| Summary: | <p> The spectral space of the adjacency matrix contains important structural information of a given network (graph), where such information can be leveraged in developing a variety of algorithms in applications such as graph partition, structural hierarchy discovery, and anomaly detection. Although many prominent works have laid the foundation for studying the graph spectra, it is still challenging to analyze the spectral space properties for directed graphs due to possible complex valued decompositions. Matrix factorization techniques such as Laplacian and normalized Laplacian have been widely adopted to study the associated spectral spaces, but network structural properties may not be well preserved in those spectral spaces due to transformations.</p><p> In this dissertation work, we explore the adjacency eigenspace of directed graphs using matrix perturbation theory and examine the relationships between graph structures and the spectral projection patterns. We study how to detect dominant structures such as clusters or anomalous nodes by establishing a connection between the connectivity of nodes and the geometric relationships in the adjacency eigenspace. We leverage selected key results from perturbation theory, linear algebra and graph theory as our tools to derive theoretical results that help to elaborate observed graph spectral projection patterns. In order to validate our theoretical results, novel algorithms including spectral clustering for both signed and unsigned networks, asymmetry analysis for network dominance, and anomaly analysis for streaming network data are developed and tested on both synthetic and real datasets. The empirical evaluation results suggest that our algorithms performs better when compared with existing state-of-the-art methods.</p><p>
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