We present novel methods to construct compact natural language lexicons within a graph-based semi-supervised learning framework, an attractive platform suited for propagating soft labels onto new natural language types from seed data. To achieve compactness, we induce sparse measures at graph vertices by incorporating sparsity-inducing penalties in Gaussian and entropic pairwise Markov networks constructed from labeled and unlabeled data. Sparse measures are desirable for high-dimensional multi-class learning problems such as the induction of labels on natural language types, which typically associate with only a few labels. Compared to standard graph-based learning methods, for two lexicon expansion problems, our approach produces significantly smaller lexicons and obtains better predictive performance.