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.

A large body of research has focused on analyzing large networks and graphs. However, network and graph data is often anonymized for reasons such as protecting data privacy. Under such circumstances, it is difficult to verify the source of network data, which leads to questions such as: Given an anonymized graph, can we identify the network from which it is collected? Or, if one claims the graph is sampled from a certain network, can we verify this claim? The intuitive approach is to check for subgraph isomophism. However, subgraph isomophism is NP-complete; hence, infeasible for most large networks. Inspired by biometrics studies, we address these challenges by formulating two new problems:network identificationandnetwork authentication. To tackle these problems, similar to research on human fingerprints, we introduce two versions of anetwork identity: (1) embedding-based identity and (2) distribution-based identity. We demonstrate the effectiveness of these network identities using extensive experiments on real-world networks. Using these identities, we propose two approaches for network identification. One method uses supervised learning and can achieve an identification accuracy of 84.4 percent, and the other, which is easier to implement, relies on distances between identities and achieves an accuracy rate of 70.8 percent. For network authentication, we propose two methods to build a network authentication system. The first is a supervised learner and yields a low false accept rate and the other method, allows one to control the false reject rate with a reasonable false accept rate across networks. We demonstrate that network authentication can also be used for biometrics, authenticating users based on their touch data on phones and tablets. Our study can help identify or verify the source of network data, validate network-based research, and be used for network-based biometrics.

Network embedding methods have been widely and successfully used in network-based applications such as node classification and link prediction. However, an ideal network embedding should not only be useful for machine learning, but interpretable. We introduce a spectral embedding method for a network, its Spectral Point, which is basically the first few spectral moments of a network. Spectral moments are interpretable, where we prove their close relationships to network structure (e.g. number of triangles and squares) and various network properties (e.g. degree distribution, clustering coefficient, and network connectivity). Using spectral points, we introduce a visualizable and bounded 3D embedding space for all possible graphs, in which one can characterize various types of graphs (e.g., cycles), or real-world networks from different categories (e.g., social or biological networks). We demonstrate that spectral points can be used for network identification (i.e., what network is this subgraph sampled from?) and that by using just the first few moments one does not lose much predictive power.

Network visualization has played a critical role in graph analysis, as it not only presents a big picture of a network but also helps reveal the structural information of a network. The most popular visual representation of networks is the node-link diagram. However, visualizing a large network with the node-link diagram can be challenging due to the difficulty in obtaining an optimal graph layout. To address this challenge, a recent advancement in network representation: network shape, allows one to compactly represent a network and its subgraphs with the distribution of their embeddings. Inspired by this research, we have designed a web platform WebShapes that enables researchers and practitioners to visualize their network data as customized 3D shapes (http://b.link/webshapes). Furthermore, we provide a case study on real-world networks to explore the sensitivity of network shapes to different graph sampling, embedding, and fitting methods, and we show examples of understanding networks through their network shapes.

Research on networks is commonly performed using anonymized network data for various reasons such as protecting data privacy. Under such circumstances, it is difficult to verify the source of network data, which leads to questions such as: Given an anonymized graph, can we identify the network from which it is collected? Or if one claims the graph is sampled from a certain network, can we verify it? The intuitive approach is to check for subgraph isomorphism. However, subgraph isomorphism is NP-complete; hence, infeasible for most large networks. Inspired by biometrics studies, we address these challenges by formulating two new problems: network identification and network authentication. To tackle these problems, similar to research on human fingerprints, we introduce two versions of a network identity: (1) embedding-based identity and (2) distribution-based identity. We demonstrate the effectiveness of these network identities on various real-world networks. Using these identities, we propose two approaches for network identification. One method uses supervised learning and can achieve an identification accuracy rate of 94.7%, and the other, which is easier to implement, relies on distances between identities and achieves an accuracy rate of 85.5%. For network authentication, we propose two methods to build a network authentication system. The first is a supervised learner and provides a low false accept rate and the other method allows one to control the false reject rate with a reasonable false accept rate across networks. Our study can help identify or verify the source of network data, validate network-based research, and be used for network-based biometrics.

There has been a surge of interest in machine learning in graphs, as graphs and networks are ubiquitous across the globe and within science and engineering: road networks, power grids, protein-protein interaction networks, scientific collaboration networks, social networks, to name a few. Recent machine learning research has focused on efficient and effective ways to represent graph structure. Existing graph representation methods such as network embedding techniques learn to map a node (or a graph) to a vector in a low-dimensional vector space. However, the mapped values are often difficult to interpret, lacking information on the structure of the network or its subgraphs. Instead of using a low-dimensional vector to represent a graph, we propose to represent a network with a 3-dimensional shape: the network shape. We introduce the first network shape, a Kronecker hull, which represents a network as a 3D convex polyhedron using stochastic Kronecker graphs. We present a linear time algorithm to build Kronecker hulls. Network shapes provide a compact representation of networks that is easy to visualize and interpret. They captures various properties of not only the network, but also its subgraphs. For instance, they can provide the distribution of subgraphs within a network, e.g., what proportion of subgraphs are structurally similar to the whole network? Using experiments on real-world networks, we show how network shapes can be used in various applications, from computing similarity between two graphs (using the overlap between network shapes of two networks) to graph compression, where a graph with millions of nodes can be represented with a convex hull with less than 40 boundary points.

Understanding the role emotions play in social interactions has been a central research question in the social sciences. However, the challenge of obtaining large-scale data on human emotions has left the most fundamental questions on emotions less explored: How do emotions vary across individuals, evolve over time, and are connected to social ties?