Clique counting in MapReduce: theory and experiments

We tackle the problem of counting the number of $k$-cliques in large-scale graphs, for any constant $k \ge 3$. Clique counting is essential in a variety of applications, among which social network analysis. Due to its computationally intensive nature, we settle for parallel solutions in the MapReduce framework, which has become in the last few years a {\em de facto} standard for batch processing of massive data sets. We give both theoretical and experimental contributions. On the theory side, we design the first exact scalable algorithm for counting (and listing) $k$-cliques. Our algorithm uses $O(m^{3/2})$ total space and $O(m^{k/2})$ work, where $m$ is the number of graph edges. This matches the best-known bounds for triangle listing when $k=3$ and is work-optimal in the worst case for any $k$, while keeping the communication cost independent of $k$. We also design a sampling-based estimator that can dramatically reduce the running time and space requirements of the exact approach, while providing very accurate solutions with high probability. We then assess the effectiveness of different clique counting approaches through an extensive experimental analysis over the Amazon EC2 platform, considering both our algorithms and their state-of-the-art competitors. The experimental results clearly highlight the algorithm of choice in different scenarios and prove our exact approach to be the most effective when the number of $k$-cliques is large, gracefully scaling to non-trivial values of $k$ even on clusters of small/medium size. Our approximation algorithm achieves extremely accurate estimates and large speedups, especially on the toughest instances for the exact algorithms. As a side effect, our study also sheds light on the number of $k$-cliques of several real-world graphs, mainly social networks, and on its growth rate as a function of $k$.