Accelerating Clique Counting in Sparse Real-World Graphs via Communication-Reducing Optimizations

Abstract

Counting instances of specific subgraphs in a larger graph is an important problem in graph mining. Finding cliques of size k (k-cliques) is one example of this NP-hard problem. Different algorithms for clique counting avoid counting the same clique multiple times by pivoting or ordering the graph. Ordering-based algorithms include an ordering step to direct the edges in the input graph, and a counting step, which is dominated by building node or edge-induced subgraphs. Of the ordering-based algorithms, kClist is the state-of-the art algorithm designed to work on sparse real-world graphs. Despite its leading overall performance, kClist's vertex-parallel implementation does not scale well in practice on graphs with a few million vertices. We present CITRON (Clique counting with Traffic Reducing Optimizations) to improve the parallel scalability and thus overall performance of clique counting. We accelerate the ordering phase by abandoning kClist's sequential core ordering and using a parallelized degree ordering. We accelerate the counting phase with our reorganized subgraph data structures that reduce memory traffic to improve scaling bottlenecks. Our sorted, compact neighbor lists improve locality and communication efficiency which results in near-linear parallel scaling. CITRON significantly outperforms kClist while counting moderately sized cliques, and thus increases the size of graph practical for clique counting. We have recently become aware of ArbCount (arXiv:2002.10047), which often outperforms us. However, we believe that the analysis included in this paper will be helpful for anyone who wishes to understand the performance characteristics of k-clique counting.