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Bipartite graphs characterize relationships between two different sets ofentities, like actor-movie, user-item, and author-paper. The butterfly, a4-vertices 4-edges (2,2)-biclique, is the simplest cohesive motif in abipartite graph and is the fundamental component of higher-order substructures.Counting and enumerating the butterflies offer significant benefits acrossvarious applications, including fraud detection, graph embedding, and communitysearch. While the corresponding motif, the triangle, in the unipartite graphshas been widely studied in both static and temporal settings, the extension ofbutterfly to temporal bipartite graphs remains unexplored. In this paper, weinvestigate the temporal butterfly counting and enumeration problem: count andenumerate the butterflies whose edges establish following a certain orderwithin a given duration. Towards efficient computation, we devise a non-trivialbaseline rooted in the state-of-the-art butterfly counting algorithm on staticgraphs, further, explore the intrinsic property of the temporal butterfly, anddevelop a new optimization framework with a compact data structure andeffective priority strategy. The time complexity is proved to be significantlyreduced without compromising on space efficiency. In addition, we generalizeour algorithms to practical streaming settings and multi-core computingarchitectures. Our extensive experiments on 11 large-scale real-world datasetsdemonstrate the efficiency and scalability of our solutions.

Efficient Temporal Butterfly Counting and Enumeration on Temporal Bipartite Graphs