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Pinning control of a complex network aims at forcing the states of all nodesto track an external signal by controlling a small number of nodes in thenetwork. In this paper, an algebraic graph-theoretic condition is introduced tooptimize pinning control. When individual node dynamics and coupling strengthof the network are given, the effectiveness of pinning scheme can be measuredby the smallest eigenvalue of the grounded Laplacian matrix obtained bydeleting the rows and columns corresponding to the pinned nodes from theLaplacian matrix of the network. The larger this smallest eigenvalue, the moreeffective the pinning scheme. Spectral properties of the smallest eigenvalueare analyzed using the network topology information, including the spectrum ofthe network Laplacian matrix, the minimal degree of uncontrolled nodes, thenumber of edges between the controlled node set and the uncontrolled node set,etc. The identified properties are shown effective for optimizing the pinningcontrol strategy, as demonstrated by illustrative examples. Finally, for bothscale-free and small-world networks, in order to maximize their correspondingsmallest eigenvalues, it is better to pin the nodes with large degrees when thepercentage of pinned nodes is relatively small, while it is better to pin nodeswith small degrees when the percentage is relatively large. This surprisingphenomenon can be explained by one of the theorems established.

Optimizing Pinning Control of Complex Dynamical Networks Based on Spectral Properties of Grounded Laplacian Matrices