  In this paper, we consider composite networks formed from the Kronecker
product of smaller networks. We find the observability and controllability
properties of the product network from those of its constituent smaller
networks. The overall network is modeled as a Linear-Structure-Invariant (LSI)
dynamical system where the underlying matrices have a fixed zero/non-zero
structure but the non-zero elements are potentially time-varying. This approach
allows to model the system parameters as free variables whose values may only
be known within a certain tolerance. We particularly look for minimal
sufficient conditions on the observability and controllability of the composite
network, which have a direct application in distributed estimation and in the
design of networked control systems. The methodology in this paper is based on
the structured systems analysis and graph theory, and therefore, the results
are generic, i.e., they apply to almost all non-zero choices of free
parameters. We show the controllability/observability results for composite
product networks resulting from full structural-rank systems and self-damped
networks. We provide an illustrative example of estimation based on Kalman
filtering over a composite network to verify our results.
