Efficient Maximum Edge-Weighted Biclique Search on Large Bipartite Graphs

Given a bipartite graph, the maximum edge biclique problem (<inline-formula><tex-math notation="LaTeX">${{\sf MEB}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MEB</mml:mi></mml:math><inline-graphic xlink:href="yang-ieq1-3220901.gif"/></alternatives></inline-formula>) aims to find a biclique with the largest number of edges. <inline-formula><tex-math notation="LaTeX">${{\sf MEB}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MEB</mml:mi></mml:math><inline-graphic xlink:href="yang-ieq2-3220901.gif"/></alternatives></inline-formula> is a fundamental problem with many real applications, such as community analysis, E-commerce services and bioinformatics. However, in some scenarios, the weight of an edge reflects valuable and important information on the relationship between two entities. Motivated by this, in this paper, we investigate the problem of maximum edge-weighted biclique search (<inline-formula><tex-math notation="LaTeX">${{\sf MEWB}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MEWB</mml:mi></mml:math><inline-graphic xlink:href="yang-ieq3-3220901.gif"/></alternatives></inline-formula>), which finds a biclique with the largest total weight of edges in a weighted bipartite graph. <inline-formula><tex-math notation="LaTeX">${{\sf MEWB}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MEWB</mml:mi></mml:math><inline-graphic xlink:href="yang-ieq4-3220901.gif"/></alternatives></inline-formula> has many real applications, including item recommendation, fraud detection, gene clustering, etc. Although we show that <inline-formula><tex-math notation="LaTeX">${{\sf MEWB}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MEWB</mml:mi></mml:math><inline-graphic xlink:href="yang-ieq5-3220901.gif"/></alternatives></inline-formula> can be resolved by adapting the search algorithm designed for <inline-formula><tex-math notation="LaTeX">${{\sf MEB}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MEB</mml:mi></mml:math><inline-graphic xlink:href="yang-ieq6-3220901.gif"/></alternatives></inline-formula>, the performance of this method is yet unsatisfactory. To improve the computation efficiency, two optimizations in terms of upper bound and search order are proposed. For the upper bound, we consider the degree distribution for vertices in the candidate set, and thus have a chance to discard a few edges to tighten the upper bound. For search order, we theoretically show that a vertex order generating the most similar search depth on vertices can achieve the least time cost for <inline-formula><tex-math notation="LaTeX">${{\sf MEWB}}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">MEWB</mml:mi></mml:math><inline-graphic xlink:href="yang-ieq7-3220901.gif"/></alternatives></inline-formula>. Guided by this fact, we propose the global summation vertex order. To further accelerate the computation, we extend our approach to a parallel environment, and develop a heuristic approach to deal with large-scale graphs by slightly sacrificing the answer quality. Extensive performance studies conducted on real datasets demonstrate that our proposals can significantly outperform the baseline method by up to two orders of magnitude. Besides, our heuristic approach gives the optimal result on 8 out of 10 real datasets, while achieving more than an order of magnitude of speed-up.