Schur-convex robustness measures in dynamical networks

We investigate robustness of networks with linear time-invariant dynamics under external stochastic disturbances. We propose a partial ordering on this class of dynamical networks and exploit their structural properties to characterize their robustness properties and fundamental limits. Then, we show that several existing and widely used scalar robustness measures are indeed Schur-convex functions on the spectrum of the Laplacian of the networks. We show that certain robustness features of the first-order consensus networks can be formulated as Schur-convex functions of their Laplacian eigenvalues. Specifically, we define the uncertainty volume and entropy based on the minimum-volume covering ellipsoid of the steady-state output points of the excited network. It is shown that the uncertainty volume is directly related to the number of spanning trees of the underlying graph of the network. Furthermore, we show that for networks with regular lattice interconnection topologies this measure scales asymptotically with network size. Finally, we propose an optimization-based method to improve robustness in linear dynamical networks.