  This paper investigates a model reduction problem for linear directed network
systems, in which the interconnections among the vertices are described by
general weakly connected digraphs. First, the definitions of pseudo
controllability and observability Gramians are proposed for semistable systems,
and their solutions are characterized by Lyapunov-like equations. Then, we
introduce a concept of vertex clusterability to guarantee the boundedness of
the approximation error and use the newly proposed Gramians to facilitate the
evaluation of the dissimilarity of each pair of vertices. An clustering
algorithm is thereto provided to generate an appropriate graph clustering,
whose characteristic matrix is employed as the projections in the
Petrov-Galerkin reduction framework. The obtained reduced-order system
preserves the weakly connected directed network structure, and the
approximation error is computed by the pseudo Gramians. Finally, the efficiency
of the proposed approach is illustrated by numerical examples.
