Achieving Network Stability via Optimal Pinning Control in Weighted Complex Networks

Interconnected systems forming complex networks are ubiquitous in many man-made and natural phenomena. When individual systems are aligned towards a desired trajectory, their synchronization's stability depends on the network's controllability, often achieved through pinning control. When optimizing controllability, unweighted Laplacian and uniform feedback gains are conventionally used for the pinned nodes, ignoring the significance of link weights, usually leading to suboptimal results. This study improves local stability of the synchronous state by addressing the controllability problem using weighted Laplacian matrices and nonuniform feedback gains. Using the master-stability function method, direct optimization of the controllability measure is formulated as a multi-objective optimization problem with a Pareto frontier and multiple optimal points. This multi-objective optimization problem is simplified into a spectral radius minimization problem, where reformulating it as a semidefinite programming (SDP) problem has led to a unique optimal point on its Pareto frontier. Many interesting analytical results have been established for different families of networks and subnetworks, including the criteria for optimal zero weights that can divide the network or optimal controllability measure of an arbitrary network and its mirrored networks. Additionally, for deterministic scale-free networks, it is demonstrated how the network can break down into smaller replicas and ultimately form a collection of path networks. Numerical simulations using the R & ouml;ssler model illustrate the feasible region and, interestingly, show that the Pareto frontier of the deterministic scale-free networks is independent of its size.