  Social network data are often constructed by incorporating reports from
multiple individuals. However, it is not obvious how to reconcile discordant
responses from individuals. There may be particular risks with
multiply-reported data if people's responses reflect normative expectations --
such as an expectation of balanced, reciprocal relationships. Here, we propose
a probabilistic model that incorporates ties reported by multiple individuals
to estimate the unobserved network structure. In addition to estimating a
parameter for each reporter that is related to their tendency of over- or
under-reporting relationships, the model explicitly incorporates a term for
``mutuality,'' the tendency to report ties in both directions involving the
same alter. Our model's algorithmic implementation is based on variational
inference, which makes it efficient and scalable to large systems. We apply our
model to data from 75 Indian villages collected with a name-generator design,
and a Nicaraguan community collected with a roster-based design. We observe
strong evidence of ``mutuality'' in both datasets, and find that this value
varies by relationship type. Consequently, our model estimates networks with
reciprocity values that are substantially different than those resulting from
standard deterministic aggregation approaches, demonstrating the need to
consider such issues when gathering, constructing, and analysing survey-based
network data.
