Fig. 1: The effect of model uncertainty in the computation of optimal network controls.
Panel (a) shows a schematic of a classic network identification procedure. The reconstructed network is affected by estimation errors δij. The symbol {aij} denote the correct network weights, whereas \(\{\hat{{a}}_{ij}\}\) the (incorrectly) reconstructed ones. Panel (b) illustrates the error in the final (output) state induced by an optimal control design based on the reconstructed network. The symbol yf denote the desired final state of (a subset of) the network nodes, whereas \({\hat{{\bf{y}}}}_{{\rm{f}}}\) the one reached by the optimal input u⋆(t). In panel (c), we consider minimum-energy controls designed from exact and incorrectly reconstructed linear dynamical networks, and compute the resulting error in the final state as the network size n varies. We consider connected Erdös–Rényi networks with edge probability \({p}_{{\rm{edge}}}=\mathrm{ln}\,n/n+0.1\), ten randomly selected control nodes, control horizon T = 2n, and a randomly chosen final state \({{\bf{x}}}_{{\rm{f}}}\in {{\mathbb{R}}}^{n}\). We assume yf = xf, i.e., the nodes that we want to control coincide with all network nodes. To mimic errors in the network reconstruction process, we add to each edge of the network a disturbance modeled as an i.i.d. random variable uniformly distributed in [ − δ, δ], δ > 0. Each curve represents the average of the (norm of the) error in the final state over 100 independent realizations of xf. To compute minimum-energy control inputs, we use the classic Gramian-based formula and standard LAPACK linear-algebra routines (see “Methods”). Notice that there is a nonzero error in the final state which grows with the size of the network even in the absence of uncertainty (δ = 0). This is due to numerical errors in the computation of the minimum-energy control which are a consequence of the ill-conditioning of the Gramian9,20.
