The adoption of network science methodology within the field of cognitive psychology has largely been restricted to descriptive network analyses. Attempts at statistical or inferential modeling have been hindered by the lack of appropriate models, differences in assumptions underlying social and cognitive phenomena, and the nature of cognitive data. Specifically, networks constructed from free-response word associations, a common type of psycholinguistic data, are often extremely sparse and dichotomized. We propose the use of geodesic distance in the cue-response network, inversely weighted by the number of co-occurrences of responses, as a method for estimating a word-association network that avoids the issue of sparsity and binary edges. Building on previous work, we constructed two normally weighted networks from free-response data drawn from a repeated word association task, comparing responses of participants with below-average and above-average vocabulary knowledge as assessed using a norm-referenced measure of vocabulary comprehension. Using latent space models (LSMs), we found that semantic similarity predicted edge weights in the above-average vocabulary network, but not the below-average network. We also found non-trivial effects of dominant part-of-speech in the above-average network and concreteness ratings in the below average network. Moreover, we found differences in network structure, with more noise in the below-average than in the above-average latent space. Model comparisons showed that the LSMs outperformed both linear and hierarchical regressions in terms of accuracy and model fit. With further refinement of this analytic approach, LSMs have the potential to shed light on how the structure of the mental lexicon changes as people develop vocabulary knowledge.