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Querying cohesive subgraphs on temporal graphs (e.g., social network, financenetwork, etc.) with various conditions has attracted intensive researchinterests 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 orconstraining an arbitrary metric $\mathcal{X}$ of $k$-core, such as size,engagement, interaction frequency, time span, burstiness, periodicity, etc. Ourobjective is to address specific TXCQ instances with conditions on different$\mathcal{X}$ in a unified algorithm framework that guarantees scalability. Forthat, this journal paper proposes a taxonomy of measurement$\mathcal{X}(\cdot)$ and achieve our objective using a two-phase frameworkwhile $\mathcal{X}(\cdot)$ is time-insensitive or time-monotonic. Specifically,Phase 1 still leverages the query processing algorithm of TCQ to induce alldistinct $k$-cores during a given time range, and meanwhile locates the ``timezones'' in which the cores emerge. Then, Phase 2 conducts fast local search and$\mathcal{X}$ evaluation in each time zone with respect to the timeinsensitivity or monotonicity of $\mathcal{X}(\cdot)$. By revealing twoinsightful concepts named tightest time interval and loosest time interval thatbound time zones, the redundant core induction and unnecessary $\mathcal{X}$evaluation in a zone can be reduced dramatically. Our experimental resultsdemonstrate that TXCQ can be addressed as efficiently as TCQ, which achievesthe latest state-of-the-art performance, by using a general algorithm frameworkthat leaves $\mathcal{X}(\cdot)$ as a user-defined function.

A Unified and Scalable Algorithm Framework of User-Defined Temporal $(k,\mathcal{X})$-Core Query