  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.
