A method of quickly calculating the number of pinning nodes on pinning synchronization in complex networks

In this paper, we propose one adaptive pinning synchronization scheme in complex networks, whose adaptive law is simple and flexible. Most of all, we find the decreasing law of maximum eigenvalues of the principal submatrices for coupling matrix, and give a method of quickly calculating pinning nodes in complex networks with binary search algorithm. Furthermore, we extend the method to the pinning synchronization scheme with linear feedback control. Numerical simulations give effectiveness of the adaptive pinning synchronization in a scale-free network, and the relations between maximum eigenvalue sequence of the principal submatrices and the number of pinning nodes under the conditions of three pinning strategies in a scale-free network and a small-world network, respectively.