Exploring Cohesive Subgraphs with Vertex Engagement and Tie Strength in Bipartite Graphs

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 ($2\times2$-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.