Generation of clusters in complex dynamical networks via pinning control

Many real-world networks show community structure, i.e., groups (or clusters) of nodes that have a high density of links within them but with a lower density of links between them. In this paper, by applying feedback injections to a fraction of network nodes, various clusters are synchronized independently according to the community structure generated by the group partition of the network (cluster synchronization). This control is achieved by pinning (i.e. applying linear feedback control) to a subset of the network nodes. Those pinned nodes are selected not randomly but according to the topological structure of communities of a given network. Specifically, for a given group partition of a network, those nodes with direct connections between groups must be pinned in order to achieve cluster synchronization. Both the local stability and global stability of cluster synchronization are investigated. Taking the tree-shaped network and the most modular network as two particular examples, we illustrate in detail how the pinning strategy influences the generation of clusters. The simulations verify the efficiency of the pinning schemes used in this paper.