Network analysis is becoming one of the most active research areas in many fields. It offers a natural way to organize data by incorporating pairwise relations. This subject is highly interdisciplinary. Researchers from physics, computer science, social science, biology and statistics have made significant contributions to network analysis in theories, methodologies and applications. Despite those recent methodological and theoretical progresses in network analysis, there have been little fundamental studies on optimal estimation. The wide range of important applications of networks ensure that the progress towards the proposed fundamental research objectives will have a great impact in a broad scientific community, which may include co-authorship networks, web networks, friendship networks, educational networks, networks with information flow, gene expression networks, political networks, and healthcare networks. Research results from this project will be disseminated through research articles and seminar series to researchers in other disciplines. The project will integrate research and education by teaching monograph courses and organizing seminars to help graduate students and postdocs, particularly minority, women, and domestic students and young researchers, who work on this topic. We will work closely with the Yale Institute for Network Science and the Yale Center for Outcomes Research and Evaluation to explore appropriate and helpful network models for social sciences and medicine, and to make valid statistical inference.
Various algorithms have been proposed and analyzed to understand the underlying generating mechanism of networks, called graphon, and to do community detection. Many consistency results are obtained. Despite these recent methodological and theoretical progresses on graphon estimation and community detection, especially on stochastic block model, there have been little fundamental studies on optimal estimation. For example, it is not clear whether the error rates for graphon estimation and community detection in those popular algorithms can be further improved. The goal of this project is to develop a coherent theory on optimal statistical network analysis. Specifically, we propose to study: 1) rate-optimal graphon estimation, 2) optimal community detection error rate, 3) computational barriers in graphon estimation and community section, 4) rate-optimal Bayesian posterior contraction, 5) generalizations to exponential family, to sparse networks, to networks of power law, to mixed membership networks, and to exchangeable high dimensional arrays or tensors, and 6) applications to social sciences and healthcare. The research in this project will significantly advance the theoretical understanding of statistical network analysis. The optimality theory will unveil the precision to what graphon estimation and community detection can be attained with or without computational constraints, and will integrate both frequentist and Bayesian perspectives for network analysis.
