Counting Induced 6-Cycles in Bipartite Graphs

Abstract

Various complex networks in real-world applications are best represented as a bipartite graph, such as user-product, paper-author, and actor-movie relations. Motif-based analysis has substantial benefits for networks and bipartite graphs are no exception. The smallest non-trivial subgraph in a bipartite graph is a (2,2)-biclique, also known as a butterfly. Although butterflies are succinct, they are limited in capturing the higher-order relations between more than two nodes from the same node set. One promising structure in this context is the induced 6-cycle which consists of three nodes on each node set forming a cycle where each node has exactly two edges. In this paper, we study the problem of counting induced 6-cycles through parallel algorithms. To the best of our knowledge, this is the first study on induced 6-cycle counting. We first consider two adaptations based on previous works for cycle counting in bipartite networks. Then, we introduce a new approach based on the node triplets and offer a systematic way to count the induced 6-cycles. Our final algorithm, BatchTripletJoin, is parallelizable across root nodes and uses minimal global storage to save memory. Our experimental evaluation on a 52 core machine shows that BatchTripletJoin is significantly faster than the other algorithms while being scalable to large graph sizes and number of cores. On a network with 112M edges, BatchTripletJoin is able to finish the computation in 78 mins by using 52 threads.