Latent Surface Models for Networks Using Aggregated Relational Data

Despite increased interest across a range of scientific applications in modeling and understanding social network structure, collecting complete network data remains logistically and financially challenging, especially in the social sciences. This article introduces a latent surface representation of social network structure for partially observed network data. We derive a multivariate measure of expected (latent) distance between an observed actor and unobserved actors with given features. We also draw novel parallels between our work and dependent data in spatial and ecological statistics. We demonstrate the contribution of our model using a random digit-dial telephone survey and a multiyear prospective study of the relationship between network structure and the spread of infectious disease. The model proposed here is related to previous network models which represents high-dimensional structure through a projection to a low-dimensional latent geometric surface-encoding dependence as distance in the space. We develop a latent surface model for cases when complete network data are unavailable. We focus specifically on aggregated relational data (ARD) which measure network structure indirectly by asking respondents how many connections they have with members of a certain subpopulation (e.g., How many individuals do you know who are HIV positive?) and are easily added to existing surveys. Instead of conditioning on the (latent) distance between two members of the network, the latent surface model for ARD conditions on the expected distance between a survey respondent and the center of a subpopulation on a latent manifold surface. A spherical latent surface and angular distance across the sphere's surface facilitate tractable computation of this expectation. This model estimates relative homogeneity between groups in the population and variation in the propensity for interaction between respondents and group members. The model also estimates features of groups which are difficult to reach using standard surveys (e.g., the homeless). Supplementary materials for this article are available online.