比最密集子图更密集:提取具有质量保证的最优拟团

Charalampos E. Tsourakakis, F. Bonchi, A. Gionis, Francesco Gullo, M. A. Tsiarli
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引用次数: 276

摘要

在许多应用中,查找密集子图是一项重要的图挖掘任务。考虑到边密度的直接优化是没有意义的,因为即使是单个边也能达到最大密度,研究的重点是优化替代密度函数。这些函数中非常流行的一个是平均度,它的最大化导致了众所周知的最密集子图概念。然而,令人惊讶的是,最密集的子图通常是大图,边缘密度小,直径大。在本文中,我们定义了一个新的密度函数,它给出了比最密集子图质量高得多的子图:通过我们的方法发现的图是紧凑的,密集的,直径更小。我们证明了所提出的函数可以从一个一般框架中推导出来,其中包括其他重要的密度函数作为子情况,并且我们展示了有趣的一般理论性质。为了优化所提出的函数,我们提供了一个加性逼近算法和一个局部搜索启发式算法。这两种算法都非常有效,并且可以很好地扩展到大型图形。我们在真实和合成数据集上评估我们的算法,并且我们还设计了几个应用研究作为原始问题的变体。与寻找最大平均度子图的方法相比,我们的算法返回的子图密度更大,直径更小。最后,我们讨论了我们的问题留下的新的有趣的研究方向。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Denser than the densest subgraph: extracting optimal quasi-cliques with quality guarantees
Finding dense subgraphs is an important graph-mining task with many applications. Given that the direct optimization of edge density is not meaningful, as even a single edge achieves maximum density, research has focused on optimizing alternative density functions. A very popular among such functions is the average degree, whose maximization leads to the well-known densest-subgraph notion. Surprisingly enough, however, densest subgraphs are typically large graphs, with small edge density and large diameter. In this paper, we define a novel density function, which gives subgraphs of much higher quality than densest subgraphs: the graphs found by our method are compact, dense, and with smaller diameter. We show that the proposed function can be derived from a general framework, which includes other important density functions as subcases and for which we show interesting general theoretical properties. To optimize the proposed function we provide an additive approximation algorithm and a local-search heuristic. Both algorithms are very efficient and scale well to large graphs. We evaluate our algorithms on real and synthetic datasets, and we also devise several application studies as variants of our original problem. When compared with the method that finds the subgraph of the largest average degree, our algorithms return denser subgraphs with smaller diameter. Finally, we discuss new interesting research directions that our problem leaves open.
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