Fast Parallel Algorithms for Counting and Listing Triangles in Big Graphs

Big graphs (networks) arising in numerous application areas pose significant challenges for graph analysts as these graphs grow to billions of nodes and edges and are prohibitively large to fit in the main memory. Finding the number of triangles in a graph is an important problem in the mining and analysis of graphs. In this article, we present two efficient MPI-based distributed memory parallel algorithms for counting triangles in big graphs. The first algorithm employs overlapping partitioning and efficient load balancing schemes to provide a very fast parallel algorithm. The algorithm scales well to networks with billions of nodes and can compute the exact number of triangles in a network with 10 billion edges in 16 minutes. The second algorithm divides the network into non-overlapping partitions leading to a space-efficient algorithm. Our results on both artificial and real-world networks demonstrate a significant space saving with this algorithm. We also present a novel approach that reduces communication cost drastically leading the algorithm to both a space- and runtime-efficient algorithm. Further, we demonstrate how our algorithms can be used to list all triangles in a graph and compute clustering coefficients of nodes. Our algorithm can also be adapted to a parallel approximation algorithm using an edge sparsification method.