We consider a variation of the Kuramoto model with dynamic coupling, where the coupling strengths are allowed to evolve in response to the phase difference between the oscillators, a model first considered by Ha, Noh, and Park. We demonstrate that the fixed points of this model, as well as their stability, can be completely expressed in terms of the fixed points and stability of the analogous classical Kuramoto problem where the coupling strengths are fixed to a constant (the same for all edges). In particular, for the "all-to-all" network, where the underlying graph is the complete graph, the problem reduces to the problem of understanding the fixed points and stability of the all-to-all Kuramoto model with equal edge weights, a problem that is well understood.

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We propose a novel cohesive subgraph model called τ -strengthened (α, β)-core (denoted as (α, β) τ -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 τ butterflies (2×2-bicliques). (α, β) τ -core requires each vertex on the upper or lower level to have at least α or β strong ties, given strength level τ . To retrieve the vertices of (α, β) τcore optimally, we construct index I α,β,τ to store all (α, β) τ -cores. Effective optimization techniques are proposed to improve index construction. To make our idea practical on large graphs, we propose 2D-indexes I α,β , I β,τ , and I α,τ that selectively store the vertices of (α, β) τ -core for some α, β, and τ . The 2D-indexes are more space-efficient and require less construction time, each of which can support (α, β) τ -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 2Dindex that minimizes the query time. Extensive experiments show that (1) (α, β) τ -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.