Performance and design of cycles in consensus networks

This work explores the role of cycles in consensus seeking networks for analysis and synthesis purposes. Cycles are critical for many reasons including improving the convergence rate of the system, resilience to link failures, and the overall performance of the system. The focus of this work examines how cycles impact the ^H2 performance of consensus networks. A first contribution shows that the addition of cycles always improves the performance of the system. We provide an analytic characterization of how the addition of edges improves the performance, and show that it is related to the inverse of the cycle lengths and the number of shared edges between independent cycles. These results are then used to consider the design of consensus networks. In this direction we present an ^ℓ1-relaxation method that leads to a convex program for adding a fixed number of edges to a consensus networks. We also demonstrate how this relaxation can be used to embed additional performance criteria, such as maximization of the algebraic connectivity of the graph.