  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.
