On the optimality of sparse long-range links in circulant consensus networks

We consider spatially invariant consensus networks in which the link weights, the directed graph describing the interconnection topology, and the temporal dynamics, are all characterized by circulant matrices. We seek the best new links, subject to budget constraints, whose addition to the network maximally improves its rate of convergence to consensus. We show that the optimal circulant link creation problem is convex and can be written as a semidefinite program. Motivated by small-world networks, we apply the link creation problem to circulant networks which possess only local communication links. We observe that the optimal new links are always sparse and long-range, and have an increasingly pronounced effect on the convergence rate of the network as its size grows. To further investigate the properties of optimal links, we restrict attention to the creation of links with small strengths, which we refer to as weak links. We employ perturbation methods to reformulate the optimal weak link creation problem, and uncover conditions on the network architecture which guarantee sparse and long-range solutions to this optimization problem.