Statistical Methods for Network Data

网络数据已经在广泛的领域中变得普遍，大量不同的研究人员已经研究了网络的各个方面，但统计方法很少应用。这个项目提出了新的理论、方法和算法，对这些问题采取了原则性的统计方法，评估了不确定性，并为一致性等理想特性创造了条件。它的重点主要是发现网络中的社区结构，这是实践中的一种常见现象，也是网络分析中的一个基本问题。提出了一种新的拟合块模型的伪似然算法，以及允许块内程度分布不均匀的几个推广，消除了经典块模型的主要局限性。基于聚合数据的伪似然性大大加快了计算速度，使这些模型能够适应比以前更大、更稀疏的网络。研究了群落检测准则的渐近分布，得到了群落结构、一致性条件和渐近正确划分阈值的显著性检验，具有重要的实际意义。还提出了新的、更强大的标准，在较弱的条件下是一致的。该提案还为比较两个网络制定了正式的非参数检验，这一问题在实践中经常出现，但目前只能通过汇总统计数据的非正式比较来解决。最后，将节点和边上的协变量合并到模型中，并用于预测网络中未观察到的链路。许多提出的方法为相应的网络问题提供了第一个统计解决方案。网络中社区检测统计方法的发展，虽然有助于核心统计理论和方法的发展，但对网络分析的跨学科领域和复杂网络的研究具有直接影响。它们的应用非常广泛，涵盖了传染病建模、国家安全、通信、社会学和基因组学等不同领域。拟议的新统计工具采取了更正式、更严格的方法，并有可能改变多少科学家采用网络分析的方法。

Network data have become common in a wide range of fields, and a large and diverse community of researchers have studied various aspects of networks, yet statistical methods are rarely applied. This project proposes new theory, methodology, and algorithms that take a principled statistical approach to these problems, assess uncertainty, and establish conditions for desirable properties such as consistency. The focus is primarily on discovering community structure in networks, a common phenomenon in practice and a fundamental question in network analysis. New pseudo-likelihood algorithms are proposed for fitting the block model for networks, as well as several generalizations that allow for non-uniform degree distribution within blocks, removing the main limitation of the classic block model. The pseudo-likelihood based on aggregated data substantially speeds up computation, allowing fitting these models to larger and sparser networks than previously possible. The asymptotic distribution of criteria used for community detection is also studied, which leads to development of significance tests for community structure, consistency conditions, and asymptotically correct partition thresholds, which have important practical implications. New, more robust criteria are also proposed, consistent under weaker conditions. The proposal also develops a formal non-parametric test for comparing two networks, a problem that arises frequently in practice but is currently addressed only through informal comparisons of summary statistics. Finally, covariates on nodes and edges are incorporated into the models and used for predicting unobserved links in the networks. Many of the proposed methods provide the first statistical solutions to the corresponding network problems. Development of statistical methods for community detection in networks, while contributing to the development of core statistical theory and methodology, has direct impact on the interdisciplinary field of network analysis and the study of complex networks. The applications of these are wide-spread, covering such diverse areas as infectious disease modeling, national security, communications, sociology, and genomics. The new statistical tools proposed take a more formal, rigorous approach, and have the potential to change how many scientists approach network analysis.