  Much of social network analysis is - implicitly or explicitly - predicated on
the assumption that individuals tend to be more similar to their friends than
to strangers. Thus, an observed social network provides a noisy signal about
the latent underlying "social space:" the way in which individuals are similar
or dissimilar. Many research questions frequently addressed via social network
analysis are in reality questions about this social space, raising the question
of inverting the process: Given a social network, how accurately can we
reconstruct the social structure of similarities and dissimilarities?
  We begin to address this problem formally. Observed social networks are
usually multiplex, in the sense that they reflect (dis)similarities in several
different "categories," such as geographical proximity, kinship, or similarity
of professions/hobbies. We assume that each such category is characterized by a
latent metric capturing (dis)similarities in this category. Each category gives
rise to a separate social network: a random graph parameterized by this metric.
For a concrete model, we consider Kleinberg's small world model and some
variations thereof. The observed social network is the unlabeled union of these
graphs, i.e., the presence or absence of edges can be observed, but not their
origins. Our main result is an algorithm which reconstructs each metric with
provably low distortion.
