We propose a novel cohesive subgraph model called $\tau$-strengthened $(\alpha ,\beta )$-core (denoted as ${(\alpha ,\beta )}_{\tau }$-core), which is the first to consider both tie strength and vertex engagement on bipartite graphs. An edge is a strong tie if contained in at least $\tau$ butterflies ($2\times 2$-bicliques). ${(\alpha ,\beta )}_{\tau }$-core requires each vertex on the upper or lower level to have at least $\alpha$ or $\beta$ strong ties, given strength level $\tau$. To retrieve the vertices of ${(\alpha ,\beta )}_{\tau }$-core optimally, we construct index $I_{\alpha ,\beta ,\tau }$ to store all ${(\alpha ,\beta )}_{\tau }$-cores. Effective optimization techniques are proposed to improve index construction. To make our idea practical on large graphs, we propose 2D-indexes $I_{\alpha ,\beta },I_{\beta ,\tau }$, and $I_{\alpha ,\tau }$ that selectively store the vertices of ${(\alpha ,\beta )}_{\tau }$-core for some $\alpha ,\beta$, and $\tau$. The 2D-indexes are more space-efficient and require less construction time, each of which can support ${(\alpha ,\beta )}_{\tau }$-core queries. As query efficiency depends on input parameters and the choice of 2D-index, we propose a learning-based hybrid computation paradigm by training a feed-forward neural network to predict the optimal choice of 2D-index that minimizes the query time. Extensive experiments show that (1) ${(\alpha ,\beta )}_{\tau }$-core is an effective model capturing unique and important cohesive subgraphs; (2) the proposed techniques significantly improve the efficiency of index construction and query processing.

我们提出了一种新的内聚子图模型，称为 $\tau$加强 $（\alpha ，\beta ）$核心（表示为 ${（\alpha ，\beta ）}_{\tau }$-core），这是第一个在二部图上同时考虑连接强度和顶点接合的情况。如果至少包含一条边，则该边是一条牢固的领带$\tau$ 蝴蝶（$\mathrm{2个}\times \mathrm{2个}$-bicliques）。 ${（\alpha ，\beta ）}_{\tau }$-core要求上层或下层的每个顶点至少具有 $\alpha$ 或者 $\beta$ 牢固的关系，给定强度水平 $\tau$。检索的顶点${（\alpha ，\beta ）}_{\tau }$最佳，我们构造索引 ${\mathrm{一世}}_{\alpha ，\beta ，\tau }$ 储存所有 ${（\alpha ，\beta ）}_{\tau }$-核心。提出了有效的优化技术来改善指标的构建。为了使我们的想法在大图上可行，我们提出了2D索引${\mathrm{一世}}_{\alpha ，\beta }，{\mathrm{一世}}_{\beta ，\tau }$， 和 ${\mathrm{一世}}_{\alpha ，\tau }$ 有选择地存储的顶点 ${（\alpha ，\beta ）}_{\tau }$核心一些 $\alpha ，\beta$， 和 $\tau$。2D索引更节省空间，并且需要更少的构建时间，每个索引都可以支持${（\alpha ，\beta ）}_{\tau }$核心查询。由于查询效率取决于输入参数和2D索引的选择，因此我们通过训练前馈神经网络来预测2D索引的最佳选择以最小化查询时间，从而提出一种基于学习的混合计算范式。大量实验表明（1）${（\alpha ，\beta ）}_{\tau }$-core是一种有效的模型，可捕获独特且重要的内聚子图；（2）所提出的技术显着提高了索引构建和查询处理的效率。