# On the relationship between pinning control effectiveness and graph topology in complex networks of dynamical systems

## Abstract

This paper concerns pinning control in complex networks of dynamical systems, where an external forcing signal is applied to the network in order to align the state of all the systems to the forcing signal. By considering the control signal as the state of a virtual dynamical system, this problem can be studied as a synchronization problem. The main focus of this paper is to study how the effectiveness of pinning control depends on the underlying graph. In particular, we look at the relationship between pinning control effectiveness and the complex network asymptotically as the number of vertices in the network increases. We show that for vertex balanced graphs, if the number of systems receiving pinning control does not grow as fast as the total number of systems, then the strength of the control needed to effect pinning control will be unbounded as the number of vertices grows. Furthermore, in order to achieve pinning control in systems coupled via locally connected graphs, as the number of systems grows, both the pinning control and the coupling among all systems need to increase. Finally, we give evidence to show that applying pinning control to minimize the distances between all systems to the pinned systems can lead to a more effective pinning control. © 2008 American Institute of Physics.