Pinning controllability analysis of the dynamical networks based on the nodes centrality

The optimal selection of the type and the number of driver nodes play basic role in the improving the controllability and the performance of the dynamical networks. In this paper, based on the derived criteria for the selection of the driver nodes, the pinning controllability of dynamical network has been studied. First, linear feedback control strategy is proposed for pinning synchronization of the networks. The Lyapunov direct method is used to analyze the stability of the closed-loop control system, where, it leads to LMI criteria for pinning synchronization. Second, subject to the various nodes centralities, the percentage of the least number of driver nodes to achieve the desired performance has been calculated for several networks topologies. The results for random networks and scale free networks show the numbers of driver nodes are decreased in the network with the high probability coefficient. In the small world network, driver nodes are increased in higher probability coefficient especially in the dense networks. These results are hold for different nodes centralities. Finally, the results for random networks show that for the betweenness centrality criterion the number of driver nodes is minimal. Similar results hold for Small World networks subject to the fact that for dense, the number of driver nodes increases. It is shown that the closseness centrality criterion is the proper choice for the State Free networks especially when the network is dense.