Querying cohesive subgraphs on temporal graphs (e.g., social network, finance network, etc.) with various conditions has attracted intensive research interests recently. In this paper, we study a novel Temporal $(k,\mathcal{X})$-Core Query (TXCQ) that extends a fundamental Temporal $k$-Core Query (TCQ) proposed in our conference paper by optimizing or constraining an arbitrary metric $\mathcal{X}$ of $k$-core, such as size, engagement, interaction frequency, time span, burstiness, periodicity, etc. Our objective is to address specific TXCQ instances with conditions on different $\mathcal{X}$ in a unified algorithm framework that guarantees scalability. For that, this journal paper proposes a taxonomy of measurement $\mathcal{X}(\cdot)$ and achieve our objective using a two-phase framework while $\mathcal{X}(\cdot)$ is time-insensitive or time-monotonic. Specifically, Phase 1 still leverages the query processing algorithm of TCQ to induce all distinct $k$-cores during a given time range, and meanwhile locates the ``time zones'' in which the cores emerge. Then, Phase 2 conducts fast local search and $\mathcal{X}$ evaluation in each time zone with respect to the time insensitivity or monotonicity of $\mathcal{X}(\cdot)$. By revealing two insightful concepts named tightest time interval and loosest time interval that bound time zones, the redundant core induction and unnecessary $\mathcal{X}$ evaluation in a zone can be reduced dramatically. Our experimental results demonstrate that TXCQ can be addressed as efficiently as TCQ, which achieves the latest state-of-the-art performance, by using a general algorithm framework that leaves $\mathcal{X}(\cdot)$ as a user-defined function.

翻译：在时序图（如社交网络、金融网络等）中查询满足多种条件的凝聚子图近期引起了广泛研究兴趣。本文研究了一种新型的时序$(k,\mathcal{X})$-核查询（TXCQ），该查询扩展了本会议论文提出的基础时序$k$-核查询（TCQ），通过优化或约束$k$-核的任意度量$\mathcal{X}$（如规模、参与度、交互频率、时间跨度、突发性、周期性等）。我们的目标是在保证可扩展性的统一算法框架中，处理具有不同$\mathcal{X}$条件的具体TXCQ实例。为此，本期刊论文提出了一种度量$\mathcal{X}(\cdot)$的分类方法，并在$\mathcal{X}(\cdot)$具有时间不敏感或时间单调性时，采用两阶段框架实现目标。具体而言，第一阶段仍利用TCQ的查询处理算法，在给定时间范围内诱导所有不同的$k$-核，同时定位这些核出现的"时区"。第二阶段则针对$\mathcal{X}(\cdot)$的时间不敏感性或单调性，在每个时区内进行快速局部搜索和$\mathcal{X}$评估。通过揭示两个名为最紧时间区间和最松时间区间的概念来约束时区，可以大幅减少时区中的冗余核诱导和不必要的$\mathcal{X}$评估。实验结果表明，通过将$\mathcal{X}(\cdot)$作为用户自定义函数的通用算法框架，TXCQ能够达到与最新最优性能的TCQ相同的处理效率。