Statistical analysis of networks with community structure and bootstrap methods for big data

This dissertation is divided into two parts, concerning two areas of statistical methodology. The first part of this dissertation concerns statistical analysis of networks with community structure. The second part of this dissertation concerns bootstrap methods for big data. Statistical analysis of networks with community structure: Networks are ubiquitous in today’s world — network data appears from varied fields such as scientific studies, sociology, technology, social media and the Internet, to name a few. An interesting aspect of many real-world networks is the presence of community structure and the problem of detecting this community structure. In the first chapter, we consider heterogeneous networks which seems to have not been considered in the statistical community detection literature. We propose a blockmodel for heterogeneous networks with community structure, and introduce a heterogeneous spectral clustering algorithm for community detection in heterogeneous networks. Theoretical properties of the clustering algorithm under the proposed model are studied, along with simulation study and data analysis. A network feature that is closely associated with community structure is the popularity of nodes in different communities. Neither the classical stochastic blockmodel nor its degree-corrected extension can satisfactorily capture the dynamics of node popularity. In the second chapter, we propose a popularity-adjusted blockmodel for flexible modeling of node popularity. We establish consistency of likelihood modularity for community detection under the proposed model, and illustrate the improved empirical insights that can be gained through this methodology by analyzing the political blogs network Popularity is defined as the number of edges between a specific node and a specific community.