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1. Parallel Algorithms for Butterfly Computations

   Shi, Jessica; Shun, Julian (January 2020, Symposium on Algorithmic Principles of Computer Systems)

   Butterflies are the smallest non-trivial subgraph in bipartite graphs, and therefore having efficient computations for analyzing them is crucial to improving the quality of certain applications on bipartite graphs. In this paper, we design a framework called ParButterfly that contains new parallel algorithms for the following problems on processing butterflies: global counting, per-vertex counting, per-edge counting, tip decomposition (vertex peeling), and wing decomposition (edge peeling). The main component of these algorithms is aggregating wedges incident on subsets of vertices, and our framework supports different methods for wedge aggregation, including sorting, hashing, histogramming, and batching. In addition, ParButterfly supports different ways of ranking the vertices to speed up counting, including side ordering, approximate and exact degree ordering, and approximate and exact complement coreness ordering. For counting, ParButterfly also supports both exact computation as well as approximate computation via graph sparsification. We prove strong theoretical guarantees on the work and span of the algorithms in ParButterfly. We perform a comprehensive evaluation of all of the algorithms in ParButterfly on a collection of real-world bipartite graphs using a 48-core machine. Our counting algorithms obtain significant parallel speedup, outperforming the fastest sequential algorithms by up to 13.6× with a self-relative speedup of up to 38.5×. Compared to general subgraph counting solutions, we are orders of magnitude faster. Our peeling algorithms achieve self-relative speedups of up to 10.7× and outperform the fastest sequential baseline by up to several orders of magnitude. 

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2. Butterfly Counting in Bipartite Networks

   https://doi.org/10.1145/3219819.3220097  

   Sanei-Mehri, Seyed-Vahid; Sariyuce, Ahmet Erdem; Tirthapura, Srikanta (August 2018, Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining)

   We consider the problem of counting motifs in bipartite affiliation networks, such as author-paper, user-product, and actor-movie relations. We focus on counting the number of occurrences of a "butterfly", a complete 2x2 biclique, the simplest cohesive higher-order structure in a bipartite graph. Our main contribution is a suite of randomized algorithms that can quickly approximate the number of butterflies in a graph with a provable guarantee on accuracy. An experimental evaluation on large real-world networks shows that our algorithms return accurate estimates within a few seconds, even for networks with trillions of butterflies and hundreds of millions of edges. 

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3. A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs

   https://doi.org/10.4230/LIPIcs.ICALP.2024.35  

   Carlson, Charlie; Davies, Ewan; Kolla, Alexandra; Potukuchi, Aditya (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph - in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to "high-temperature" problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. Our contributions include a novel extension of the method of graph containers that differs considerably from other recent low-temperature algorithms. The additional key insights come from spectral graph theory and have previously been successful in approximation algorithms. As a result, we can overcome some limitations that seem inherent to the aforementioned class of algorithms. In particular, we exploit the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e., bounded threshold rank), which, via subspace enumeration, lets us enumerate small cuts efficiently. 

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4. Fast Counting and Utilizing Induced 6-Cycles in Bipartite Networks

   https://doi.org/10.1109/TKDE.2025.3546516  

   Niu, Jason; Zola, Jaroslaw; Sarıyüce, Ahmet Erdem (June 2025, IEEE Transactions on Knowledge and Data Engineering)

   Bipartite graphs are a powerful tool for modeling the interactions between two distinct groups. These bipartite relationships often feature small, recurring structural patterns called motifs which are building blocks for community structure. One promising structure 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 and utilizing induced 6-cycles in large bipartite networks. We first consider two adaptations inspired by previous works for cycle counting in bipartite networks. Then, we introduce a new approach for node triplets which offer a systematic way to count the induced 6-cycles, used in BATCHTRIPLETJOIN. Our experimental evaluation 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. In addition, we provide a new way to identify anomalous node triplets by comparing and contrasting the butterfly and induced 6-cycle counts of the nodes. We showcase several case studies on real-world networks from Amazon Kindle ratings, Steam game reviews, and Yelp ratings. 

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5. YACC: A Framework Generalizing TuránShadow for Counting Large Cliques

   Shweta Jain, Hanghang Tong (April 2022, SDM2022)

   Clique-counting is a fundamental problem that has application in many areas eg. dense subgraph discovery, community detection, spam detection, etc. The problem of k-clique-counting is difficult because as k increases, the number of k-cliques goes up exponentially. Enumeration algorithms (even parallel ones) fail to count k-cliques beyond a small k. Approximation algorithms, like TuránShadow have been shown to perform well upto k = 10, but are inefficient for larger cliques. The recently proposed Pivoter algorithm significantly improved the state-of-the-art and was able to give exact counts of all k-cliques in a large number of graphs. However, the clique counts of some graphs (for example, com-lj) are still out of reach of these algorithms. We revisit the TuránShadow algorithm and propose a generalized framework called YACC that leverages several insights about real-world graphs to achieve faster clique-counting. The bottleneck in TuránShadow is a recursive subroutine whose stopping condition is based on a classic result from extremal combinatorics called Turán's theorem. This theorem gives a lower bound for the k-clique density in a subgraph in terms of its edge density. However, this stopping condition is based on a worst-case graph that does not reflect the nature of real-world graphs. Using techniques for quickly discovering dense subgraphs, we relax the stopping condition in a systematic way such that we get a smaller recursion tree while still maintaining the guarantees provided by TuránShadow. We deploy our algorithm on several real-world data sets and show that YACC reduces the size of the recursion tree and the running time by over an order of magnitude. Using YACC, we are able to obtain clique counts for several graphs for which clique-counting was infeasible before, including com-lj. 

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