Pinning a Complex Dynamical Network to Its Equilibrium

TLDR

Network topology complexity strongly influences the stabilization of complex dynamical networks. The study investigates control strategies for random and scale‑free networks, exploring conditions for global or local stabilization and introducing virtual control with varied pinning schemes. The authors apply local feedback control to a small subset of nodes, introduce virtual control with diverse pinning schemes, and analyze synchronization analytically and numerically. Pinning highly connected nodes in scale‑free networks reduces the number of required controllers, while random pinning performs similarly in random networks.

Abstract

It is now known that the complexity of network topology has a great impact on the stabilization of complex dynamical networks. In this work, we study the control of random networks and scale-free networks. Conditions are investigated for globally or locally stabilizing such networks. Our strategy is to apply local feedback control to a small fraction of network nodes. We propose the concept of virtual control for microscopic dynamics throughout the process with different pinning schemes for both random networks and scale-free networks. We explain the main reason why significantly less local controllers are required by specifically pinning the most highly connected nodes in a scale-free network than those required by the randomly pinning scheme, and why there is no significant difference between specifically and randomly pinning schemes for controlling random dynamical networks. We also study the synchronization phenomenon of controlled dynamical networks in the stabilization process, both analytically and numerically.