On Optimality of Sparse Long-Range Links in Circulant Consensus Networks

We consider spatially invariant consensus networks in which the directed graph describing the interconnection topology, the link weights, 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. Motivated by small-world networks, we apply the optimal link creation problem to circulant networks with local communication links. We observe that the optimal new links are 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 analytically, we restrict attention to the creation of links with small weights, referred to as weak links. We employ perturbation methods to reformulate the problem of optimal weak link creation, and uncover conditions on the network architecture which guarantee sparse and long-range solutions to this optimization problem.