Optimal network design for synchronization of coupled oscillators

This paper studies the problem of designing networks of nonidentical coupled oscillators in order to achieve a desired level of phase cohesiveness, defined as the maximum asymptotic phase difference across the edges of the network. In particular, we consider the following two design problems: (i) the nodal-frequency design problem, in which we tune the natural frequencies of the oscillators given the topology of the network, and (ii) the (robust) edge-weight design problem, in which we design the edge weights assuming that the natural frequencies are given (or belong to a given convex uncertainty set). For both problems, we optimize an objective function of the design variables while considering a desired level of phase cohesiveness as our design constraint. This constraint defines a convex set in the nodal-frequency design problem. In contrast, in the edge-weight design problem, the phase cohesiveness constraint yields a non-convex set, unless the underlying network is either a tree or an arbitrary graph with identical edge weights. We then propose a convex semidefinite relaxation to approximately solve the (non-convex) edge-weight design problem for general (possibly cyclic) networks with nonidentical edge weights. We illustrate the applicability of our results by analyzing several network design problems of practical interest, such as power redispatch in power grids, sparse network design, (robust) network design for distributed wireless analog clocks, and the detection of edges leading to the Braess' paradox in power grids.