On the Trade-off Between Controllability and Robustness in Networks of Diffusively Coupled Agents

In this paper, we demonstrate a conflicting relationship between two crucial propertiescontrollability and robustnessin 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.