Global stabilization of complex networks with digraph topologies via a local pinning algorithm

This paper concerns the global stability of controlling a complex network with digraph topology to a homogeneous trajectory of the uncoupled system by the local pinning control algorithm. The derived stability condition indicates that the smallest real part of eigenvalues of the Laplacian sub-matrix corresponding to the unpinned vertices can be used to measure the stabilizability of a digraph with a given pinned vertex set. A pinned vertex set can stabilize a directed network to some unstable trajectories, for instance, to a chaotic trajectory of the uncoupled systems, if and only if the pinned vertex set can access all other vertices in the digraph. Furthermore, in the bigraph case, the analytical estimation of the stabilizability’s lower bound suggests that an optimal pinning strategy should take not only the vertex degree, but also the shortest path between pairs of vertices into considerations.