  Given a weighted graph $G$, a $(\beta,\varepsilon)$-hopset $H$ is an edge set
such that for any $s,t \in V(G)$, where $s$ can reach $t$ in $G$, there is a
path from $s$ to $t$ in $G \cup H$ which uses at most $\beta$ hops whose length
is in the range $[dist_G(s,t), (1+\varepsilon)dist_G(s,t)]$. We break away from
the traditional question that asks for a hopset that achieves small $|H|$ and
instead study its sensitivity, a new quality measure which, informally, is the
maximum number of times a vertex (or edge) is bypassed by an edge in $H$. The
highlights of our results are:
  (i) $(\widetilde{O}(\sqrt{n}),0)$-hopsets on undirected graphs with $O(\log
n)$ sensitivity, complemented with a lower bound showing that
$\widetilde{O}(\sqrt{n})$ is tight up to polylogarithmic factors for any
construction with polylogarithmic sensitivity.
  (ii) $(n^{o(1)},\varepsilon)$-hopsets on undirected graphs with $n^{o(1)}$
sensitivity for any $\varepsilon > 0$ that is at least inverse polylogarithmic,
complemented with a lower bound on the tradeoff between $\beta, \varepsilon$,
and the sensitivity.
  (iii) $\widetilde{O}(\sqrt{n})$-shortcut sets on directed graphs with $O(\log
n)$ sensitivity, complemented with a lower bound showing that $\beta =
\widetilde{\Omega}(n^{1/3})$ for any construction with polylogarithmic
sensitivity.
  We believe hopset sensitivity is a natural measure in and of itself, and
could potentially find use in a diverse range of contexts. More concretely, the
notion of hopset sensitivity is also directly motivated by the Differentially
Private All Sets Range Queries problem. Our result for $O(\log n)$ sensitivity
$(\widetilde{O}(\sqrt{n}),0)$-hopsets on undirected graphs immediately improves
the current best-known upper bound on utility from $\widetilde{O}(n^{1/3})$ to
$\widetilde{O}(n^{1/4})$ in the pure-DP setting, which is tight up to
polylogarithmic factors.
