Spectral Methods for Contextualizing relational data

网络（或图）可以表示具有无数交互元素的复杂系统中的关系。两个主要的例子是代表一群人之间友谊的社会网络和代表活细胞中蛋白质之间功能关系的生物网络。许多实质性问题可以表述为(a)网络结构问题和(b)对行动者及其关系的补充测量问题。该项目将提供一个统计框架，同时分析关系（即网络）数据及其背景测量。主要目标是研究网络中参与者的关联数据和协变量测量之间的联合可变性。第二个目标是开始研究同一组参与者的网络样本之间的联合可变性。在这两个目标中，本项目将(1)提出一个一般的非参数模型和一组简单的参数模型，(2)设计快速的光谱估计器，以及(3)提供估计理论，检验非参数和参数模型下光谱估计器的统计性能。在大数据时代，数据集更大、更复杂，通常来自对具有无数相互作用元素的复杂系统的测量；社会和生物网络可以代表复杂系统中的关系，这些实质性问题本质上是关于网络的问题。ENCODE研究中的生物网络就是一个例子。此外，复杂系统中的关系通常是用一组丰富的关于参与者及其关系的补充信息来测量的。该研究计划将提供一个统计框架，包括模型、算法和理论，以研究与网络串联的补充信息，从而将网络和关系置于环境中。

Networks (or graphs) can represent the relationships in complex systems with myriad interacting elements. Two primary examples are social networks that represents the set of friendships in a group of people and biological networks that represent the functional relationships between proteins in a living cell. Many substantive questions can be phrased as questions of (a) the network structure and (b) supplementary measurements on the actors and their relationships. This project will provide a statistical framework to simultaneously analyze relational (i.e. network) data and its contextualizing measurements. The primary objective is to study the joint variability between the relational data and covariate measurements on the actors in the network. A secondary objective is to begin studying the joint variability among a sample of networks on the same set of actors. In both objectives, this project will (1) propose a general nonparametric model and a set of simple parametric models, (2) devise fast spectral estimators, and (3) provide estimation theory that examines the statistical performance of the spectral estimators under the nonparametric and parametric models. In the age of big data, data sets are both larger and more complex, often coming from measurements on complex systems with myriad interacting elements; social and biological networks can represent the relationships in complex systems and these substantive questions are, in essence, questions regarding networks. The biological networks in the ENCODE research are an example. Moreover, the relationships in complex systems are often measured with a rich set of supplemental information on the actors and their relationships. This research program will provide a statistical framework, including models, algorithms, and theory, to study the supplemental information in tandem with the network, thereby contextualizing the network and the relationships.