some important limitations of state-of-the-art graph pooling techniques and achieves the best performance in several supervised and unsupervised tasks.
In this work, we present graph star net (GraphStar), a novel and unified graph neural net architecture which utilizes message-passing relay and attention mechanism for multiple prediction tasks - node classification, graph classification and link prediction. GraphStar addresses many earlier challenges facing graph neural nets and achieves non-local representation without increasing the model depth or bearing heavy computational costs. We also propose a new method to tackle topic-specific sentiment analysis based on node classification and text classification as graph classification. Our work shows that 'star nodes' can learn effective graph-data representation and improve on current methods for the three tasks. Specifically, for graph classification and link prediction, GraphStar outperforms the current state-of-the-art models by 2-5% on several key benchmarks.
Graph Neural Networks (GNNs) have boosted the performance of many graph related tasks such as node classification and graph classification. Recent researches show that graph neural networks are vulnerable to adversarial attacks, which deliberately add carefully created unnoticeable perturbation to the graph structure. The perturbation is usually created by adding/deleting a few edges, which might be noticeable even when the number of edges modified is small. In this paper, we propose a graph rewiring operation which affects the graph in a less noticeable way compared to adding/deleting edges. We then use reinforcement learning to learn the attack strategy based on the proposed rewiring operation. Experiments on real world graphs demonstrate the effectiveness of the proposed framework. To understand the proposed framework, we further analyze how its generated perturbation to the graph structure affects the output of the target model.
As a new approach to train generative models, \emph{generative adversarial networks} (GANs) have achieved considerable success in image generation. This framework has also recently been applied to data with graph structures. We propose labeled-graph generative adversarial networks (LGGAN) to train deep generative models for graph-structured data with node labels. We test the approach on various types of graph datasets, such as collections of citation networks and protein graphs. Experiment results show that our model can generate diverse labeled graphs that match the structural characteristics of the training data and outperforms all alternative approaches in quality and generality. To further evaluate the quality of the generated graphs, we use them on a downstream task of graph classification, and the results show that LGGAN can faithfully capture the important aspects of the graph structure.
Graph data widely exist in many high-impact applications. Inspired by the success of deep learning in grid-structured data, graph neural network models have been proposed to learn powerful node-level or graph-level representation. However, most of the existing graph neural networks suffer from the following limitations: (1) there is limited analysis regarding the graph convolution properties, such as seed-oriented, degree-aware and order-free; (2) the node's degree-specific graph structure is not explicitly expressed in graph convolution for distinguishing structure-aware node neighborhoods; (3) the theoretical explanation regarding the graph-level pooling schemes is unclear. To address these problems, we propose a generic degree-specific graph neural network named DEMO-Net motivated by Weisfeiler-Lehman graph isomorphism test that recursively identifies 1-hop neighborhood structures. In order to explicitly capture the graph topology integrated with node attributes, we argue that graph convolution should have three properties: seed-oriented, degree-aware, order-free. To this end, we propose multi-task graph convolution where each task represents node representation learning for nodes with a specific degree value, thus leading to preserving the degree-specific graph structure. In particular, we design two multi-task learning methods: degree-specific weight and hashing functions for graph convolution. In addition, we propose a novel graph-level pooling/readout scheme for learning graph representation provably lying in a degree-specific Hilbert kernel space. The experimental results on several node and graph classification benchmark data sets demonstrate the effectiveness and efficiency of our proposed DEMO-Net over state-of-the-art graph neural network models.
We present a novel framework based on optimal transport for the challenging problem of comparing graphs. Specifically, we exploit the probabilistic distribution of smooth graph signals defined with respect to the graph topology. This allows us to derive an explicit expression of the Wasserstein distance between graph signal distributions in terms of the graph Laplacian matrices. This leads to a structurally meaningful measure for comparing graphs, which is able to take into account the global structure of graphs, while most other measures merely observe local changes independently. Our measure is then used for formulating a new graph alignment problem, whose objective is to estimate the permutation that minimizes the distance between two graphs. We further propose an efficient stochastic algorithm based on Bayesian exploration to accommodate for the non-convexity of the graph alignment problem. We finally demonstrate the performance of our novel framework on different tasks like graph alignment, graph classification and graph signal prediction, and we show that our method leads to significant improvement with respect to the-state-of-art algorithms.
Most graph kernels are an instance of the class of $\mathcal{R}$-Convolution kernels, which measure the similarity of objects by comparing their substructures. Despite their empirical success, most graph kernels use a naive aggregation of the final set of substructures, usually a sum or average, thereby potentially discarding valuable information about the distribution of individual components. Furthermore, only a limited instance of these approaches can be extended to continuously attributed graphs. We propose a novel method that relies on the Wasserstein distance between the node feature vector distributions of two graphs, which allows to find subtler differences in data sets by considering graphs as high-dimensional objects, rather than simple means. We further propose a Weisfeiler-Lehman inspired embedding scheme for graphs with continuous node attributes and weighted edges, enhance it with the computed Wasserstein distance, and thus improve the state-of-the-art prediction performance on several graph classification tasks.
This paper focuses on the discrimination capacity of aggregation functions: these are the permutation invariant functions used by graph neural networks to combine the features of nodes. Realizing that the most powerful aggregation functions suffer from a dimensionality curse, we consider a restricted setting. In particular, we show that the standard sum and a novel histogram-based function have the capacity to discriminate between any fixed number of inputs chosen by an adversary. Based on our insights, we design a graph neural network aiming, not to maximize discrimination capacity, but to learn discriminative graph representations that generalize well. Our empirical evaluation provides evidence that our choices can yield benefits to the problem of structural graph classification.
While graph kernels (GKs) are easy to train and enjoy provable theoretical guarantees, their practical performances are limited by their expressive power, as the kernel function often depends on hand-crafted combinatorial features of graphs. Compared to graph kernels, graph neural networks (GNNs) usually achieve better practical performance, as GNNs use multi-layer architectures and non-linear activation functions to extract high-order information of graphs as features. However, due to the large number of hyper-parameters and the non-convex nature of the training procedure, GNNs are harder to train. Theoretical guarantees of GNNs are also not well-understood. Furthermore, the expressive power of GNNs scales with the number of parameters, and thus it is hard to exploit the full power of GNNs when computing resources are limited. The current paper presents a new class of graph kernels, Graph Neural Tangent Kernels (GNTKs), which correspond to infinitely wide multi-layer GNNs trained by gradient descent. GNTKs enjoy the full expressive power of GNNs and inherit advantages of GKs. Theoretically, we show GNTKs provably learn a class of smooth functions on graphs. Empirically, we test GNTKs on graph classification datasets and show they achieve strong performance.