Spectral Methods for Contextualizing relational data

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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.