A Fast and Provable Method for Estimating Clique Counts Using Tur\'an's   Theorem
  Clique counts reveal important properties about the structure of massive graphs, especially social networks. The simple setting of just 3-cliques (triangles) has received much attention from the research community. For larger cliques (even, say 6-cliques) the problem quickly becomes intractable because of combinatorial explosion. Most methods used for triangle counting do not scale for large cliques, and existing algorithms require massive parallelism to be feasible.   We present a new randomized algorithm that provably approximates the number of k-cliques, for any constant k. The key insight is the use of (strengthenings of) the classic Tur\'an's theorem: this claims that if the edge density of a graph is sufficiently high, the k-clique density must be non-trivial. We define a combinatorial structure called a Tur\'an shadow, the construction of which leads to fast algorithms for clique counting.   We design a practical heuristic, called TUR\'AN-SHADOW, based on this theoretical algorithm, and test it on a large class of test graphs. In all cases,TUR\'AN-SHADOW has less than 2% error, in a fraction of the time used by well-tuned exact algorithms. We do detailed comparisons with a range of other sampling algorithms, and find that TUR\'AN-SHADOW is generally much faster and more accurate. For example, TUR\'AN-SHADOW estimates all cliques numbers up to size 10 in social network with over a hundred million edges. This is done in less than three hours on a single commodity machine. 
Shweta Jain, C. Seshadhri
cs.SI cs.DS