Effective information analysis generally boils down to the geometry of the data represented by a graph. Typical applications include social networks, transportation networks, the spread of epidemic disease, brain's neuronal networks, gene data on biological regulatory networks, telecommunication networks, knowledge graph, which are lying on the non-Euclidean graph domain. To describe the geometric structures, graph matrices such as adjacency matrix or graph Laplacian can be employed to reveal latent patterns. This thesis focuses on the theoretical analysis of graph neural networks and the development of methods for specific applications using graph representation. Four methods are proposed, including rational neural networks for jump graph signal estimation, RemezNet for robust attribute prediction in the graph, ICNet for integrated circuit security, and CNF-Net for dynamic circuit deobfuscation.

For the first method, a recent important state-of-art method is the graph convolutional networks (GCN) nicely integrate local vertex features and graph topology in the spectral domain. However, current studies suffer from drawbacks: graph CNNs rely on Chebyshev polynomial approximation which results in oscillatory approximation at jump discontinuities since Chebyshev polynomials require degree Omega(poly(1/epsilon)) to approximate a jump signal such as |x|. To reduce complexity, RatioanlNet is proposed to integrate rational function and neural networks for graph node level embeddings. For the second method, we propose a method for function approximation which suffers from several drawbacks: non-robustness and infeasibility issue; neural networks are incapable of extracting analytical representation; there is no study reported to integrate the superiorities of neural network and Remez. This work proposes a novel neural network model to address the above issues. Specifically, our method utilizes the characterizations of Remez to design objective functions. To avoid the infeasibility issue and deal with the non-robustness, a set of constraints are imposed inspired by the equioscillation theorem of best rational approximation. The third method proposes an approach for circuit security. Circuit obfuscation is a recently proposed defense mechanism to protect digital integrated circuits (ICs) from reverse engineering. Estimating the deobfuscation runtime is a challenging task due to the complexity and heterogeneity of graph-structured circuit, and the unknown and sophisticated mechanisms of the attackers for deobfuscation. To address the above-mentioned challenges, this work proposes the first graph-based approach that predicts the deobfuscation runtime based on graph neural networks. The fourth method proposes a representation for dynamic size of circuit graph. By analyzing SAT attack method, a conjunctive normal form (CNF) bipartite graph is utilized to characterize the complexity of this SAT problem. To overcome the difficulty in capturing the dynamic size of the CNF graph, an energy-based kernel is proposed to aggregate dynamic features.