  Dimension reduction techniques for dynamical systems on networks are
considered to promote our understanding of the original high-dimensional
dynamics. One strategy of dimension reduction is to derive a low-dimensional
dynamical system whose behavior approximates the observables of the original
dynamical system that are weighted linear summations of the state variables at
the different nodes. Recently proposed methods use the leading eigenvector of
the adjacency matrix of the network as the mixture weights to obtain such
observables. In the present study, we explore performances of this type of
one-dimensional reductions of dynamical systems on networks when we use
non-leading eigenvectors of the adjacency matrix as the mixture weights. Our
theory predicts that non-leading eigenvectors can be more efficient than the
leading eigenvector and enables us to select the eigenvector minimizing the
error. We numerically verify that the optimal non-leading eigenvector
outperforms the leading eigenvector for some dynamical systems and networks. We
also argue that, despite our theory, it is practically better to use the
leading eigenvector as the mixture weights to avoid misplacing the bifurcation
point too distantly and to be resistant against dynamical noise.
