Knowing the structure of an offline social network facilitates a variety of analyses, including studying the rate at which infectious diseases may spread and identifying a subset of actors to immunize in order to reduce, as much as possible, the rate of spread. Offline social network topologies are typically estimated by surveying actors and asking them to list their neighbours. While identifying close friends and family (i.e., strong ties) can typically be done reliably, listing all of one's acquaintances (i.e., weak ties) is subject to error due to respondent fatigue. This issue is commonly circumvented through the use of so-called "fixed choice" surveys where respondents are asked to name a fixed, small number of their weak ties (e.g., two or ten). Of course, the resulting crude observed network will omit many ties, and using this crude network to infer properties of the network, such as its degree distribution or clustering coefficient, will lead to biased estimates. This paper develops estimators, based on the method of moments, for a number of network characteristics including those related to the first and second moments of the degree distribution as well as the network size, using fixed-choice survey data. Experiments with simulated data illustrate that the proposed estimators perform well across a variety of network topologies and measurement scenarios, and the resulting estimates are significantly more accurate than those obtained directly using the crude observed network, which are commonly used in the literature. We also describe a variation of the Jackknife procedure that can be used to obtain an estimate of the estimator variance.