Colorful triangle counting and a MapReduce implementation

In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, let G be a graph with n vertices, t triangles and let Δ be the maximum number of triangles an edge of G is contained in. Our randomized algorithm colors the vertices of G   with N=1/pN=1/p colors uniformly at random, counts monochromatic triangles, i.e., triangles whose vertices have the same color, and scales that count appropriately. We show that if p⩾max(Δlognt,lognt) then for any constant ϵ>0ϵ>0 our unbiased estimate T   is concentrated around its expectation, i.e., Pr[|T−E[T]|⩾ϵE[T]]=o(1)Pr[|T−E[T]|⩾ϵE[T]]=o(1). Finally, our algorithm is amenable to being parallelized. We present a simple MapReduce implementation of our algorithm.

► We study the problem of triangle counting in large graphs (e.g., social networks).
► We provide a new efficient randomized algorithm.
► We provide tight conditions which guarantee concentrated estimates.
► Our algorithm provides the best speedups for a (1±ϵ1±ϵ) approximation.
► Our algorithm is easily parallelizable. We give an implementation in MapReduce.