  In this paper, we demonstrate a conflicting relationship between two crucial
properties---controllability and robustness---in linear dynamical networks of
diffusively coupled agents. In particular, for any given number of nodes $N$
and diameter $D$, we identify networks that are maximally robust using the
notion of Kirchhoff index and then analyze their strong structural
controllability. For this, we compute the minimum number of leaders, which are
the nodes directly receiving external control inputs, needed to make such
networks controllable under all feasible coupling weights between agents. Then,
for any $N$ and $D$, we obtain a sharp upper bound on the minimum number of
leaders needed to design strong structurally controllable networks with $N$
nodes and diameter $D$. We also discuss that the bound is best possible for
arbitrary $N$ and $D$. Moreover, we construct a family of graphs for any $N$
and $D$ such that the graphs have maximal edge sets (maximal robustness) while
being strong structurally controllable with the number of leaders in the
proposed sharp bound. We then analyze the robustness of this graph family. The
results suggest that optimizing robustness increases the number of leaders
needed for strong structural controllability. Our analysis is based on
graph-theoretic methods and can be applied to exploit network structure to
co-optimize robustness and controllability in networks.
