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Counting the frequency of small subgraphs is a fundamental technique innetwork analysis across various domains, most notably in bioinformatics andsocial networks. The special case of triangle counting has received muchattention. Getting results for 4-vertex patterns is highly challenging, andthere are few practical results known that can scale to massive sizes. Indeed,even a highly tuned enumeration code takes more than a day on a graph withmillions of edges. Most previous work that runs for truly massive graphs employclusters and massive parallelization.  We provide a sampling algorithm that provably and accurately approximates thefrequencies of all 4-vertex pattern subgraphs. Our algorithm is based on anovel technique of 3-path sampling and a special pruning scheme to decrease thevariance in estimates. We provide theoretical proofs for the accuracy of ouralgorithm, and give formal bounds for the error and confidence of ourestimates. We perform a detailed empirical study and show that our algorithmprovides estimates within 1% relative error for all subpatterns (over a largeclass of test graphs), while being orders of magnitude faster than enumerationand other sampling based algorithms. Our algorithm takes less than a minute (ona single commodity machine) to process an Orkut social network with 300 millionedges.

Path Sampling: A Fast and Provable Method for Estimating 4-Vertex Subgraph Counts