  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
re-dispatch 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.
