Structured Latent Space Models for Multiplex Networks

Abstract

Network data arises in fields such as neuroimaging, sociology, and medicine, to represent pairwise connections (edges) between units (nodes), or the interconnected behavior of a complex system. While classical statistical network analysis has most often considered a single large network, more complex network structures with rich auxiliary information are becoming increasingly common in practice. In this thesis we consider modeling, estimation, and inference for multiplex network data: multiple, heterogeneous networks (layers) observed on a shared set of nodes. Multiplex networks can represent a sample of networks with shared nodes, a network evolving over time, or a network with multiple types of connections. All of these examples of multiplex networks may have additional layer structure, such as a grouping or time ordering.

Latent space models are frequently used for modeling single layer networks and include models such as the stochastic block model and the random dot product graph. These models are interpretable, and efficiently share information as the number of nodes in the network increases. In the Chapters 2 and 3 of this thesis, we develop new latent space modeling frameworks tailored to two important multiplex layer structures. First, we detail a latent space model for multiplex networks with shared structure (MultiNeSS), with the goal of decomposing multiplex layers in to their common and individual components. Second, we introduce a new latent space embedding approach for multiplex layers indexed by a continuous covariate, the functional adjacency spectral embedding (FASE). Both MultiNeSS and FASE can adaptively choose the embedding dimensions or functional smoothness based on the data at hand, show good empirical performance on real and synthetic multiplex networks, and come with statistical guarantees on recovery of model parameters.

In Chapter 4, we leverage the dimension reduction induced by latent space modeling to design powerful two-sample testing methodology for multiplex networks with grouped layers. We address the problem of testing for statistically significant differences in a prespecified subset of the node pairs, allowing an analyst to focus their attention on a specific region of interest in the network. Using ideas based on random projection and low rank matrix modeling, we develop statistically sound tests which improve power locally by taking advantage of the underlying patterns in the entirety of the network.