Optimizing Pinning Control of Directed Networks Using Spectral Graph Theory

Abstract

Pinning control of a complex network aims at aligning the states of all the nodes to an external forcing signal by controlling a small number of nodes in the network. An algebraic graph-theoretic condition has been proposed to optimize pinning control for both undirected and directed network. The problem we are trying to solve in this paper is the optimization of pinning control in directed networks, where the effectiveness of pinning control can be measured by the smallest eigenvalue of the submatrix obtained by deleting the rows and columns corresponding to the pinned nodes from the symmetrized Laplacian matrix of the network when coupling strength and individual node dynamics are given. By analysing the spectral properties using the topology information of the directed network, a necessary condition that ensure the smallest eigenvalue greater than 0 is obtained. Upper bounds and lower bounds of the smallest eigenvalue are proved, and then an algorithm that optimize pinning scheme to improve the smallest eigenvalue is proposed. Illustrative examples are shown to demonstrate our theoretical results in the paper.