Enhancing network resilience through topological switching

Abstract

This work studies how to preemptively increase the resilience of a network by means of time-varying topological actuation. To do this, we focus on linear dynamical systems that are compatible with a given network, and consider policies that switch periodically between the given one and an alternative, topologically-compatible dynamics. In particular, we seek to solve design problems aimed at finding a) the optimal switching schedule between two preselected topologies, and b) an optimal topology and optimal switching schedule. By imposing periodicity, we first provide a metric of resilience in terms of the spectral abscissa of the averaged linear time-invariant dynamics. By restricting our policies to commutative networks, we then show how the optimal scheduling problem reduces to a convex optimization, providing a bound on the net resilience that can be achieved. After this, we find that the optimal, sparse commutative network to switch with is fully disconnected and allocates the spectral sum among the nodes of the network equally. We then impose additional restrictions on topology edge selection, which leads to a biconvex optimization for which certain matrix rank conditions guide the choice of weighting parameters to obtain desirable solutions. Finally, we provide two methods to solve this problem efficiently (based on a McCormick relaxation, and alternating minimization), and illustrate the results in simulations.