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Given the prevalence of large-scale graphs in real-world applications, the storage and time for training neural models have raised increasing concerns. To alleviate the concerns, we propose and study the problem of graph condensation for graph neural networks (GNNs). Specifically, we aim to condense the large, original graph into a small, synthetic and highly-informative graph, such that GNNs trained on the small graph and large graph have comparable performance. We approach the condensation problem by imitating the GNN training trajectory on the original graph through the optimization of a gradient matching loss and design a strategy to condense node futures and structural information simultaneously. Extensive experiments have demonstrated the effectiveness of the proposed framework in condensing different graph datasets into informative smaller graphs. In particular, we are able to approximate the original test accuracy by 95.3% on Reddit, 99.8% on Flickr and 99.0% on Citeseer, while reducing their graph size by more than 99.9%, and the condensed graphs can be used to train various GNN architectures.Code is released at //github.com/ChandlerBang/GCond.

The presence of noise is an intrinsic problem in acquisition processes for digital images. One way to enhance images is to combine the forward and backward diffusion equations. However, the latter problem is well known to be exponentially unstable with respect to any small perturbations on the final data. In this scenario, the final data can be regarded as a blurred image obtained from the forward process, and that image can be pixelated as a network. Therefore, we study in this work a regularization framework for the backward diffusion equation on graphs. Our aim is to construct a spectral graph-based solution based upon a cut-off projection. Stability and convergence results are provided together with some numerical experiments.

Over the years, many graph problems specifically those in NP-complete are studied by a wide range of researchers. Some famous examples include graph colouring, travelling salesman problem and subgraph isomorphism. Most of these problems are typically addressed by exact algorithms, approximate algorithms and heuristics. There are however some drawback for each of these methods. Recent studies have employed learning-based frameworks such as machine learning techniques in solving these problems, given that they are useful in discovering new patterns in structured data that can be represented using graphs. This research direction has successfully attracted a considerable amount of attention. In this survey, we provide a systematic review mainly on classic graph problems in which learning-based approaches have been proposed in addressing the problems. We discuss the overview of each framework, and provide analyses based on the design and performance of the framework. Some potential research questions are also suggested. Ultimately, this survey gives a clearer insight and can be used as a stepping stone to the research community in studying problems in this field.

This paper studies node classification in the inductive setting, i.e., aiming to learn a model on labeled training graphs and generalize it to infer node labels on unlabeled test graphs. This problem has been extensively studied with graph neural networks (GNNs) by learning effective node representations, as well as traditional structured prediction methods for modeling the structured output of node labels, e.g., conditional random fields (CRFs). In this paper, we present a new approach called the Structured Proxy Network (SPN), which combines the advantages of both worlds. SPN defines flexible potential functions of CRFs with GNNs. However, learning such a model is nontrivial as it involves optimizing a maximin game with high-cost inference. Inspired by the underlying connection between joint and marginal distributions defined by Markov networks, we propose to solve an approximate version of the optimization problem as a proxy, which yields a near-optimal solution, making learning more efficient. Extensive experiments on two settings show that our approach outperforms many competitive baselines.

The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.

Graph Neural Networks (GNN) is an emerging field for learning on non-Euclidean data. Recently, there has been increased interest in designing GNN that scales to large graphs. Most existing methods use "graph sampling" or "layer-wise sampling" techniques to reduce training time. However, these methods still suffer from degrading performance and scalability problems when applying to graphs with billions of edges. This paper presents GBP, a scalable GNN that utilizes a localized bidirectional propagation process from both the feature vectors and the training/testing nodes. Theoretical analysis shows that GBP is the first method that achieves sub-linear time complexity for both the precomputation and the training phases. An extensive empirical study demonstrates that GBP achieves state-of-the-art performance with significantly less training/testing time. Most notably, GBP can deliver superior performance on a graph with over 60 million nodes and 1.8 billion edges in less than half an hour on a single machine.

Meta-learning extracts the common knowledge acquired from learning different tasks and uses it for unseen tasks. It demonstrates a clear advantage on tasks that have insufficient training data, e.g., few-shot learning. In most meta-learning methods, tasks are implicitly related via the shared model or optimizer. In this paper, we show that a meta-learner that explicitly relates tasks on a graph describing the relations of their output dimensions (e.g., classes) can significantly improve the performance of few-shot learning. This type of graph is usually free or cheap to obtain but has rarely been explored in previous works. We study the prototype based few-shot classification, in which a prototype is generated for each class, such that the nearest neighbor search between the prototypes produces an accurate classification. We introduce "Gated Propagation Network (GPN)", which learns to propagate messages between prototypes of different classes on the graph, so that learning the prototype of each class benefits from the data of other related classes. In GPN, an attention mechanism is used for the aggregation of messages from neighboring classes, and a gate is deployed to choose between the aggregated messages and the message from the class itself. GPN is trained on a sequence of tasks from many-shot to few-shot generated by subgraph sampling. During training, it is able to reuse and update previously achieved prototypes from the memory in a life-long learning cycle. In experiments, we change the training-test discrepancy and test task generation settings for thorough evaluations. GPN outperforms recent meta-learning methods on two benchmark datasets in all studied cases.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.

Traditional methods for link prediction can be categorized into three main types: graph structure feature-based, latent feature-based, and explicit feature-based. Graph structure feature methods leverage some handcrafted node proximity scores, e.g., common neighbors, to estimate the likelihood of links. Latent feature methods rely on factorizing networks' matrix representations to learn an embedding for each node. Explicit feature methods train a machine learning model on two nodes' explicit attributes. Each of the three types of methods has its unique merits. In this paper, we propose SEAL (learning from Subgraphs, Embeddings, and Attributes for Link prediction), a new framework for link prediction which combines the power of all the three types into a single graph neural network (GNN). GNN is a new type of neural network which directly accepts graphs as input and outputs their labels. In SEAL, the input to the GNN is a local subgraph around each target link. We prove theoretically that our local subgraphs also reserve a great deal of high-order graph structure features related to link existence. Another key feature is that our GNN can naturally incorporate latent features and explicit features. It is achieved by concatenating node embeddings (latent features) and node attributes (explicit features) in the node information matrix for each subgraph, thus combining the three types of features to enhance GNN learning. Through extensive experiments, SEAL shows unprecedentedly strong performance against a wide range of baseline methods, including various link prediction heuristics and network embedding methods.