In this paper we are interested in finding communities with bipartite structure. A bipartite community is a pair of disjoint vertex sets S, \(S'\) such that the number of edges with one endpoint in S and the other endpoint in \(S'\) is “significantly more than expected.” This additional structure is natural to some applications of community detection. In fact, using other terminology, they have already been used to study correlation networks, social networks, and two distinct biological networks. In 2012 two groups independently [(1) Lee, Oveis Gharan, and Trevisan and (2) Louis, Raghavendra, Tetali, and Vempala] used higher eigenvalues of the normalized Laplacian to find an approximate solution to the k-sparse-cuts problem. In 2015 Liu generalized spectral methods for finding k communities to find k bipartite communities. Our approach improves the bounds on bipartite conductance (measure of strength of a bipartite community) found by Liu and also implies improvements to the original spectral methods by Lee et al. and Louis et al. We also highlight experimental results found when applying a practical algorithm derived from our theoretical results to three distinct real-world networks.

在本文中，我们有兴趣寻找具有二分结构的社区。二分社区是一对不相交的顶点集S，\（S'\），这样，一个端点在S中，另一个端点在\（S'\）中的边数“大大超过预期”。对于社区检测的某些应用来说，这种附加结构是很自然的。实际上，使用其他术语，它们已被用于研究关联网络，社交网络和两个不同的生物网络。在2012年，两个独立的小组[（1）Lee，Oveis Gharan和Trevisan和（2）Louis，Raghavendra，Tetali和Vempala]使用归一化Laplacian的较高特征值来找到k的近似解。-稀疏削减问题。在2015年，Liu推广了用于找到k个社区和k个二分社区的光谱方法。我们的方法改善了Liu发现的二分法电导的界限（衡量两分共同体的强度），也暗示了Lee等人对原始光谱方法的改进。和路易斯等。我们还重点介绍了将根据我们的理论结果得出的实用算法应用于三个不同的真实世界网络时发现的实验结果。