  Principal submatrices of the controllability Gramian and their inverses are
examined, for a network-consensus model with inputs at a subset of network
nodes. Specifically, several properties of the Gramian submatrices and their
inverses -- including dominant eigenvalues and eigenvectors, diagonal entries,
and sign patterns -- are characterized by exploiting the special
doubly-nonnegative structure of the matrices. In addition, majorizations for
these properties are obtained in terms of cutsets in the network's graph, based
on the diffusive form of the model. The asymptotic (long time horizon)
structure of the controllability Gramian is also analyzed. The results on the
Gramian are used to study metrics for target control of the network-consensus
model.

Systems and Control;Signal Processing;Dynamical Systems;Physics and Society