Graph representation learning with the family of graph convolution networks (GCN) provides powerful tools for prediction on graphs. As graphs grow with more edges, the GCN family suffers from sub-optimal generalization performance due to task-irrelevant connections. Recent studies solve this problem by using graph sparsification in neural networks. However, graph sparsification cannot generate ultra-sparse graphs while simultaneously maintaining the performance of the GCN family. To address this problem, we propose Graph Ultra-sparsifier, a semi-supervised graph sparsifier with dynamically-updated regularization terms based on the graph convolution. The graph ultra-sparsifier can generate ultra-sparse graphs while maintaining the performance of the GCN family with the ultra-sparse graphs as inputs. In the experiments, when compared to the state-of-the-art graph sparsifiers, our graph ultra-sparsifier generates ultra-sparse graphs and these ultra-sparse graphs can be used as inputs to maintain the performance of GCN and its variants in node classification tasks.

Spatial-temporal graph have been widely observed in various domains such as neuroscience, climate research, and transportation engineering. The state-of-the-art models of spatialtemporal graphs rely on Graph Neural Networks (GNNs) to obtain explicit representations for such networks and to discover hidden spatial dependencies in them. These models have demonstrated superior performance in various tasks. In this paper, we propose a sparse adversarial attack framework AdverSparse to illustrate that when only a few key connections are removed in such graphs, hidden spatial dependencies learned by such spatial-temporal models are significantly impacted, leading to various issues such as increasing prediction errors. We formulate the adversarial attack as an optimization problem and solve it by the Alternating Direction Method of Multipliers (ADMM). Experiments show that AdverSparse can find and remove key connections in these graphs, leading to malfunctioning models, even in models capable of learning hidden spatial dependencies.

Network representation learning has played a critical role in studying networks. One way to study a graph is to focus on its spectrum, i.e., the eigenvalue distribution of its associated matrices. Recent advancements in spectral graph theory show that spectral moments of a network can be used to capture the network structure and various graph properties. However, sometimes networks with different structures or sizes can have the same or similar spectral moments, not to mention the existence of the cospectral graphs. To address such problems, we propose a 3D network representation that relies on the spectral information of subgraphs: the Spectral Path, a path connecting the spectral moments of the network and those of its subgraphs of different sizes. We show that the spectral path is interpretable and can capture relationship between a network and its subgraphs, for which we present a theoretical foundation. We demonstrate the effectiveness of the spectral path in applications such as network visualization and network identification.