Social networks have received much attention these days. Researchers have de- veloped different methods to study the structure and characteristics of the network topology. Our focus is on spectral analysis of the adjacency matrix of the underlying network. Recent work showed good properties in the adjacency spectral space but there are few theoretical and systematical studies to support their findings.In this dissertation, we conduct an in-depth theoretical study to show the close relationship between algebraic spectral properties of the adjacency matrix and vari- ous patterns in a broad range of social networks such as friendship networks, alliance and war networks, and distrusted networks. In our framework, we apply matrix per- turbation theory and approximate the eigenvectors of real graphs by those of the ideal cases. Our theoretical results show that the principal eigenvectors capture the structure of major communities and exhibit them as orthogonal lines/clusters rotated with certain angles from canonical axes. Our results also show that the minor eigen- vectors with skew distributions in values capture weak or subtle signals hidden in local communities. We utilize our theoretical results to develop algorithms for several problems in social network analysis including community partition, anomaly detec- tion and privacy preserving social network reconstruction. Empirical evaluations on various synthetic data and real-world social networks validate our theoretical findings and show the effectiveness of our algorithms.In a nutshell, we theoretically study the patterns in the adjacency spectral space as well as conditions for their existence and explore the application of the spectral properties of the adjacency matrix in different tasks of social network analysis.
