Many real-world binary relations can be modelled as bipartite graphs, which can be inherently temporal and each edge is associated with a timestamp. The ( α, β )-core, a popular structure that requires minimum degrees over two layers of vertices, is useful for understanding the organisation of bipartite networks. However, the temporal property has rarely been considered in cohesive subgraph mining in bipartite graphs. This gap prevents the finding of time-sensitive ( α, β )-cores in real-world applications. In this paper, we aim at finding ( α, β )-cores within any time window over a temporal bipartite graph. To address this problem, we propose a novel DAG (Directed Acyclic Graph)-like hierarchy with qualified time windows to describe the temporal containment property of the ( α, β )-core. Furthermore, we construct the superior-optimized index which significantly optimizes space complexity and guarantees efficient query performance. We also propose a maintenance approach that can efficiently update the index by removing stale information and incorporating newly inserted temporal edges. Extensive experiments are conducted on eight real-world graphs and the results show the effectiveness and efficiency of our indexes.

Community search (CS) aims at personalized subgraph discovery which is the key to understanding the organisation of many real-world networks. CS in undirected networks has attracted significant attention from researchers, including many solutions for various cohesive subgraph structures and for different levels of dynamism with edge insertions and deletions, while they are much less considered for directed graphs. In this paper, we propose incremental solutions of CS based on the D-truss in dynamic directed graphs, where the D-truss is a cohesive subgraph structure defined based on two types of triangles in directed graphs. We first analyze the theoretical boundedness of D-truss given edge insertions and deletions, then we present basic single-update algorithms. To improve the efficiency, we propose an order-based D-Index, associated batch-update algorithms and a fully-dynamic query algorithm. Our extensive experiments on real-world graphs show that our proposed solution achieves a significant speedup compared to the SOTA solution, the scalability over updates is also verified.