Statistical Inference for Networks with Complex Topological Structures

了解网络的结构对于理解和预测涉及网络的应用现象至关重要，例如叛乱和恐怖主义网络的弹性以及疾病传播网络对流行病的影响。在过去的十年中，对于具有简单拓扑特征的网络的统计推断，如网络中的连接数和网络成员形成连接的倾向，已经取得了巨大的进展。然而，对于具有复杂拓扑特征的网络的统计推断，例如各种形式的网络关闭被认为是至关重要的，仍然不发达，因为复杂的拓扑特征提出了具有挑战性的概念，理论和计算问题。该项目解决了支持此类网络的统计推断的基本问题。开发的统计模型和方法将以R包的形式公开提供。本研究项目关注具有复杂拓扑特征的网络的统计推断基础。它从一个统计推断的核心问题开始：从同一来源观察到更多数据意味着什么？传统的答案是，通过观察一个越来越大的网络，可以观察到更多的数据。通过研究不断增长的网络，出现了一些有趣的见解。其中，具有复杂拓扑特征的模型可能取决于网络的大小，对于大小网络使用相同的模型、相同的参数可能没有意义。尽管有这样的见解，如何对大范围的复杂拓扑特征进行建模以及如何进行合理的统计推断的问题仍然没有答案。该项目试图提供答案，并基于以下想法：如果具有复杂拓扑特征的模型依赖于网络的大小，那么统计推断应该基于相同数量级的网络。换句话说，统计推断应该基于复制。至少有两种形式的复制是可能的：基于由许多子网组成的单个网络的复制，或者基于相同数量级的多个网络的复制，即最大网络的大小是最小网络大小的常数倍。这种网络数据被称为多层网络数据，具有重要的应用；例子包括由单位和子单位组成的武装部队网络和由学校和班级组成的学校网络。多层网络数据为建模、方法和理论提供了绝佳的机会。该项目将利用这些机会精心设计新的多层网络模型，以捕获网络中的复杂拓扑特征、节点的重叠子集和时间网络。统计理论将利用多层次网络数据的复制特性，这为具有复杂拓扑特征的网络提供了第一个通用统计理论的途径。统计计算将利用多层网络模型的条件独立结构，并将发展大规模并行计算程序。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Understanding the structure of networks is critical to understanding and predicting application phenomena involving networks, such as the resilience of insurgent and terrorist networks and the impact of disease-transmission networks on epidemics. In the past decade, tremendous progress has been made on statistical inference for networks with simple topological features, such as the number of connections in networks and the propensities of network members to form connections. However, statistical inference for networks with complex topological features, such as various forms of network closure that are believed to be crucial, remains underdeveloped, because complex topological features raise challenging conceptual, theoretical, and computational issues. This project addresses fundamental questions underpinning statistical inference for such networks. The statistical models and methods that are developed will be made publicly available in the form of R packages.This research project is concerned with the foundations of statistical inference for networks with complex topological features. It starts with a question that is at the heart of statistical inference: What does it mean to observe more data from the same source? The conventional answer is that more data are observed by observing a larger and larger network. Some interesting insights have emerged by studying growing networks. Among them is that models with complex topological features may depend on the size of the network and that using the same model, with the same parameters, for small and large networks may not be meaningful. Despite such insights, the question of how to model a wide range of complex topological features and how to conduct sound statistical inference remains unanswered. This project attempts to provide answers and is based on the following idea: If models with complex topological features depend on the size of the network, then statistical inference should be based on networks of the same order of magnitude. In other words, statistical inference should be based on replication. At least two forms of replication are possible: replication based on a single network consisting of many subnetworks or many networks of the same order of magnitude, i.e., the size of the largest network is a constant multiple of the size of the smallest network. Such network data, called multilevel network data, has important applications; examples include networks of armed forces consisting of units and subunits and school networks consisting of schools and school classes. Multilevel network data offer outstanding opportunities for modeling, methods, and theory. This project will take advantage of these opportunities to elaborate novel multilevel network models capturing complex topological features in networks, overlapping subsets of nodes, and temporal networks. Statistical theory will take advantage of the replicative nature of multilevel network data, which provides a path to the first general statistical theory for networks with complex topological features. Statistical computing will take advantage of the conditional independence structure of multilevel network models and will develop large-scale parallel computing procedures.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.