(p,q)-biclique Counting and Enumeration for Large Sparse Bipartite Graphs

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Abstract

In this paper, we study the problem of (p,q)-biclique counting and enumeration for large sparse bipartite graphs.Given a bipartite graph $G=(U,V,E)$, and two integer parameters $p$ and $q$,we aim to efficiently count and enumerate all (p,q)-bicliques in $G$, where a (p,q)-biclique B(L,R) is a complete subgraph of G with $L \subseteq U$, $R \subseteq V$, $|L|=p$, and $|R|=q$.The problem of (p,q)-biclique counting and enumeration has many applications, such as graph neural network information aggregating, densest subgraph detection, and cohesive subgroup analysis.Despite the wide range of applications,to the best of our knowledge, we note that there is no efficient and scalable solution to this problem in the literature. This problem is computationally challenging, due to the worst-case exponential number of (p,q)-bicliques. In this paper, we propose a competitive branch-and-bound baseline method, namely BCList,which explores the search space in a depth-first manner, together with a variety of pruning techniques.Although BCList offers a useful computation framework to our problem,its worst-case time complexity is exponential to $p+q$.To alleviate this, we propose an advanced approach, called BCList++.Particularly, BCList++ applies a layer based exploring strategy to enumerate (p,q)-bicliques by anchoring the search on either $U$ or $V$ only, which has a worst-case time complexity exponential to either $p$ or $q$ only.Consequently, a vital task is to choose a layer with the least computation cost. To this end, we develop a cost model, which is built uponan unbiased estimator for the density of $2$-hop graph induced by $U$ or $V$.To improve computation efficiency, BCList++ exploits pre-allocated arrays and vertex labeling techniques such that the frequent subgraph creating operations can be substituted by array element switching operations.We conduct extensive experiments on $16$ real-life datasets, and the experimental results demonstratethat BCList++ significantly outperforms the baseline methods by up to $3$ orders of magnitude.We show via a case study that (p,q)-bicliques optimizes the efficiency of graph neural networks.