  We study an optimal control problem aimed at achieving a desired tradeoff
between the network coherence and communication requirements in the distributed
controller. Our objective is to add a certain number of edges to an undirected
network, with a known graph Laplacian, in order to optimally enhance
closed-loop performance. To promote controller sparsity, we introduce
$\ell_1$-regularization into the optimal ${\cal H}_2$ formulation and cast the
design problem as a semidefinite program. We derive a Lagrange dual, provide
interpretation of dual variables, and exploit structure of the optimality
conditions for undirected networks to develop customized proximal gradient and
Newton algorithms that are well-suited for large problems. We illustrate that
our algorithms can solve the problems with more than million edges in the
controller graph in a few minutes, on a PC. We also exploit structure of
connected resistive networks to demonstrate how additional edges can be
systematically added in order to minimize the ${\cal H}_2$ norm of the
closed-loop system.
