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