Estimating Graphlet Statistics via Lifting

Exploratory analysis over network data is often limited by our ability to efficiently calculate graph statistics, which can provide a model-free understanding of macroscopic properties of a network. This work introduces a framework for estimating the graphlet count - the number of occurrences of a small subgraph motif (e.g. a wedge or a triangle) in the network. For massive graphs, where accessing the whole graph is not possible, the only viable algorithms are those which act locally by making a limited number of vertex neighborhood queries. We introduce a Monte Carlo sampling technique for graphlet counts, called lifting, which can simultaneously sample all graphlets of size up to $k$ vertices. We outline three variants of lifted graphlet counts: the ordered, unordered, and shotgun estimators. We prove that our graphlet count updates are unbiased for the true graphlet count, have low correlation between samples, and have a controlled variance. We compare the experimental performance of lifted graphlet counts to the state-of-the art graphlet sampling procedures: Waddling and the pairwise subgraph random walk.