Recent advances in temporal graph research have redefined traditional static graph concepts such as triangles, motifs, and $k$-cores. Inspired by this, we introduce a novel $(k,δ)$-truss for temporal graphs, requiring triangles to exist within sufficiently short time windows. The $(k,δ)$-truss ensures both static and temporal cohesion, while the original $k$-truss is a special case when $δ= \infty$. To address $(k,δ)$-truss queries, we propose index-free and index-based approaches. Utilizing the dual containment relation of $(k,δ)$-trusses, our indexes losslessly compress all $(k,δ)$-trusses into map or tree structures, significantly reducing space while enabling optimal-time retrieval. To scale to large temporal graphs, we develop two index construction algorithms based on truss decomposition and truss maintenance, respectively, which substantially reduce redundant computations. Moreover, we present techniques for the dynamic maintenance of the proposed indexes. The experimental results demonstrate that index-based approaches process queries in interactive time and outperform the index-free approach by 2$\sim$4 orders of magnitude, while the indexes achieve compression ratios of up to $10^{-4}$ and can be updated efficiently without rebuilding from scratch.

翻译：近期时间图研究的进展重新定义了传统静态图概念，如三角形、子图模式及$k$-核。受此启发，我们针对时间图提出了一种新型$(k,δ)$-桁架结构，要求三角形存在于足够短的时间窗口内。$(k,δ)$-桁架同时保证了静态与时间上的凝聚性，而原始$k$-桁架是$δ= \infty$时的特例。为解决$(k,δ)$-桁架查询问题，我们提出了无索引与基于索引两种方法。利用$(k,δ)$-桁架的双重包含关系，我们的索引将全部$(k,δ)$-桁架无损压缩为映射或树形结构，在显著降低存储空间的同时实现最优时间检索。为扩展至大规模时间图，我们分别基于桁架分解与桁架维护开发了两种索引构建算法，大幅减少了冗余计算。此外，我们提出了所提索引的动态维护技术。实验结果表明，基于索引的方法能以交互时间处理查询，性能优于无索引方法2至4个数量级，同时索引压缩比可达$10^{-4}$，且无需从头重建即可高效更新。