Figure 4.

Panel (a) shows, in a logarithmic scale, the fragility versus responsiveness tradeoff for networks with matrices $A_{\alpha }=A_{\mathrm{skew}}-\alpha I$, with $A_{\mathrm{skew}}=-A_{\mathrm{skew}}^{\top }\in {ℝ}^{40\times 40}$ and $1\le \alpha \le 2$. Panel (b) shows, in a logarithmic scale, the fragility versus responsiveness tradeoff for networks with matrices $A_{\beta }=D_{\beta }^{-1}{A}_{\mathrm{sym}}{D}_{\beta }$, with $A_{\mathrm{sym}}=A_{\mathrm{sym}}^{\top }\in {ℝ}^{40\times 40}$ and $1\le \beta \le 2$. Panel (c) shows the fragility versus responsiveness tradeoff for networks with matrices $A_{\gamma }=A+B{K}_{\gamma }$, where $A\in {ℝ}^{40\times 40}$ is a randomly-generated symmetric matrix, $1\le \gamma \le 2$, $B=I$, and $K_{\gamma }$ solves the infinite-horizon Linear Quadratic Regulator³² problem with matrices Q = I and $R=\gamma I$. The values of $\alpha$, $\beta$, and $\gamma$ are color coded. It should be noted that (i) fragility (1/$r(A)$) and responsiveness ($\overline{\sigma }(G)$) are directly related, and (ii) the value of $\alpha$ (stability margin), $\beta$ (non-normality degree) and $\gamma$ (control cost) determine fragility and responsiveness. In all figures we use $n_{\mathrm{c}}=n$ control nodes.