Testing and Modeling Dependencies Between a Network and Nodal Attributes

Network analysis is often focused on characterizing the dependencies between network relations and node-level attributes. Potential relationships are typically explored by modeling the network as a function of the nodal attributes or by modeling the attributes as a function of the network. These methods require specification of the exact nature of the association between the network and attributes, reduce the network data to a small number of summary statistics, and are unable provide predictions simultaneously for missing attribute and network information. Existing methods that model the attributes and network jointly also assume the data are fully observed. In this article we introduce a unified approach to analysis that addresses these shortcomings. We use a latent variable model to obtain a low dimensional representation of the network in terms of node-specific network factors and use a test of dependence between the network factors and attributes as a surrogate for a test of dependence between the network and attributes. We propose a formal testing procedure to determine if dependencies exists between the network factors and attributes. We also introduce a joint model for the network and attributes, for use if the test rejects, that can capture a variety of dependence patterns and be used to make inference and predictions for missing observations.