  The $\mathsf{HYBRID}$ model, introduced in [Augustine et al., SODA '20],
provides a theoretical foundation for networks that allow multiple
communication modes. The model follows the principles of synchronous message
passing, whereas nodes are allowed to use \textit{two} fundamentally different
communication modes. First, a local mode where nodes may exchange arbitrary
information per round over edges of a local communication graph $G$ (akin to
the $\mathsf{LOCAL}$ model). Second, a global mode where every node may
exchange $O(\log n)$ messages of size $O(\log n)$ bits per round with arbitrary
nodes in the network. The $\mathsf{HYBRID}$ model intends to reflect the
conditions of many real hybrid networks, where high-bandwidth but inherently
local communication is combined with highly flexible global communication with
restricted bandwidth.
  We continue to explore the power and limitations of the $\mathsf{HYBRID}$
model by investigating the complexity of computing shortest paths and diameter
of the local communication graph $G$. We improve on the known upper bound for
the exact all pairs shortest paths problem (APSP) from [Augustine et al., SODA
'20] and provide algorithms to approximate solutions for the $k$ source
shortest paths problem ($k$-SSSP). We demonstrate that our results for APSP and
$k$-SSP are almost tight (up to poly-logarithmic factors). Furthermore, we give
an improved algorithm for the exact single source shortest paths problem for
graphs with large diameter. For the problem of approximating the diameter of
the local communication network we give the first non-trivial upper bound. This
upper bound is complemented by a lower bound for the exact diameter problem.
