Estimation and Inference for Network Data

Abstract

Networks play a key role in many scientific domains. In this thesis, we analyze several important questions in network analysis. The first question we analyze concerns how to understand latent structure in networks. Specifically, we propose a method that estimates the latent type, dimension, and curvature of the latent space model. The second problem we consider concerns network data collection. Collecting full network data is often prohibitively expensive and time-consuming. A common form of cheaper network data, known as Aggregated Relational Data (ARD), asks respondents ``How many people do you know with trait X?" for various pre-determined traits. In the second project, we show that ARD is sufficient to recover many statistics of the unobserved graph, which shows that researchers can simply collect ARD instead of full network data. The third question we analyze concerns model selection for network data. We derive a testing procedure that allows researchers to select the most appropriate model from a collection of candidate models, using the eigenvalues of the normalized adjacency matrix. We also show how this testing method is applicable to cases where the researcher only has access to ARD. Finally, we consider the problem of obtaining low-dimensional representations of objects from dissimilarity data. We propose a Bayesian procedure that uses dissimilarity data to obtain a representation of the objects in a Hyperbolic space, and show that this procedure obtains useful representations for the objects in several down-stream tasks in gene expression data.