  Collecting complete network data is expensive, time-consuming, and often
infeasible. Aggregated Relational Data (ARD), which capture information about a
social network by asking a respondent questions of the form ``How many people
with trait X do you know?'' provide a low-cost option when collecting complete
network data is not possible. Rather than asking about connections between each
pair of individuals directly, ARD collects the number of contacts the
respondent knows with a given trait. Despite widespread use and a growing
literature on ARD methodology, there is still no systematic understanding of
when and why ARD should accurately recover features of the unobserved network.
This paper provides such a characterization by deriving conditions under which
statistics about the unobserved network (or functions of these statistics like
regression coefficients) can be consistently estimated using ARD. We do this by
first providing consistent estimates of network model parameters for three
commonly used probabilistic models: the beta-model with node-specific
unobserved effects, the stochastic block model with unobserved community
structure, and latent geometric space models with unobserved latent locations.
A key observation behind these results is that cross-group link probabilities
for a collection of (possibly unobserved) groups identifies the model
parameters, meaning ARD is sufficient for parameter estimation. With these
estimated parameters, it is possible to simulate graphs from the fitted
distribution and analyze the distribution of network statistics. We can then
characterize conditions under which the simulated networks based on ARD will
allow for consistent estimation of the unobserved network statistics, such as
eigenvector centrality or response functions by or of the unobserved network,
such as regression coefficients.
