Optimal Estimation of Statistical Networks

网络分析正在成为许多领域中最活跃的研究领域之一。它提供了一种通过合并成对关系来组织数据的自然方法。这门学科是高度交叉的。物理学、计算机科学、社会科学、生物学和统计学等领域的研究人员在理论、方法和应用方面对网络分析做出了重要贡献。尽管近年来在网络分析方法和理论方面取得了一些进展，但关于最优估计的基础研究却很少。网络的广泛重要应用确保了所提出的基础研究目标的进展将在广泛的科学社区中产生巨大影响，其中可能包括合作网络、网络网络、友谊网络、教育网络、信息流网络、基因表达网络、政治网络和医疗网络。该项目的研究成果将通过研究文章和系列研讨会传播给其他学科的研究人员。该项目将通过教授专题课程和组织研讨会来整合研究和教育，以帮助从事这一课题的研究生和博士后，特别是少数民族、妇女、国内学生和年轻研究人员。我们将与耶鲁大学网络科学研究所和耶鲁大学成果研究与评估中心密切合作，探索适合社会科学和医学的、有用的网络模型，并做出有效的统计推断。已经提出并分析了各种算法，以了解网络的底层生成机制，称为graphon，并进行社区检测。得到了许多一致性结果。尽管近年来在图元估计和社区检测方面，特别是在随机块模型方面取得了方法和理论上的进步，但在最优估计方面的基础研究却很少。例如，目前尚不清楚这些流行算法中graphon估计和社区检测的错误率是否可以进一步提高。这个项目的目标是发展一个关于最优统计网络分析的连贯理论。具体而言，我们建议研究：1)速率最优图元估计，2)最优社区检测错误率，3)图元估计和社区部分的计算障碍，4)速率最优贝叶斯后验收缩，5)推广到指数族，稀疏网络，幂律网络，混合隶属度网络，可交换高维数组或张量，以及6)在社会科学和医疗保健中的应用。本课题的研究将对统计网络分析的理论认识有重要的推动作用。最优性理论将揭示在有或没有计算约束的情况下，图形估计和社区检测可以达到的精度，并将频率论和贝叶斯观点整合到网络分析中。

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.