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Algorithms for mining very large graphs, such as those representing onlinesocial networks, to discover the relative frequency of small subgraphs withinthem are of high interest to sociologists, computer scientists and marketeersalike. However, the computation of these network motif statistics via naiveenumeration is infeasible for either its prohibitive computational costs oraccess restrictions on the full graph data. Methods to estimate the motifstatistics based on random walks by sampling only a small fraction of thesubgraphs in the large graph address both of these challenges. In this paper,we present a new algorithm, called the Waddling Random Walk (WRW), whichestimates the concentration of motifs of any size. It derives its name from thefact that it sways a little to the left and to the right, thus also samplingnodes not directly on the path of the random walk. The WRW algorithm achievesits computational efficiency by not trying to enumerate subgraphs around therandom walk but instead using a randomized protocol to sample subgraphs in theneighborhood of the nodes visited by the walk. In addition, WRW achievessignificantly higher accuracy (measured by the closeness of its estimate to thecorrect value) and higher precision (measured by the low variance in itsestimations) than the current state-of-the-art algorithms for mining subgraphstatistics. We illustrate these advantages in speed, accuracy and precisionusing simulations on well-known and widely used graph datasets representingreal networks.

Waddling Random Walk: Fast and Accurate Mining of Motif Statistics in Large Graphs