How can we quickly find the number of triangles in a large graph, without actually counting them? Triangles are important for real world social networks, lying at the heart of the clustering coefficient and of the transitivity ratio. However, straight-forward and even approximate counting algorithms can be slow, trying to execute or approximate the equivalent of a 3-way database join. In this paper, we provide two algorithms, the EIGENTRIANGLE for counting the total number of triangles in a graph, and the EIGENTRIANGLELOCAL algorithm that gives the count of triangles that contain a desired node. Additional contributions include the following: (a) We show that both algorithms achieve excellent accuracy, with up to 1000x faster execution time, on several, real graphs and (b) we discover two new power laws (DEGREE-TRIANGLE and TRIANGLEPARTICIPATION laws) with surprising properties.

20 pages

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