  Network models are widely used to represent relational information among
interacting units and the structural implications of these relations. Recently,
social network studies have focused a great deal of attention on random graph
models of networks whose nodes represent individual social actors and whose
edges represent a specified relationship between the actors. Most inference for
social network models assumes that the presence or absence of all possible
links is observed, that the information is completely reliable, and that there
are no measurement (e.g., recording) errors. This is clearly not true in
practice, as much network data is collected though sample surveys. In addition
even if a census of a population is attempted, individuals and links between
individuals are missed (i.e., do not appear in the recorded data). In this
paper we develop the conceptual and computational theory for inference based on
sampled network information. We first review forms of network sampling designs
used in practice. We consider inference from the likelihood framework, and
develop a typology of network data that reflects their treatment within this
frame. We then develop inference for social network models based on information
from adaptive network designs. We motivate and illustrate these ideas by
analyzing the effect of link-tracing sampling designs on a collaboration
network.
