Estimating Clique Composition and Size Distributions from Sampled Network Data

Cliques are defined as complete graphs or subgraphs; they are the strongest form of cohesive subgroup, and are of interest in both social science and engineering contexts. In this paper we show how to efficiently estimate the distribution of clique sizes from a probability sample of nodes obtained from a graph (e.g., by independence or link-trace sampling). We introduce two types of unbiased estimators, one of which exploits labeling of sampled nodes neighbors and one of which does not require this information. We compare the estimators on a variety of real-world graphs and provide suggestions for their use. We generalize our estimators to cases in which cliques are distinguished not only by size but also by node attributes, allowing us to estimate clique composition by size. Finally, we apply our methodology to a sample of Facebook users to estimate the clique size distribution by gender over the social graph.