Butterfly Counting in Bipartite Networks

Graph motifs are used to analyze networks from diverse domains. We consider the problem of counting motifs in bipartite affiliation networks, such as author-paper, user-product, and actor-movie relations. The substantial prior work on counting motifs in unipartite graphs, such as triangle counting, does not apply here, since bipartite graphs do not have triangles. Unlike the solution of projecting bipartite graphs onto unipartite graphs, which leads to a substantial increase in the size of the network, we directly address counting motifs in bipartite networks. We focus on counting the number of occurrences of a “butterfly”, a complete 2× 2 subgraph, which is also the simplest cohesive and bipartite higher-order structure. Our main contribution is a suite of randomized algorithms for approximating the number of butterflies in a graph with provable accuracy guarantees. An experimental evaluation on large real-world networks shows that our algorithms can return accurate estimates within a few seconds, even for networks with trillions of butterflies and hundreds of millions of edges.