We introduce a new generalization of the $k$-core decomposition for temporal networks that respects temporal dynamics. In contrast to the standard definition and previous core-like decompositions for temporal graphs, our $(k,\Delta)$-core decomposition is an edge-based decomposition founded on the new notion of $\Delta$-degree. The $\Delta$-degree of an edge is defined as the minimum number of edges incident to one of its endpoints that have a temporal distance of at most~$\Delta$. Moreover, we define a new notion of temporal connectedness leading to an efficiently computable equivalence relation between so-called $\Delta$-connected components of the temporal network. We provide efficient algorithms for the $(k,\Delta)$-core decomposition and $\Delta$-connectedness, and apply them to solve community search problems, where we are given a query node and want to find a densely connected community containing the query node. Such a community is an edge-induced temporal subgraph representing densely connected groups of nodes with frequent interactions, which also captures changes over time. We provide an efficient algorithm for community search for the case without restricting the number of nodes. If the number of nodes is restricted, we show that the decision version is NP-complete. In our evaluation, we show how in a real-world social network, the inner $(k,\Delta)$-cores contain only the spreading of misinformation and that the $\Delta$-connected components of the cores are highly edge-homophilic, i.e., the majorities of the edges in the $\Delta$-connected components represent either misinformation or fact-checking. Moreover, we demonstrate how our algorithms for $\Delta$-community search successfully and efficiently identify informative structures in collaboration networks.

翻译：我们提出了一种新的面向时间网络的$k$-核心分解泛化方法，该分解尊重时间动态特性。与标准定义及现有时间图的核心类分解不同，我们的$(k,\Delta)$-核心分解是一种基于边的新型分解，其基础是$\Delta$-度的概念。边$e$的$\Delta$-度定义为：与$e$的一个端点相关联且时间距离不超过$\Delta$的最小边数。此外，我们定义了一种新的时间连通性概念，从而能够在时间网络的所谓$\Delta$-连通分量之间建立可高效计算的等价关系。我们为$(k,\Delta)$-核心分解和$\Delta$-连通性设计了高效算法，并将其应用于解决社区搜索问题——给定一个查询节点，需找到包含该查询节点的密集连通社区。此类社区由边诱导的时间子图构成，代表了节点间频繁交互的密集连通群组，同时能捕捉其随时间的变化。在节点数量不受限的场景下，我们提供了一种高效的社区搜索算法；当节点数量受限时，我们证明其判定问题为NP完全问题。实验中，我们展示了真实社交网络中，内部$(k,\Delta)$-核心仅包含虚假信息的传播，且这些核心的$\Delta$-连通分量具有高度边同质性——即$\Delta$-连通分量中的绝大多数边代表虚假信息或事实核查。此外，我们展示了$\Delta$-社区搜索算法如何高效且成功地识别协作网络中的信息性结构。