The title for the dissertation is "Topics in Network Analysis and Multivariate Statistics".  An overarching objective in contemporary statistical network analysis is extracting salient information from datasets consisting of multiple networks. While numerous methods have been developed for analyzing network-valued data, key challenges remain in network property estimation, integrating multiple networks for classification and constructing networks. In this thesis, we develop novel methodologies to address these challenges. First, while considerable efforts have been devoted to node and network clustering, comparatively less attention has been given to connectivity estimation and parsimonious embedding dimension selection. We propose a method to simultaneously estimate a latent connectivity matrix and its embedding dimensionality (rank) after first pre-estimating the number of communities and node cluster memberships. The proposed method provides accurate and robust dimensionality estimates. When exact membership recovery is possible and dimensionality is much smaller than the number of communities, it outperforms averaging-based methods for estimating connectivity and dimensionality. Second, we explore the integration of functional and structural brain networks for classification, showing that network embeddings obtained from the MultiVERSE algorithm can improve classification accuracy over using a single network type. Finally, motivated in part by the observation that correlation-based gene networks may not exhibit the expected scale-free property, while networks constructed after adjusting for latent factors are more likely to be scale-free, we develop a novel longitudinal factor analysis method. The proposed method accurately estimates latent factors and captures dynamic changes in both factors and loadings over time, particularly in response to treatment effects. The method uses an Empirical Bayes matrix factorization approach, allowing both factors and loadings to evolve over time.

Advisor and Committee Chair: Dr. Chris McKennan