# Temporal Network Core Decomposition and Community Search

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.

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